# Dakota V6.7 User's Manual

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SAND2014-4633 Unlimited Release July 2014 Updated November 13, 2017 Dakota, A Multilevel Parallel Object-Oriented Framework for Design Optimization, Parameter Estimation, Uncertainty Quantification, and Sensitivity Analysis: Version 6.7 User’s Manual Brian M. Adams, Mohamed S. Ebeida, Michael S. Eldred, Gianluca Geraci, John D. Jakeman, Kathryn A. Maupin, Jason A. Monschke, J. Adam Stephens, Laura P. Swiler, Dena M. Vigil, Timothy M. Wildey Optimization and Uncertainty Quantification Department William J. Bohnhoff Radiation Transport Department Keith R. Dalbey Software Simulation and Data Processing Department John P. Eddy System Readiness and Sustainment Technologies Department Joseph R. Frye, Russell W. Hooper Multiphysics Applications Department Kenneth T. Hu W76-0/W88 Stockpile Systems Engineering Department Patricia D. Hough, Mohammad Khalil Quantitative Modeling and Analysis Department Elliott M. Ridgway Data-driven and Neural Computing Sandia National Laboratories P.O. Box 5800 Albuquerque, New Mexico 87185 Ahmad Rushdi Electrical and Computer Engineering University of California One Shields Avenue Davis, CA 95616-5294 4 Abstract The Dakota toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. Dakota contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic expansion methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogatebased optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the Dakota toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a user’s manual for the Dakota software and provides capability overviews and procedures for software execution, as well as a variety of example studies. Dakota Version 6.7 User’s Manual generated on November 13, 2017 Contents Preface 17 1 19 2 Introduction 1.1 Motivation for Dakota Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2 Dakota Capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3 Coupling Dakota to a Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.4 User’s Manual Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.5 Files Referenced in this Manual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Dakota Tutorial 2.1 25 Quickstart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.1 Acquiring and Installing Dakota . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.1.2 Running Dakota with a simple input file . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1.3 Examples of Dakota output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2 Dakota Input File Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.1 Rosenbrock Test Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.2 Two-Dimensional Grid Parameter Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3.3 Gradient-based Unconstrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3.4 Uncertainty Quantification with Monte Carlo Sampling . . . . . . . . . . . . . . . . . . . 34 2.3.5 User Supplied Simulation Code Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.3.5.1 Optimization with a User-Supplied Simulation Code - Case 1 . . . . . . . . . . 37 2.3.5.2 Optimization with a User-Supplied Simulation Code - Case 2 . . . . . . . . . . 41 2.4 Dakota Command-Line Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.5 Next Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6 3 CONTENTS 2.5.2 Key Getting Started References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 47 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1.1 Initial Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.1.2 Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2 Vector Parameter Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3 List Parameter Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.4 Centered Parameter Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.5 Multidimensional Parameter Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.6 Parameter Study Usage Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.7 Example: Vector Parameter Study with Rosenbrock . . . . . . . . . . . . . . . . . . . . . . . . . 54 Design of Experiments Capabilities 57 4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.2 Design of Computer Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.3 DDACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3.1 Central Composite Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3.2 Box-Behnken Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3.3 Orthogonal Array Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.3.4 Grid Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3.5 Monte Carlo Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3.6 LHS Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.3.7 OA-LHS Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.4 FSUDace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 4.5 PSUADE MOAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.6 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.7 5 Problem Exploration and Method Selection . . . . . . . . . . . . . . . . . . . . . . . . . 43 Parameter Study Capabilities 3.1 4 2.5.1 4.6.1 Sensitivity Analysis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.6.2 Assessing Sensitivity with DACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 DOE Usage Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Uncertainty Quantification Capabilities 5.1 69 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.1.1 Summary of Dakota UQ Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Dakota Version 6.7 User’s Manual generated on November 13, 2017 CONTENTS 7 5.1.2 Variables and Responses for UQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2 5.3 Sampling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5.2.1 Uncertainty Quantification Example using Sampling Methods . . . . . . . . . . . . . . . 74 5.2.2 Incremental Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2.3 Principal Component Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.2.4 Wilks-based Sample Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Reliability Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3.1 Local Reliability Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.3.1.1 5.4 5.3.2 Global Reliability Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 5.3.3 Uncertainty Quantification Examples using Reliability Analysis . . . . . . . . . . . . . . 83 5.6 5.7 5.8 5.3.3.1 Mean-value Reliability with Textbook . . . . . . . . . . . . . . . . . . . . . . 83 5.3.3.2 FORM Reliability with Lognormal Ratio . . . . . . . . . . . . . . . . . . . . . 84 Stochastic Expansion Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.4.1 5.5 Method mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 Uncertainty Quantification Examples using Stochastic Expansions . . . . . . . . . . . . . 90 5.4.1.1 Polynomial Chaos Expansion for Rosenbrock . . . . . . . . . . . . . . . . . . 90 5.4.1.2 Uncertainty Quantification Example using Stochastic Collocation . . . . . . . . 90 Importance Sampling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.5.1 Importance Sampling Method based on Reliability Approach . . . . . . . . . . . . . . . . 96 5.5.2 Gaussian Process Adaptive Importance Sampling Method . . . . . . . . . . . . . . . . . 96 Adaptive Sampling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.6.1 Adaptive sampling based on surrogates . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.6.2 Adaptive sampling based on dart throwing . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Epistemic Nondeterministic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.7.1 Interval Methods for Epistemic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5.7.2 Dempster-Shafer Theory of Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Bayesian Calibration Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5.8.1 QUESO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.8.2 DREAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.8.3 GPMSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.8.4 WASABI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.8.5 Feature Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 5.8.6 Bayesian Calibration Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.8.7 Calibrating the Observation Error Model . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Dakota Version 6.7 User’s Manual generated on November 13, 2017 8 CONTENTS 5.8.8 Model Discrepancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5.8.9 Bayesian Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.8.9.1 5.9 6 Uncertainty Quantification Usage Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Optimization Capabilities 6.1 6.2 Constraint Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Optimizing with Dakota: Choosing a Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.2.1 6.2.2 6.2.3 6.3 121 Optimization Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.1.1 Gradient-Based Local Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.2.1.1 Method Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.2.1.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 Derivative-Free Local Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.2.2.1 Method Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6.2.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Derivative-Free Global Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.2.3.1 Method Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 6.2.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 Additional Optimization Capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.3.1 6.3.2 Multiobjective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.3.1.1 Multiobjective Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.3.1.2 Multiobjective Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Optimization with User-specified or Automatic Scaling . . . . . . . . . . . . . . . . . . . 139 6.3.2.1 7 One-at-a-time Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Scaling Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.4 Optimization Usage Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.5 Optimization Third Party Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 Nonlinear Least Squares Capabilities 147 7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 7.2 Nonlinear Least Squares Fomulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.3 Nonlinear Least Squares with Dakota . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 7.4 Solution Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.4.1 Gauss-Newton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.4.2 NLSSOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.4.3 NL2SOL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 Dakota Version 6.7 User’s Manual generated on November 13, 2017 CONTENTS 9 7.4.4 Additional Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 8 7.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 7.6 Usage Guidelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Models 153 8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 8.2 Single Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 8.3 Recast Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 8.4 Surrogate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 8.4.1 Overview of Surrogate Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 8.4.2 Correction Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 8.4.3 Data Fit Surrogate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.4.3.1 Procedures for Surface Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 8.4.3.2 Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 8.4.3.3 Two Point Adaptive Nonlinearity Approximation . . . . . . . . . . . . . . . . . 158 8.4.3.4 Linear, Quadratic, and Cubic Polynomial Models . . . . . . . . . . . . . . . . 159 8.4.3.5 Kriging/Gaussian-Process Spatial Interpolation Models . . . . . . . . . . . . . 160 8.4.3.6 Artificial Neural Network (ANN) Models . . . . . . . . . . . . . . . . . . . . 163 8.4.3.7 Multivariate Adaptive Regression Spline (MARS) Models . . . . . . . . . . . . 163 8.4.3.8 Radial Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 8.4.3.9 Moving Least Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 8.4.3.10 Piecewise Decomposition Option for Global Surrogate Models . . . . . . . . . 164 8.4.3.11 Surrogate Diagnostic Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 9 8.4.4 Multifidelity Surrogate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 8.4.5 Reduced Order Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 8.4.6 Surrogate Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.5 Nested Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 8.6 Random Field Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 8.7 Active Subspace Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 Variables 171 9.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 9.2 Design Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 9.2.1 Continuous Design Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 9.2.2 Discrete Design Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Dakota Version 6.7 User’s Manual generated on November 13, 2017 10 CONTENTS 9.3 9.2.2.1 Discrete Design Integer Variables . . . . . . . . . . . . . . . . . . . . . . . . . 172 9.2.2.2 Discrete Design String Variables . . . . . . . . . . . . . . . . . . . . . . . . . 173 9.2.2.3 Discrete Design Real Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Uncertain Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 9.3.1 9.3.2 Aleatory Uncertain Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 9.3.1.1 Continuous Aleatory Uncertain Variables . . . . . . . . . . . . . . . . . . . . . 174 9.3.1.2 Discrete Aleatory Uncertain Variables . . . . . . . . . . . . . . . . . . . . . . 174 Epistemic Uncertain Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 9.3.2.1 Continuous Epistemic Uncertain Variables . . . . . . . . . . . . . . . . . . . . 175 9.3.2.2 Discrete Epistemic Uncertain Variables . . . . . . . . . . . . . . . . . . . . . . 175 9.4 State Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 9.5 Management of Mixed Variables by Iterator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 9.6 9.7 9.5.1 View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 9.5.2 Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 9.5.3 Precedence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Dakota Parameters File Data Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 9.6.1 Parameters file format (standard) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 9.6.2 Parameters file format (APREPRO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 The Active Set Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 9.7.1 Active set vector control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 10 Interfaces 183 10.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 10.2 Simulation Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 10.2.1 Direct Simulation Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 10.2.2 System Call Simulation Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 10.2.3 Fork Simulation Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 10.2.4 Syntax for Filter and Driver Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 10.2.5 Fork or System Call: Which to Use? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 10.3 Building a Black-Box Interface to a Simulation Code . . . . . . . . . . . . . . . . . . . . . . . . 186 10.3.1 Generic Script Interface Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 10.3.2 Adapting These Scripts to Another Simulation . . . . . . . . . . . . . . . . . . . . . . . 192 10.3.3 Additional Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 10.4 Simulation Interface Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Dakota Version 6.7 User’s Manual generated on November 13, 2017 CONTENTS 11 10.4.1 Single analysis driver without filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 10.4.2 Single analysis driver with filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 10.4.3 Multiple analysis drivers without filters . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 10.4.4 Multiple analysis drivers with filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 10.5 Simulation File Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 10.5.1 File Saving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 10.5.2 File Tagging for Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 10.5.3 Temporary Files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 10.5.4 File Tagging for Analysis Drivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 10.5.5 Work Directories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 10.6 Parameter to Response Mapping Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 10.7 Parameters and Results Files with dakota.interfacing . . . . . . . . . . . . . . . . . . . . . . . . 206 10.7.1 Creating Parameters and Results objects . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 10.7.2 Parameters objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.7.3 Results objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 10.7.4 Response object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 10.7.5 dakota.interfacing Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 11 Responses 211 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 11.1.1 Response function types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 11.1.2 Gradient availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 11.1.3 Hessian availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 11.1.4 Field Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 11.2 Dakota Results File Data Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 11.3 Active Variables for Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 12 Inputs to Dakota 217 12.1 Overview of Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 12.1.1 Tabular Data Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 12.2 Data Imports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 12.2.1 AMPL algebraic mappings: stub.nl, stub.row, and stub.col . . . . . . . . . . . . . . . . . 218 12.2.2 Genetic algorithm population import . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 12.2.3 Calibration data import . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 12.2.4 PCE coefficient import . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Dakota Version 6.7 User’s Manual generated on November 13, 2017 12 CONTENTS 12.2.5 Surrogate construction and evaluation data . . . . . . . . . . . . . . . . . . . . . . . . . 220 12.2.6 Variables/responses import to post-run . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 13 Output from Dakota 223 13.1 Overview of Output Formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 13.2 Standard Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 13.3 Tabular Output Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 13.4 Graphics Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 13.5 Error Messages Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 13.6 Stochastic expansion exports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 13.7 Surrogate Model Exports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 13.8 Variables Output from Pre-run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 14 Advanced Methods 233 14.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 14.2 Hybrid Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 14.3 Multistart Local Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 14.4 Pareto Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 14.5 Mixed Integer Nonlinear Programming (MINLP) . . . . . . . . . . . . . . . . . . . . . . . . . . 236 14.5.1 Example MINLP Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 14.6 Surrogate-Based Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 14.6.1 Surrogate-Based Local Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 14.6.1.1 SBO with Data Fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 14.6.1.2 SBO with Multifidelity Models . . . . . . . . . . . . . . . . . . . . . . . . . . 244 14.6.1.3 SBO with Reduced Order Models . . . . . . . . . . . . . . . . . . . . . . . . . 244 14.6.2 Surrogate-Based Global Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 15 Advanced Model Recursions 249 15.1 Mixed Aleatory-Epistemic UQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 15.1.1 Interval-valued probability (IVP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 15.1.2 Second-order probability (SOP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 15.1.3 Dempster-Shafer Theory of Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 15.2 Optimization Under Uncertainty (OUU) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 15.2.1 Nested OUU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 15.2.2 Surrogate-Based OUU (SBOUU) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Dakota Version 6.7 User’s Manual generated on November 13, 2017 CONTENTS 13 15.2.3 Trust-Region Surrogate-Based OUU (TR-SBOUU) . . . . . . . . . . . . . . . . . . . . . 256 15.2.4 RBDO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 15.2.5 Stochastic Expansion-Based Design Optimization . . . . . . . . . . . . . . . . . . . . . . 257 15.2.6 Epistemic OUU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 15.3 Surrogate-Based Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 16 Advanced Simulation Code Interfaces 261 16.1 Algebraic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 16.2 Developing a Direct Simulation Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 16.2.1 Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 16.2.2 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 16.2.3 Sandwich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 16.3 Existing Direct Interfaces to External Simulators . . . . . . . . . . . . . . . . . . . . . . . . . . 266 16.3.1 Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 16.3.1.1 Dakota/Matlab input file specification . . . . . . . . . . . . . . . . . . . . . . . 266 16.3.1.2 Matlab .m file specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 16.3.2 Python . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 16.4 Scilab Script and Direct Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 16.4.1 Scilab Script Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 16.4.2 Scilab Linked Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 16.4.3 Scilab Compiled Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 17 Parallel Computing 273 17.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 17.1.1 Categorization of parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 17.1.2 Parallel Dakota algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 17.1.2.1 Parallel iterators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 17.1.2.2 Advanced methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 17.1.2.3 Parallel models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 17.2 Single-level parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 17.2.1 Asynchronous Local Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 17.2.1.1 Direct function synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . 279 17.2.1.2 System call synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 17.2.1.3 Fork synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 17.2.1.4 Asynchronous Local Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Dakota Version 6.7 User’s Manual generated on November 13, 2017 14 CONTENTS 17.2.1.5 Local evaluation scheduling options . . . . . . . . . . . . . . . . . . . . . . . 283 17.2.2 Message Passing Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 17.2.2.1 Partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 17.2.2.2 Scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 17.2.2.3 Message Passing Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 17.2.3 Hybrid Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 17.2.3.1 Hybrid Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 17.3 Multilevel parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 17.3.1 Asynchronous Local Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 17.3.2 Message Passing Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 17.3.2.1 Partitioning of levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 17.3.2.2 Scheduling within levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 17.3.3 Hybrid Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 17.4 Capability Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 17.5 Running a Parallel Dakota Job . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 17.5.1 Single-processor execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 17.5.2 Multiprocessor execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 17.6 Specifying Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 17.6.1 The interface specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 17.6.2 The meta-iterator and nested model specifications . . . . . . . . . . . . . . . . . . . . . . 295 17.6.3 Single-processor Dakota specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 17.6.3.1 Example 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 17.6.3.2 Example 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 17.6.4 Multiprocessor Dakota specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 17.6.4.1 Example 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 17.6.4.2 Example 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 17.6.4.3 Example 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 17.7 Application Parallelism Use Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 17.7.1 Case 1: Massively Serial — Multiple serial analysis jobs . . . . . . . . . . . . . . . . . . 303 17.7.2 Case 2: Sequential Parallel — One parallel analysis job at a time . . . . . . . . . . . . . . 304 17.7.3 Case 3: Evaluation Tiling — Multiple simultaneous parallel analysis jobs . . . . . . . . . 304 17.7.3.1 Mpiexec server mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 17.7.3.2 Relative node scheduling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 17.7.3.3 Machinefile management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 Dakota Version 6.7 User’s Manual generated on November 13, 2017 CONTENTS 15 17.7.4 Case 4: Evaluation Submission — Parallel analysis jobs submitted to a queue . . . . . . . 306 18 Restart Capabilities and Utilities 307 18.1 Restart Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 18.2 The Dakota Restart Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 18.2.1 Print . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 18.2.2 To/From Neutral File Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 18.2.3 To Tabular Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 18.2.4 Concatenation of Multiple Restart Files . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 18.2.5 Removal of Corrupted Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 19 Simulation Failure Capturing 315 19.1 Failure detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 19.2 Failure communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 19.3 Failure mitigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 19.3.1 Abort (default) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 19.3.2 Retry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 19.3.3 Recover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 19.3.4 Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 19.4 Special values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 20 Additional Examples 319 20.1 Textbook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 20.1.1 Gradient-based Constrained Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 321 20.2 Rosenbrock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 20.2.1 Least-Squares Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 20.3 Herbie, Smooth Herbie, and Shubert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 20.3.1 Efficient Global Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 20.4 Sobol and Ishigami Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 20.5 Cylinder Head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 20.5.1 Constrained Gradient Based Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 333 20.6 Container . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 20.6.1 Constrained Gradient Based Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 337 20.7 Cantilever . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 20.7.1 Constrained Gradient Based Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 339 Dakota Version 6.7 User’s Manual generated on November 13, 2017 16 CONTENTS 20.7.2 Optimization Under Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 20.8 Multiobjective Test Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 20.8.1 Multiobjective Test Problem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 20.8.2 Multiobjective Test Problem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 20.9 Morris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 20.9.1 Morris One-at-a-Time Sensitivity Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 20.10Test Problems for Reliability Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 20.10.1 Log Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 20.10.2 Steel Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 20.10.3 Portal Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 20.10.4 Short Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 20.10.5 Steel Column . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 20.11Test Problems for Forward Uncertainty Quantification . . . . . . . . . . . . . . . . . . . . . . . . 353 20.11.1 Genz functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 20.11.2 Elliptic Partial differential equation with uncertain coefficients . . . . . . . . . . . . . . . 354 20.11.3 Damped Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 20.11.4 Non-linear coupled system of ODEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 20.11.5 Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 20.11.6 Bayes linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 Dakota Version 6.7 User’s Manual generated on November 13, 2017 Preface The Dakota project started in 1994 as an internal research and development activity at Sandia National Laboratories in Albuquerque, New Mexico. The original goal was to provide a common set of optimization tools for a group of engineers solving structural analysis and design problems. Prior to the Dakota project, there was no focused effort to archive optimization methods for reuse on other projects. Thus, engineers found themselves repeatedly building new custom interfaces between the engineering analysis software and optimization software. This was especially burdensome when using parallel computing, as each project developed a unique master program to coordinate concurrent simulations on a network of workstations or a parallel computer. The initial Dakota toolkit provided the engineering and analysis community at Sandia access to a variety of optimization algorithms, hiding the complexity of the optimization software interfaces from the users. Engineers could readily switch between optimization software packages by simply changing a few lines in a Dakota input file. In addition to structural analysis, Dakota has been applied to computational fluid dynamics, nonlinear dynamics, shock physics, heat transfer, electrical circuits, and many other science and engineering models. Dakota has grown significantly beyond an optimization toolkit. In addition to its state-of-the-art optimization methods, Dakota includes methods for global sensitivity and variance analysis, parameter estimation, uncertainty quantification, and verification, as well as meta-level strategies for surrogate-based optimization, hybrid optimization, and optimization under uncertainty. Available to all these algorithms is parallel computation support; ranging from desktop multiprocessor computers to massively parallel computers typically found at national laboratories and supercomputer centers. As of Version 5.0, Dakota is publicly released as open source under a GNU Lesser General Public License and is available for free download world-wide. See http://www.gnu.org/licenses/lgpl.html for more information on the LGPL software use agreement. Dakota Versions 3.0 through 4.2+ were licensed under the GNU General Public License. Dakota public release facilitates research and software collaborations among Dakota developers at Sandia National Laboratories and other institutions, including academic, government, and corporate entities. See the Dakota FAQ at http://dakota.sandia.gov/faq-page for more information on the public release rationale and ways to contribute. Dakota leadership includes Brian Adams (project lead), Mike Eldred (founder and research lead), Adam Stephens (support manager), Dena Vigil (product owner), and Jim Stewart (business manager). For a listing of current and former contributors and third-party library developers, visit the Dakota webpage at http://dakota. sandia.gov. Contact Information: Brian M. Adams, Dakota Project Lead Sandia National Laboratories P.O. Box 5800, Mail Stop 1318 Albuquerque, NM 87185-1318 Web (including support information): http://dakota.sandia.gov 18 CONTENTS Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 1 Introduction 1.1 Motivation for Dakota Development Computational models are commonly used in engineering design and scientific discovery activities for simulating complex physical systems in disciplines such as fluid mechanics, structural dynamics, heat transfer, nonlinear structural mechanics, shock physics, and many others. These simulators can be an enormous aid to engineers who want to develop an understanding and/or predictive capability for complex behaviors typically observed in the corresponding physical systems. Simulators often serve as virtual prototypes, where a set of predefined system parameters, such as size or location dimensions and material properties, are adjusted to improve the performance of a system, as defined by one or more system performance objectives. Such optimization or tuning of the virtual prototype requires executing the simulator, evaluating performance objective(s), and adjusting the system parameters in an iterative, automated, and directed way. System performance objectives can be formulated, for example, to minimize weight, cost, or defects; to limit a critical temperature, stress, or vibration response; or to maximize performance, reliability, throughput, agility, or design robustness. In addition, one would often like to design computer experiments, run parameter studies, or perform uncertainty quantification (UQ). These approaches reveal how system performance changes as a design or uncertain input variable changes. Sampling methods are often used in uncertainty quantification to calculate a distribution on system performance measures, and to understand which uncertain inputs contribute most to the variance of the outputs. A primary goal for Dakota development is to provide engineers and other disciplinary scientists with a systematic and rapid means to obtain improved or optimal designs or understand sensitivity or uncertainty using simulationbased models. These capabilities generally lead to improved designs and system performance in earlier design stages, alleviating dependence on physical prototypes and testing, shortening design cycles, and reducing product development costs. In addition to providing this practical environment for answering system performance questions, the Dakota toolkit provides an extensible platform for the research and rapid prototyping of customized methods and meta-algorithms [34]. 1.2 Dakota Capabilities Dakota delivers a variety of iterative methods and meta-algorithms, and the ability to flexibly interface them to your simulation code. While Dakota was originally conceived to more readily interface simulation codes and optimization algorithms, recent versions go beyond optimization to include other iterative analysis methods such 20 CHAPTER 1. INTRODUCTION as uncertainty quantification with nondeterministic propagation methods, parameter estimation with nonlinear least squares solution methods, and sensitivity/variance analysis with general-purpose design of experiments and parameter study capabilities. These capabilities may be used on their own or as building blocks within more sophisticated meta-algorithms such as hybrid optimization, surrogate-based optimization, optimization under uncertainty, or mixed aleatory/epistemic UQ. The principal classes of Dakota algorithms, with brief descriptions, are summarized here. For details, formulations, and usage guidelines, see the referenced chapters. • Parameter Studies (Chapter 3): Parameter studies employ deterministic designs to explore the effect of parametric changes within simulation models, yielding one form of sensitivity analysis. They can help assess simulation characteristics such as smoothness, multi-modality, robustness, and nonlinearity, which affect the choice of algorithms and controls in follow-on optimization and UQ studies. Typical examples include centered, one-at-a-time variations or joint variation on a grid. • Design of Experiments (Chapter 4): Design and analysis of computer experiments (DACE) techniques are often used to explore the parameter space of an engineering design problem, for example to perform global sensitivity analysis. DACE methods can help reach conclusions similar to parameter studies, but the primary goal of these methods is to generate good coverage of the input parameter space. Representative methods include Latin hypercube sampling, orthogonal arrays, and Box-Behnken designs. • Uncertainty Quantification (Chapter 5): Uncertainty quantification methods (also referred to as nondeterministic analysis methods) compute probabilistic information about response functions based on simulations performed according to specified input parameter probability distributions. Put another way, these methods perform a forward uncertainty propagation in which probability information for input parameters is mapped to probability information for output response functions. Common approaches include Monte Carlo sampling, reliability methods, and polynomial chaos expansions. • Optimization (Chapter 6): Optimization solvers seek to minimize cost or maximize system performance, as predicted by the simulation model, subject to constraints on input variables or secondary simulation responses. Categories of algorithms include gradient-based, derivative-free, and global optimization. Dakota also includes capabilities for multi-objective trade-off optimization and automatic scaling of problem formulations. Advanced Dakota approaches include hybrid (multi-method), multi-start local, and Pareto-set optimization. • Calibration (Chapter 7): Calibration algorithms seek to maximize agreement between simulation outputs and experimental data (or desired outputs). They are used to solve inverse problems (often referred to as parameter estimation or least-squares problems). Dakota approaches include nonlinear least squares and Bayesian calibration. Dakota includes a number of related advanced capabilities. Surrogate models are inexpensive approximate models that are intended to capture the salient features of an expensive high-fidelity model and include data fits, multifidelity, and reduced-order model surrogates. They can be used to explore the variations in response quantities over regions of the parameter space, or they can serve as inexpensive stand-ins for optimization or uncertainty quantification studies. Section 8.4 summarizes surrogate model mechanics in Dakota, while optimization methods tailored to particular surrogate approaches are surveyed in Section 14.6. Nested models permit layering one Dakota method over another, enabling algorithms like mixed epistemicaleatory or second-order UQ, optimization under uncertainty, or surrogate-based UQ. Additional information on these nested approaches is provided in Section 8.5 and Chapter 15. The methods and algorithms in Dakota are designed to exploit parallel computing resources such as those found in a desktop multiprocessor workstation, a network of workstations, or a massively parallel computing platform. Dakota Version 6.7 User’s Manual generated on November 13, 2017 1.3. COUPLING DAKOTA TO A SIMULATION 21 Figure 1.1: The loosely-coupled or “black-box” interface between Dakota and a user-supplied simulation code. This parallel computing capability is a critical technology for rendering real-world engineering design problems computationally tractable. See Chapter 17. Dakota also has emerging capabilities in solution verification and Bayesian calibration/UQ, which are documented briefly in the Dakota Reference Manual, and in later sections of this manual. In addition to its iterative methods and algorithms, Dakota also provides a graphical user interface that allows you to manage your Dakota studies visually. Among other features, the GUI allows you to intuitively link your Dakota study to a simulation model (see Section 1.3 below) as well as graphically plot output data from your Dakota study. For the full Dakota GUI tutorial, please visit this link: https://dakota.sandia.gov/ content/dakota-gui-user-manual. 1.3 Coupling Dakota to a Simulation A key Dakota advantage is access to a broad range of iterative capabilities through a single, relatively simple, interface between Dakota and your simulator. Trying a different iterative method or meta-algorithm typically requires changing only a few commands in the Dakota text input file and starting the new analysis. It does not require intimate knowledge of the underlying software package integrated in Dakota, with its unique command syntax and interfacing requirements. In addition, Dakota will manage concurrent executions of your computational model in parallel, whether on a desktop or high-performance cluster computer. Figure 1.1 depicts a typical loosely-coupled relationship between Dakota and the simulation code(s). Such coupling is often referred to as “black-box,” as Dakota has no (or little) awareness of the internal details of the computational model, obviating any need for its source code. Such loose coupling is the simplest and most common interfacing approach Dakota users employ. Dakota and the simulation code exchange data by reading and writing short data files. Dakota is executed with commands that the user supplies in a text input file (not shown in Figure 1.1) which specify the type of analysis to be performed (e.g., parameter study, optimization, uncertainty quantification, etc.), along with the file names associated with the user’s simulation code. During operation, Dakota automatically executes the user’s simulation code by creating a separate process external to Dakota. The solid lines in Figure 1.1 denote file input/output (I/O) operations inherent to Dakota or the user’s simulation code. The dotted lines indicate passing or converting information that must be implemented by the user. As Dakota Dakota Version 6.7 User’s Manual generated on November 13, 2017 22 CHAPTER 1. INTRODUCTION runs, it writes out a parameters file containing the current variable values. Dakota then starts the user’s simulation code (or, often, a short driver script wrapping it), and when the simulation completes, reads the response data from a results file. This process is repeated until all of the simulations required by the iterative study are complete. In some cases it is advantageous to have a close coupling between Dakota and the simulation code. This close coupling is an advanced feature of Dakota and is accomplished through either a direct interface or a SAND (simultaneous analysis and design) interface. For the direct interface, the user’s simulation code is modified to behave as a function or subroutine under Dakota. This interface can be considered to be “semi-intrusive” in that it requires relatively minor modifications to the simulation code. Its major advantage is the elimination of the overhead resulting from file I/O and process creation. It can also be a useful tool for parallel processing, by encapsulating all computation in a single executable. For details on direct interfacing, see Section 16.2. A SAND interface approach is “fully intrusive” in that it requires further modifications to the simulation code so that an optimizer has access to the internal residual vector and Jacobian matrices computed by the simulation code. In a SAND approach, both the optimization method and a nonlinear simulation code are converged simultaneously. While this approach can greatly reduce the computational expense of optimization, considerable software development effort must be expended to achieve this intrusive coupling between SAND optimization methods and the simulation code. SAND may be supported in future Dakota releases. 1.4 User’s Manual Organization The Dakota User’s Manual is organized into the following major categories. New users should consult the Tutorial to get started, then likely the Method Tour and Interfacing to select a Dakota method and build an interface to your code. • Tutorial (Chapter 2): How to obtain, install, and use Dakota, with a few introductory examples. • Method Tour (Chapters 3 through 7): Survey of the major classes of iterative methods included in Dakota, with background, mathematical formulations, usage guidelines, and summary of supporting third-party software. • Models (Chapters 8 through 11): Explanation of Dakota models, which manage the mapping from variables through interfaces to responses, as well as details on parameter and response file formats for simulation code interfacing. • Input/Output (Chapters 12 and 13): Summary of input to Dakota, including tabular data, and outputs generated by Dakota. • Advanced Topics: – Recursion with Components: Chapter 14 addresses component-based method recursions, and Chapter 15 addresses component-based model recursions. – Interfacing: Chapter 16 describes interfacing Dakota with engineering simulation codes in both loose- and tightly-coupled modes. – Parallelism: Chapter 17 describes Dakota’s parallel computing capabilities, with a summary of major application parallel modes in Section 17.7. – Fault Tolerance: Chapter 18 describes restart capabilities and utilities, and Chapter 19 explains ways to detect and mitigate simulation failures. • Additional Examples (Chapter 20): Supplemental example analysis problems and discussion. Dakota Version 6.7 User’s Manual generated on November 13, 2017 1.5. FILES REFERENCED IN THIS MANUAL 1.5 23 Files Referenced in this Manual Dakota input files are shown in figures throughout the Manual. The filename is specified in the comments and unless specified otherwise, these files are available in the Dakota/examples/users directory, where Dakota refers to the directory where Dakota was installed. Some of the input files have associated files, such as output or tabular data, with the same base filename, and .sav appended to the names. Additional files are referenced, and if the location differs then it will be specified in the text. A small number of examples refer to files included only in the source directory, which is labeled Dakota Source. You will need a copy of the source to view these files - see Section 2.1.1. 1.6 Summary Dakota is both a production tool for engineering design and analysis activities and a research tool for the development of new algorithms in optimization, uncertainty quantification, and related areas. Because of the extensible, object-oriented design of Dakota, it is relatively easy to add new iterative methods, meta-algorithms, simulation interfacing approaches, surface fitting methods, etc. In addition, Dakota can serve as a rapid prototyping tool for algorithm development. That is, by having a broad range of building blocks available (i.e., parallel computing, surrogate models, simulation interfaces, fundamental algorithms, etc.), new capabilities can be assembled rapidly which leverage the previous software investments. For additional discussion on framework extensibility, refer to the Dakota Developers Manual [2]. The capabilities of Dakota have been used to solve engineering design and optimization problems at Sandia Labs, at other Department of Energy labs, and by our industrial and academic collaborators. Often, this real-world experience has provided motivation for research into new areas of optimization. The Dakota development team welcomes feedback on the capabilities of this software toolkit, as well as suggestions for new areas of research. Dakota Version 6.7 User’s Manual generated on November 13, 2017 24 CHAPTER 1. INTRODUCTION Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 2 Dakota Tutorial 2.1 Quickstart This section provides an overview of acquiring and installing Dakota, running a simple example, and looking at the basic output available. More detailed information about downloads and installation can be found on the Dakota website http://dakota.sandia.gov. 2.1.1 Acquiring and Installing Dakota Dakota operates on most systems running Unix or Linux operating systems as well as on Windows, natively in a Command Prompt window, and (optionally) with the help of a Cygwin emulation layer. Dakota is developed and most extensively tested on Redhat Enterprise Linux with GNU compilers, but additional operating systems / compiler combinations are tested nightly as well. See the Dakota website for more information on supported platforms for particular Dakota versions. Department of Energy users: Dakota may already be available on your target system. Sandia users should visit http://dakota.sandia.gov/ for information on supported Dakota installations on engineering networks and cluster computers, as well as for Sandia-specific downloads. At other DOE institutions, contact your system administrator about Dakota availability. If Dakota is not available for your target platform, you may still download Dakota as described below. New users should visit https://dakota.sandia.gov/quickstart.html to get started with Dakota. This typically involves the following steps: 1. Download Dakota. You may download binary executables for your preferred platforms or you can compile Dakota from source code. Downloads are available from http://dakota.sandia.gov/download.html. 2. Install Dakota. Instructions are available from http://dakota.sandia.gov/content/install-dakota. Guidance is also included in the Dakota source files, including Dakota Source/INSTALL. Further platform/operating system-specific guidance can be found in Dakota Source/examples/platforms. 3. Verify that Dakota runs. To perform a quick check that your Dakota executable runs, open a terminal window (in Windows, cmd.exe), 26 CHAPTER 2. DAKOTA TUTORIAL and type: dakota -v Dakota version information should display in your terminal window. For a more detailed description of Dakota command line options, see Section 2.4. 4. Participate in Dakota user communities. Join Dakota mail lists to get the most up-to-date guidance for downloading, compiling, installing, or running. For information about mail lists, getting help, and other available help resources, see http://dakota.sandia.gov/content/get-help. 2.1.2 Running Dakota with a simple input file This section is intended for users who are new to Dakota, to demonstrate the basics of running a simple example. First Steps 1. Make sure Dakota runs. You should see Dakota version information when you type: dakota -v 2. Create a working directory 3. Copy rosen multidim.in from the Dakota/examples/users/ directory to the working directory – see Section 1.5 for help. 4. From the working directory, run dakota -i rosen multidim.in -o rosen multidim.out > rosen multidim.stdout What should happen Dakota outputs a large amount of information to help users track progress. Four files should have been created: 1. The screen output has been redirected to the file rosen multidim.stdout. The contents are messages from Dakota and notes about the progress of the iterator (i.e. method/algorithm). 2. The output file rosen multidim.out contains information about the function evaluations. 3. rosen multidim.dat is created due to the specification of tabular graphics data and tabular graphics file. This summarizes the variables and responses for each function evaluation. 4. dakota.rst is a restart file. If a Dakota analysis is interrupted, it can be often be restarted without losing all progress. In addition to the files, some plots are created due to the specification of graphics. These can be helpful when processing the data or diagnosing unexpected results. If your particular installation or build of Dakota does not support graphics, you will instead get a warning to this effect. Dakota has some data processing capabilities for output analysis. The output file will contain the relevant results. In this case, the output file has details about each of the 81 function evaluations. For more advanced or customized data processing or visualization, the tabular data file can be imported into another analysis tool. What now? • Assuming Dakota ran successfully, skim the three text files (restart files are in a binary format). These are described further in Section 2.1.3. Dakota Version 6.7 User’s Manual generated on November 13, 2017 2.1. QUICKSTART 27 • This example used a parameter study method, and the rosenbrock test problem. More details about the example are in Section 2.3.2 and the test problem is described in Sections 2.3.1 and 20.2. • Explore the many methods available in Dakota in Chapters 3– 7. • Try running the other examples in the same directory. These are mentioned throughout the manual and are listed in Table 2.1 for convenience. • Learn the syntax needed to use these methods. For help running Dakota, see Section 2.4 and for input file information, see Section 2.2. • Learn how to use your own analysis code with Dakota in Chapter 16. 2.1.3 Examples of Dakota output Beyond numerical results, all output files provide information that allows the user to check that the actual analysis was the intended analysis. More details on all outputs can be found in Chapter 13. Screen output saved to a file Whenever an output file is specified for a Dakota run, the screen output itself becomes quite minimal consisting of version statements, environment statements and execution times. Output file The output file is much more extensive, because it contains information on every function evaluation (See Figure 2.1). It is organized into three basic parts: 1. Information on the problem For this example, we see that a new restart file is being created and Dakota has carried out a multidim parameter study with 8 partitions for each of two variables. 2. Information on each function evaluation Each function evaluation is numbered. Details for function evaluation 1 show that at input variable values x1 = −2.0 and x2 = −2.0, the direct rosenbrock function is being evaluated. There is one response with a value of 3.609e+03. 3. Summary statistics The function evaluation summary is preceded by <<<<<. For this example 81 total evaluations were assessed; all were new, none were read in from the restart file. Correlation matrices complete the statistics and output for this problem. Successful runs will finish with <<<<< Iterator study type completed. Tabular output file For this example, the default name for the tabular output file dakota tabular.dat was changed in the input file to rosen multidim.dat. This tab-delimited text file (Figure 2.1.3) summarizes the inputs and outputs to the function evaluator. The first line contains the names of the variables and responses, as well as headers for the evaluation id and interface columns. %eval_id interface x1 x2 Dakota Version 6.7 User’s Manual generated on November 13, 2017 response_fn_1 28 CHAPTER 2. DAKOTA TUTORIAL The number of function evaluations will match the number of evaluations listed in the summary part of the output file for single method approaches; the names of inputs and outputs will match the descriptors specified in the input file. The interface column is useful when a Dakota input file contains more than one simulation interface. In this instance, there is only one, and it has no id interface specified, so Dakota has supplied a default value of NO ID. This file is ideal for import into other data analysis packages. {Writing new restart file dakota.rst methodName = multidim_parameter_study gradientType = none hessianType = none >>>>> Running multidim_parameter_study iterator. Multidimensional parameter study for variable partitions of 8 8 -----------------------------Begin Function Evaluation 1 -----------------------------Parameters for function evaluation 1: -2.0000000000e+00 x1 -2.0000000000e+00 x2 Direct function: invoking rosenbrock Active response data for function evaluation 1: Active set vector = { 1 } 3.6090000000e+03 response_fn_1 . . . <<<<< Function evaluation summary: 81 total (81 new, 0 duplicate) Simple Correlation Matrix among all inputs and outputs: x1 x2 response_fn_1 x1 1.00000e+00 x2 1.73472e-17 1.00000e+00 response_fn_1 -3.00705e-03 -5.01176e-01 1.00000e+00 . . . <<<<< Iterator multidim_parameter_study completed.} Figure 2.1: Rosenbrock 2-D parameter study example: excerpt from output file Dakota Version 6.7 User’s Manual generated on November 13, 2017 2.2. DAKOTA INPUT FILE FORMAT %eval_id 1 2 3 4 interface NO_ID NO_ID NO_ID NO_ID 29 x1 -2 -1.5 -1 -0.5 x2 -2 -2 -2 -2 response_fn_1 3609 1812.5 904 508.5 Figure 2.2: Rosenbrock 2-D parameter study example: excerpt from tabular data file 2.2 Dakota Input File Format See Section 1.5 for location of all files referenced in this manual. There are six specification blocks that may appear in Dakota input files. These are identified in the input file using the following keywords: variables, interface, responses, model, method, and environment. While, these keyword blocks can appear in any order in a Dakota input file, there is an inherent relationship that ties them together. The simplest form of that relationship is shown in Figure 2.3 and can be summarized as follows: In each iteration of its algorithm, a method block requests a variables-to-responses mapping from its model, which the model fulfills through an interface. While most Dakota analyses satisfy this relationship, where a single method runs a single model, advanced cases are possible and are discussed in Chapter 14. Figure 2.3: Relationship between the six blocks, for a simple study. As a concrete example, a simple Dakota input file, rosen multidim.in, is shown in Figure 2.4 for a twodimensional parameter study on Rosenbrock’s function. This input file will be used to describe the basic format and syntax used in all Dakota input files. The results are shown later, in Section 2.3.2. The first block of the input file shown in Figure 2.4 is the environment block. This keyword block is used to specify the general Dakota settings such as Dakota’s graphical output (via the graphics flag) and the tabular data output (via the tabular graphics data keyword). In advanced cases, it also identifies the top method pointer that will control the Dakota study. The environment block is optional, and at most one such block can appear in a Dakota input file. Dakota Version 6.7 User’s Manual generated on November 13, 2017 30 CHAPTER 2. DAKOTA TUTORIAL # Dakota Input File: rosen_multidim.in # Usage: # dakota -i rosen_multidim.in -o rosen_multidim.out > rosen_multidim.stdout environment graphics tabular_data tabular_data_file = ’rosen_multidim.dat’ method multidim_parameter_study partitions = 8 8 model single variables continuous_design = 2 lower_bounds -2.0 upper_bounds 2.0 descriptors ’x1’ -2.0 2.0 "x2" interface analysis_drivers = ’rosenbrock’ direct responses response_functions = 1 no_gradients no_hessians Figure 2.4: Rosenbrock 2-D parameter study example: the Dakota input file. The method block of the input file specifies which iterative method Dakota will employ and associated method options. The keyword multidim parameter study in Figure 2.4 calls for a multidimensional parameter study, while the keyword partitions specifies the number of intervals per variable (a method option). In this case, there will be eight intervals (nine data points) evaluated between the lower and upper bounds of both variables (bounds provided subsequently in the variables section), for a total of 81 response function evaluations. At least one method block is required, and multiple blocks may appear in Dakota input files for advanced studies. The model block of the input file specifies the model that Dakota will use. A model provides the logical unit for determining how a set of variables is mapped through an interface into a set of responses when needed by an iterative method. In the default case, the model allows one to specify a single set of variables, interface, and responses. The model block is optional in this simple case. Alternatively, it can be explicitly defined as in Figure 2.4, where the keyword single specifies the use of a single model in the parameter study. If one wishes to perform more sophisticated studies such as surrogate-based analysis or optimization under uncertainty, the logical organization specified in the model block becomes critical in informing Dakota on how to manage the different components of such studies, and multiple model blocks are likely needed. See Chapter 8 for relevant advanced model specification details. The variables block of the input file specifies the number, type, and characteristics of the parameters that will Dakota Version 6.7 User’s Manual generated on November 13, 2017 2.3. EXAMPLES 31 be varied by Dakota. The variables can be classified as design variables, uncertain variables, or state variables. Design variables are typically used in optimization and calibration, uncertain variables are used in UQ and sensitivity studies, and state variables are usually fixed. In all three cases, variables can be continuous or discrete, with discrete having real, integer, and string subtypes. See Chapter 9 for more information on the types of variables supported by Dakota. The variables section shown in Figure 2.4 specifies that there are two continuous design variables. The sub-specifications for continuous design variables provide the descriptors “x1” and “x2” as well as lower and upper bounds for these variables. The information about the variables is organized in column format for readability. So, both variables x1 and x2 have a lower bound of -2.0 and an upper bound of 2.0. At least one variables block is required, and multiple blocks may appear in Dakota input files for advanced studies. The interface block of the input file specifies the simulation code that will be used to map variables into responses as well as details on how Dakota will pass data to and from that code. In this example, the keyword direct is used to indicate the use of a function linked directly into Dakota, and data is passed directly between the two. The name of the function is identified by the analysis driver keyword. Alternatively, fork or system executions can be used to invoke instances of a simulation code that is external to Dakota as explained in Section 2.3.5.2 and Chapter 16. In this case, data is passed between Dakota and the simulation via text files. At least one interface block is required, and multiple blocks may appear in Dakota input files for advanced studies. The responses block of the input file specifies the types of data that the interface will return to Dakota. They are categorized primarily according to usage. Objective functions are used in optimization, calibration terms in calibration, and response functions in sensitivity analysis and UQ. For the example shown in Figure 2.4, the assignment response functions = 1 indicates that there is only one response function. The responses block can include additional information returned by the interface. That includes constraints and derivative information, both discussed in Chapter 11. In this example, there are no constraints associated with Rosenbrock’s function, so the keywords for constraint specifications are omitted. The keywords no gradients and no hessians indicate that no derivatives will be provided to the method; none are needed for a parameter study. At least one responses block is required, and multiple blocks may appear in Dakota input files for advanced studies. We close this section with a list of rules regarding the formatting of the Dakota input file. • “Flat” text only. • Whitespace is ignored. • Comments begin with # and continue to the end of the line. • Keyword order is largely unimportant as long as major sections are respected and there is no ambiguity. • Equal signs are optional. • Strings can be surrounded by single or double quotes (but not “fancy” quotes). • Scientific notation is fine. Please see the Dakota Reference Manual [3] for additional details on this input file syntax. 2.3 Examples This section serves to familiarize users with how to perform parameter studies, optimization, and uncertainty quantification through their common Dakota interface. The initial examples utilize simple built in driver functions; later we show how to utilize Dakota to drive the evaluation of user supplied black box code. The examples presented in this chapter are intended to show the simplest use of Dakota for methods of each type. More advanced examples of using Dakota for specific purposes are provided in subsequent, topic-based, chapters. Dakota Version 6.7 User’s Manual generated on November 13, 2017 32 2.3.1 CHAPTER 2. DAKOTA TUTORIAL Rosenbrock Test Problem The examples shown later in this chapter use the Rosenbrock function [115] (also described in [57], among other places), which has the form: f (x1 , x2 ) = 100(x2 − x21 )2 + (1 − x1 )2 (2.1) A three-dimensional plot of this function is shown in Figure 2.5(a), where both x1 and x2 range in value from −2 to 2. Figure 2.5(b) shows a contour plot for Rosenbrock’s function. An optimization problem using Rosenbrock’s function is formulated as follows: minimize f (x1 , x2 ) x ∈ <2 subject to −2 ≤ x1 ≤ 2 (2.2) −2 ≤ x2 ≤ 2 (a) (b) Figure 2.5: Rosenbrock’s function: (a) 3-D plot and (b) contours with x1 on the bottom axis. Note that there are no linear or nonlinear constraints in this formulation, so this is a bound constrained optimization problem. The unique solution to this problem lies at the point (x1 , x2 ) = (1, 1), where the function value is zero. Several other test problems are available. See Chapter 20 for a description of these test problems as well as further discussion of the Rosenbrock test problem. 2.3.2 Two-Dimensional Grid Parameter Study Parameter study methods in the Dakota toolkit involve the computation of response data sets at a selection of points in the parameter space. These response data sets are not linked to any specific interpretation, so they may consist of any allowable specification from the responses keyword block, i.e., objective and constraint functions, Dakota Version 6.7 User’s Manual generated on November 13, 2017 2.3. EXAMPLES 33 least squares terms and constraints, or generic response functions. This allows the use of parameter studies in direct coordination with optimization, least squares, and uncertainty quantification studies without significant modification to the input file. An example of a parameter study is the 2-D parameter study example problem listed in Figure 2.4. This is executed by Dakota using the command noted in the comments: dakota -i rosen_multidim.in -o rosen_multidim.out > rosen_multidim.stdout The output of the Dakota run is written to the file named rosen multidim.out while the screen output, or standard output, is redirect to rosen multidim.stdout. For comparison, files named rosen multidim.out.sav and rosen multidim.stdout.sav are included in the Dakota/examples/users directory. As for many of the examples, Dakota provides a report on the best design point located during the study at the end of these output files. This 2-D parameter study produces the grid of data samples shown in Figure 2.6. In general, a multidimensional parameter study lets one generate a grid in multiple dimensions. The keyword multidim parameter study indicates that a grid will be generated over all variables. The keyword partitions indicates the number of grid partitions in each dimension. For this example, the number of the grid partitions are the same in each dimension (8 partitions) but it would be possible to specify (partitions = 8 2), and have only two partitions over the second input variable. Note that the graphics flag in the environment block of the input file could be commented out since, for this example, the iteration history plots created by Dakota are not particularly instructive. More interesting visualizations can be created by importing Dakota’s tabular data into an external graphics/plotting package. Common graphics and plotting packages include Mathematica, Matlab, Microsoft Excel, Origin, Tecplot, Gnuplot and many others. (Sandia National Laboratories and the Dakota developers do not endorse any of these commercial products.) Figure 2.6: Rosenbrock 2-D parameter study example: location of the design points (dots) evaluated. 2.3.3 Gradient-based Unconstrained Optimization Dakota’s optimization capabilities include a variety of gradient-based and nongradient-based optimization methods. This subsection demonstrates the use of one such method through the Dakota interface. Dakota Version 6.7 User’s Manual generated on November 13, 2017 34 CHAPTER 2. DAKOTA TUTORIAL A Dakota input file for a gradient-based optimization of Rosenbrock’s function is listed in Figure 2.7. The format of the input file is similar to that used for the parameter studies, but there are some new keywords in the responses and method sections. First, in the responses block of the input file, the keyword block starting with numerical gradients specifies that a finite difference method will be used to compute gradients for the optimization algorithm. Note that the Rosenbrock function evaluation code inside Dakota has the ability to give analytical gradient values. (To switch from finite difference gradient estimates to analytic gradients, uncomment the analytic gradients keyword and comment out the four lines associated with the numerical gradients specification.) Next, in the method block of the input file, several new keywords have been added. In this block, the keyword conmin frcg indicates the use of the Fletcher-Reeves conjugate gradient algorithm in the CONMIN optimization software package [139] for bound-constrained optimization. The keyword max iterations is used to indicate the computational budget for this optimization (in this case, a single iteration includes multiple evaluations of Rosenbrock’s function for the gradient computation steps and the line search steps). The keyword convergence tolerance is used to specify one of CONMIN’s convergence criteria (under which CONMIN terminates if the objective function value differs by less than the absolute value of the convergence tolerance for three successive iterations). The Dakota command is noted in the file, and copies of the outputs are in the Dakota/examples/users directory, with .sav appended to the name. When this example problem is executed using Dakota with graphics support enabled, Dakota creates some iteration history graphics similar to the screen capture shown in Figure 2.8(a). These plots show how the objective function and design parameters change in value during the optimization steps. The scaling of the horizontal and vertical axes can be changed by moving the scroll knobs on each plot. Also, the “Options” button allows the user to plot the vertical axes using a logarithmic scale. Note that log-scaling is only allowed if the values on the vertical axis are strictly greater than zero. Figure 2.8(b) shows the iteration history of the optimization algorithm. The optimization starts at the point (x1 , x2 ) = (−1.2, 1.0) as given in the Dakota input file. Subsequent iterations follow the banana-shaped valley that curves around toward the minimum point at (x1 , x2 ) = (1.0, 1.0). Note that the function evaluations associated with the line search phase of each CONMIN iteration are not shown on the plot. At the end of the Dakota run, information is written to the output file to provide data on the optimal design point. These data include the optimum design point parameter values, the optimum objective and constraint function values (if any), plus the number of function evaluations that occurred and the amount of time that elapsed during the optimization study. 2.3.4 Uncertainty Quantification with Monte Carlo Sampling Uncertainty quantification (UQ) is the process of determining the effect of input uncertainties on response metrics of interest. These input uncertainties may be characterized as either aleatory uncertainties, which are irreducible variabilities inherent in nature, or epistemic uncertainties, which are reducible uncertainties resulting from a lack of knowledge. Since sufficient data is generally available for aleatory uncertainties, probabilistic methods are commonly used for computing response distribution statistics based on input probability distribution specifications. Conversely, for epistemic uncertainties, data is generally sparse, making the use of probability theory questionable and leading to nonprobabilistic methods based on interval specifications. The subsection demonstrates the use of Monte Carlo random sampling for Uncertainty Quantification. Figure 2.9 shows the Dakota input file for an example problem that demonstrates some of the random sampling capabilities available in Dakota. In this example, the design parameters, x1 and x2, will be treated as uncertain parameters that have uniform distributions over the interval [-2, 2]. This is specified in the variables block of the input file, beginning with the keyword uniform uncertain. Another difference from earlier input files such as Figure 2.7 occurs in the responses block, where the keyword response functions is used in place of objective functions. The final changes to the input file occur in the method block, where the keyword Dakota Version 6.7 User’s Manual generated on November 13, 2017 2.3. EXAMPLES 35 # Dakota Input File: rosen_grad_opt.in # Usage: # dakota -i rosen_grad_opt.in -o rosen_grad_opt.out > rosen_grad_opt.stdout environment graphics tabular_data tabular_data_file = ’rosen_grad_opt.dat’ method conmin_frcg convergence_tolerance = 1e-4 max_iterations = 100 model single variables continuous_design = 2 initial_point -1.2 lower_bounds -2.0 upper_bounds 2.0 descriptors ’x1’ 1.0 -2.0 2.0 "x2" interface analysis_drivers = ’rosenbrock’ direct responses objective_functions = 1 # analytic_gradients numerical_gradients method_source dakota interval_type forward fd_step_size = 1.e-5 no_hessians Figure 2.7: Rosenbrock gradient-based unconstrained optimization example: the Dakota input file. Dakota Version 6.7 User’s Manual generated on November 13, 2017 36 CHAPTER 2. DAKOTA TUTORIAL (a) (b) Figure 2.8: Rosenbrock gradient-based unconstrained optimization example: (a) screen capture of the Dakota graphics and (b) sequence of design points (dots) evaluated (line search points omitted). sampling is used. The other keywords in the methods block of the input file specify the number of samples (200), the seed for the random number generator (17), the sampling method (random), and the response threshold (100.0). The seed specification allows a user to obtain repeatable results from multiple runs. If a seed value is not specified, then Dakota’s sampling methods are designed to generate nonrepeatable behavior (by initializing the seed using a system clock). The keyword response levels allows the user to specify threshold values for which Dakota will output statistics on the response function output. Note that a unique threshold value can be specified for each response function. In this example, Dakota will select 200 design points from within the parameter space, evaluate the value of Rosenbrock’s function at all 200 points, and then perform some basic statistical calculations on the 200 response values. The Dakota command is noted in the file, and copies of the outputs are in the Dakota/examples/users Dakota Version 6.7 User’s Manual generated on November 13, 2017 2.3. EXAMPLES 37 directory, with .sav appended to the name. Figure 2.10 shows example results from this sampling method. Note that your results will differ from those in this file if your seed value differs or if no seed is specified. As shown in Figure 2.10, the statistical data on the 200 Monte Carlo samples is printed at the end of the output file in the section that starts with “Statistics based on 200 samples.” In this section summarizing moment-based statistics, Dakota outputs the mean, standard deviation, skewness, and kurtosis estimates for each of the response functions. For example, the mean of the Rosenbrock function given uniform input uncertainties on the input variables is 455.4 and the standard deviation is 536.8. This is a very large standard deviation, due to the fact that the Rosenbrock function varies by three orders of magnitude over the input domain. The skewness is positive, meaning this is a right-tailed distribution, not a symmetric distribution. Finally, the kurtosis (a measure of the “peakedness” of the distribution) indicates that this is a strongly peaked distribution (note that we use a central, standardized kurtosis so that the kurtosis of a normal is zero). After the moment-related statistics, the 95% confidence intervals on the mean and standard deviations are printed. This is followed by the fractions (“Probability Level”) of the response function values that are below the response threshold values specified in the input file. For example, 34 percent of the sample inputs resulted in a Rosenbrock function value that was less than or equal to 100, as shown in the line listing the cumulative distribution function values. Finally, there are several correlation matrices printed at the end, showing simple and partial raw and rank correlation matrices. Correlations provide an indication of the strength of a monotonic relationship between input and outputs. More detail on correlation coefficients and their interpretation can be found in Section 5.2.1. More detail about sampling methods in general can be found in Section 5.2. Finally, Figure 2.11 shows the locations of the 200 sample sites within the parameter space of the Rosenbrock function for this example. 2.3.5 User Supplied Simulation Code Examples This subsection provides examples of how to use Dakota to drive user supplied black box code. 2.3.5.1 Optimization with a User-Supplied Simulation Code - Case 1 Many of the previous examples made use of the direct interface to access the Rosenbrock and textbook test functions that are compiled into Dakota. In engineering applications, it is much more common to use the fork interface approach within Dakota to manage external simulation codes. In both of these cases, the communication between Dakota and the external code is conducted through the reading and writing of short text files. For this example, the C++ program rosenbrock.cpp in Dakota Source/test is used as the simulation code. This file is compiled to create the stand-alone rosenbrock executable that is referenced as the analysis driver in Figure 2.12. This stand-alone program performs the same function evaluations as Dakota’s internal Rosenbrock test function. Figure 2.12 shows the text of the Dakota input file named rosen syscall.in that is provided in the directory Dakota/examples/users. The only differences between this input file and the one in Figure 2.7 occur in the interface keyword section. The keyword fork indicates that Dakota will use fork calls to create separate Unix processes for executions of the user-supplied simulation code. The name of the simulation code, and the names for Dakota’s parameters and results file are specified using the analysis driver, parameters file, and results file keywords, respectively. The Dakota command is noted in the file, and copies of the outputs are in the Dakota/examples/users directory, with .sav appended to the name. This run of Dakota takes longer to complete than the previous gradient-based optimization example since the fork interface method has additional process creation and file I/O overhead, as compared to the internal communication that occurs when the direct interface method is used. Dakota Version 6.7 User’s Manual generated on November 13, 2017 38 CHAPTER 2. DAKOTA TUTORIAL # Dakota Input File: rosen_sampling.in # Usage: # dakota -i rosen_sampling.in -o rosen_sampling.out > rosen_sampling.stdout environment graphics tabular_data tabular_data_file = ’rosen_sampling.dat’ method sampling sample_type random samples = 200 seed = 17 response_levels = 100.0 model single variables uniform_uncertain = 2 lower_bounds -2.0 -2.0 upper_bounds 2.0 2.0 descriptors ’x1’ ’x2’ interface analysis_drivers = ’rosenbrock’ direct responses response_functions = 1 no_gradients no_hessians Figure 2.9: Monte Carlo sampling example: the Dakota input file. To gain a better understanding of what exactly Dakota is doing with the fork interface approach, add the keywords file tag and file save to the interface specification and re-run Dakota. Check the listing of the local directory and you will see many new files with names such as params.in.1, params.in.2, etc., and results.out.1, results.out.2, etc. There is one params.in.X file and one results.out.X file for each of the function evaluations performed by Dakota. This is the file listing for params.in.1: 2 -1.200000000000000e+00 1.000000000000000e+00 1 1 2 1 2 0 variables x1 x2 functions ASV_1:obj_fn derivative_variables DVV_1:x1 DVV_2:x2 analysis_components Dakota Version 6.7 User’s Manual generated on November 13, 2017 2.3. EXAMPLES 39 Statistics based on 200 samples: Moment-based statistics for each response function: Mean Std Dev Skewness response_fn_1 4.5540183516e+02 5.3682678089e+02 1.6661798252e+00 Kurtosis 2.7925726822e+00 95% confidence intervals for each response function: LowerCI_Mean UpperCI_Mean LowerCI_StdDev response_fn_1 3.8054757609e+02 5.3025609422e+02 4.8886795789e+02 UpperCI_StdDev 5.9530059589e+02 Level mappings for each response function: Cumulative Distribution Function (CDF) for response_fn_1: Response Level Probability Level Reliability Index -------------- ----------------- ----------------1.0000000000e+02 3.4000000000e-01 General Rel Index ----------------- Probability Density Function (PDF) histograms for each response function: PDF for response_fn_1: Bin Lower Bin Upper Density Value ----------------------------1.1623549854e-01 1.0000000000e+02 3.4039566059e-03 1.0000000000e+02 2.7101710856e+03 2.5285698843e-04 Simple Correlation Matrix among all inputs and outputs: x1 x2 response_fn_1 x1 1.00000e+00 x2 -5.85097e-03 1.00000e+00 response_fn_1 -9.57746e-02 -5.08193e-01 1.00000e+00 Partial Correlation Matrix between input and output: response_fn_1 x1 -1.14659e-01 x2 -5.11111e-01 Simple Rank Correlation Matrix among all inputs and outputs: x1 x2 response_fn_1 x1 1.00000e+00 x2 -6.03315e-03 1.00000e+00 response_fn_1 -1.15360e-01 -5.04661e-01 1.00000e+00 Partial Rank Correlation Matrix between input and output: response_fn_1 x1 -1.37154e-01 x2 -5.08762e-01 Figure 2.10: Results of Monte Carlo Sampling on the Rosenbrock Function Dakota Version 6.7 User’s Manual generated on November 13, 2017 40 CHAPTER 2. DAKOTA TUTORIAL Figure 2.11: Monte Carlo sampling example: locations in the parameter space of the 200 Monte Carlo samples using a uniform distribution for both x1 and x2 . The basic pattern is that of array lengths and string identifiers followed by listings of the array entries, where the arrays consist of the variables, the active set vector (ASV), the derivative values vector (DVV), and the analysis components (AC). For the variables array, the first line gives the total number of variables (2) and the “variables” string identifier, and the subsequent two lines provide the array listing for the two variable values (-1.2 and 1.0) and descriptor tags (“x1” and “x2” from the Dakota input file). The next array conveys the ASV, which indicates what simulator outputs are needed. The first line of the array gives the total number of response functions (1) and the “functions” string identifier, followed by one ASV code and descriptor tag (“ASV 1”) for each function. In this case, the ASV value of 1 indicates that Dakota is requesting that the simulation code return the response function value in the file results.out.X. (Possible ASV values: 1 = value of response function value, 2 = response function gradient, 4 = response function Hessian, and any sum of these for combinations up to 7 = response function value, gradient, and Hessian; see 9.7 for more detail.) The next array provides the DVV, which defines the variable identifiers used in computing derivatives. The first line of the array gives the number of derivative variables (2) and the “derivative variables” string identifier, followed by the listing of the two DVV variable identifiers (the first and second variables) and descriptor tags (“DVV 1” and “DVV 2”). The final array provides the AC array used to provide additional strings for use by the simulator (e.g., to provide the name of a particular mesh file). The first line of the array gives the total number of analysis components (0) and the “analysis components” string identifier, followed by the listing of the array, which is empty in this case. The executable program rosenbrock reads in the params.in.X file and evaluates the objective function at the given values for x1 and x2 . Then, rosenbrock writes out the objective function data to the results.out.X file. Here is the listing for the file results.out.1: 2.420000000000000e+01 f The value shown above is the value of the objective function, and the descriptor ‘f’ is an optional tag returned by the simulation code. When the fork call has completed, Dakota reads in the data from the results.in.X file and processes the results. Dakota then continues with additional executions of the rosenbrock program until the optimization process is complete. Dakota Version 6.7 User’s Manual generated on November 13, 2017 2.3. EXAMPLES 2.3.5.2 41 Optimization with a User-Supplied Simulation Code - Case 2 In many situations the user-supplied simulation code cannot be modified to read and write the params.in.X file and the results.out.X file, as described above. Typically, this occurs when the simulation code is a commercial or proprietary software product that has specific input file and output file formats. In such cases, it is common to replace the executable program name in the Dakota input file with the name of a Unix shell script containing a sequence of commands that read and write the necessary files and run the simulation code. For example, the executable program named rosenbrock listed in Figure 2.12 could be replaced by a Unix Bourne or C-shell script named simulator script, with the script containing a sequence of commands to perform # Dakota Input File: rosen_syscall.in # Usage: # dakota -i rosen_syscall.in -o rosen_syscall.out > rosen_syscall.stdout environment graphics tabular_data tabular_data_file = ’rosen_syscall.dat’ method conmin_frcg convergence_tolerance = 1e-4 max_iterations = 100 model single variables continuous_design = 2 initial_point -1.2 lower_bounds -2.0 upper_bounds 2.0 descriptors ’x1’ 1.0 -2.0 2.0 "x2" interface analysis_drivers = ’rosenbrock’ fork parameters_file = ’params.in’ results_file = ’results.out’ responses objective_functions = 1 numerical_gradients method_source dakota interval_type forward fd_step_size = 1.e-5 no_hessians Figure 2.12: Dakota input file for gradient-based optimization using the fork call interface to an external rosenbrock simulator. Dakota Version 6.7 User’s Manual generated on November 13, 2017 42 CHAPTER 2. DAKOTA TUTORIAL the following steps: insert the data from the parameters.in.X file into the input file of the simulation code, execute the simulation code, post-process the files generated by the simulation code to compute response data, and return the response data to Dakota in the results.out.X file. The steps that are typically used in constructing and using a Unix shell script are described in Section 10.3. 2.4 Dakota Command-Line Options The Dakota executable file is named dakota (dakota.exe on Windows). If this command is entered at the command prompt without any arguments, a usage message similar to the following appears: usage: dakota [options and] -help (Print this summary) -version (Print Dakota version number) -input <$val> (REQUIRED Dakota input file $val) -output <$val> (Redirect Dakota standard output to file $val) -error <$val> (Redirect Dakota standard error to file $val) -parser <$val> (Parsing technology: nidr[strict][:dumpfile]) -no_input_echo (Do not echo Dakota input file) -check (Perform input checks) -pre_run [$val] (Perform pre-run (variables generation) phase) -run [$val] (Perform run (model evaluation) phase) -post_run [$val] (Perform post-run (final results) phase) -read_restart [$val] (Read an existing Dakota restart file $val) -stop_restart <$val> (Stop restart file processing at evaluation $val) -write_restart [$val] (Write a new Dakota restart file $val) Of these available command line inputs, only the “-input” option is required, and “-input” can be omitted if the input file name is the final item on the command line; all other command-line inputs are optional. The “-help” option prints the usage message above. The “-version” option prints the version number of the executable. The “-check” option invokes a dry-run mode in which the input file is processed and checked for errors, but the study is not performed. The “-input” option provides the name of the Dakota input file. The “-output” and “-error” options provide file names for redirection of the Dakota standard output (stdout) and standard error (stderr), respectively. By default, Dakota will echo the input file to the output stream, but “-no input echo” can override this behavior. The “-parser” input is for debugging and will not be further described here. The “-read restart” and “-write restart” options provide the names of restart databases to read from and write to, respectively. The “-stop restart” option limits the number of function evaluations read from the restart database (the default is all the evaluations) for those cases in which some evaluations were erroneous or corrupted. Restart management is an important technique for retaining data from expensive engineering applications. This advanced topic is discussed in detail in Chapter 18. Note that these command line options can be abbreviated so long as the abbreviation is unique. Accordingly, the following are valid, unambiguous specifications: “-h”, “-v”, “-c”, “-i”, “-o”, “-e”, “-re”, “-s”, “-w”, “-pr”, “-ru”, and “-po” and can be used in place of the longer forms of the command line options. To run Dakota with a particular input file, the following syntax can be used: dakota -i dakota.in or more simply Dakota Version 6.7 User’s Manual generated on November 13, 2017 2.5. NEXT STEPS 43 dakota dakota.in This will echo the standard output (stdout) and standard error (stderr) messages to the terminal. To redirect stdout and stderr to separate files, the -o and -e command line options may be used: dakota -i dakota.in -o dakota.out -e dakota.err or dakota -o dakota.out -e dakota.err dakota.in Alternatively, any of a variety of Unix redirection variants can be used. The simplest of these redirects stdout to another file: dakota dakota.in > dakota.out To run the dakota process in the background, append an ampersand symbol (&) to the command with an embedded space, e.g., dakota dakota.in > dakota.out & Refer to [6] for more information on Unix redirection and background commands. The “-pre run”, “-run”, and “-post run” options instruct Dakota to run one or more execution phases, excluding others. For example pre-run might generate variable sets, run (core run) might invoke the simulation to evaluate variables and produce responses, and post-run might accept variable/response sets and analyzes the results (for example, calculate correlations from a set of samples). Currently only two modes are supported and only for sampling, parameter study, and DACE methods: (1) pre-run only with optional tabular output of variables: dakota -i dakota.in -pre_run [::myvariables.dat] and (2) post-run only with required tabular input of variables/responses: dakota -i dakota.in -post_run myvarsresponses.dat:: 2.5 Next Steps After reading this chapter, you should understand the mechanics of acquiring, installing, and executing Dakota to perform simple studies. You should have a high-level appreciation for what inputs Dakota requires, how it behaves during interfacing and operation for a few kinds of studies, and what representative output is obtained. To effectively use Dakota, you will need to understand the character of your problem, select a Dakota method to help you meet your analysis goals, and develop an interface to your computational model. 2.5.1 Problem Exploration and Method Selection Section 1.2 provides a high-level overview of the analysis techniques available in Dakota, with references to more details and usage guidelines in the following chapters. Selecting an appropriate method to meet your analysis goals requires understanding problem characteristics. For example, in optimization, typical questions that should Dakota Version 6.7 User’s Manual generated on November 13, 2017 44 CHAPTER 2. DAKOTA TUTORIAL be addressed include: Are the design variables continuous, discrete, or mixed? Is the problem constrained or unconstrained? How expensive are the response functions to evaluate? Will the response functions behave smoothly as the design variables change or will there be nonsmoothness and/or discontinuities? Are the response functions likely to be multimodal, such that global optimization may be warranted? Is analytic gradient data available, and if not, can gradients be calculated accurately and cheaply? Questions pertinent for uncertainty quantification may include: Can I accurately model the probabilistic distributions of my uncertain variables? Are the response functions relatively linear? Am I interested in a full random process characterization of the response functions, or just statistical results? If there is not sufficient information from the problem description and prior knowledge to answer these questions, then additional problem characterization activities may be warranted. Dakota parameter studies and design of experiments methods can help answer these questions by systematically interrogating the model. The resulting trends in the response functions can be evaluated to determine if these trends are noisy or smooth, unimodal or multimodal, relatively linear or highly nonlinear, etc. In addition, the parameter studies may reveal that one or more of the parameters do not significantly affect the results and can be omitted from the problem formulation. This can yield a potentially large savings in computational expense for the subsequent studies. Refer to Chapters 3 and 4 for additional information on parameter studies and design of experiments methods. For a list of all the example Dakota input files, see Table 2.1. All of these input files can be found in Dakota/examples/users. 2.5.2 Key Getting Started References The following references address many of the most common questions raised by new Dakota users: • Dakota documentation and training materials are available from the Dakota website http://dakota. sandia.gov. • Dakota input file syntax (valid keywords and settings) is described in the Dakota Reference Manual [3]. • Example input files are included throughout this manual, and are included in Dakota distributions and installations. See Section 1.5 for help finding these files. • Detailed method descriptions appear in the Method Tour in Chapters 3 through 7. • Building an interface to a simulation code: Section 10.3, and related information on parameters file formats (Section 9.6) and results file format (Section 11.2). • Chapter 13 describes the different Dakota output file formats, including commonly encountered error messages. • Chapter 18 describes the file restart and data re-use capabilities of Dakota. • Documentation for getting started with the Dakota Graphical User Interface may be found here: https: //dakota.sandia.gov/content/dakota-gui-user-manual. Dakota Version 6.7 User’s Manual generated on November 13, 2017 2.5. NEXT STEPS 45 Table 2.1: List of Example Input Files Method Category parameter study parameter study design of comp exp uncertainty quantification uncertainty quantification uncertainty quantification uncertainty quantification uncertainty quantification uncertainty quantification uncertainty quantification uncertainty quantification optimization optimization optimization optimization optimization optimization optimization optimization optimization optimization optimization optimization optimization optimization calibration calibration advanced methods advanced methods advanced methods advanced methods advanced methods advanced models advanced models advanced models Specific Method multidimensional parameter study vector parameter study moat (Morris) random sampling LHS sampling polynomial chaos expansion stochastic collocation reliability Mean Value reliability FORM global interval analysis global evidence analysis gradient-based, unconstrained gradient-based, unconstrained gradient-based, constrained gradient-based, constrained gradient-based, constrained evolutionary algorithm pattern search efficient global optimization (EGO) efficient global optimization (EGO) multiobjective Pareto opt., moga Pareto opt., moga Pareto opt., moga optimization with scaling nonlinear least squares NLS with datafile hybrid minimization Pareto minimization multistart minimization surrogate based global surrogate based local opt. under uncertainty (OUU) second order probability sampling on a surrogate model Dakota Version 6.7 User’s Manual generated on November 13, 2017 Input File Name rosen multidim.in rosen ps vector.in morris ps moat.in rosen sampling.in textbook uq sampling.in rosen uq pce.in rosen uq sc.in textbook uq meanvalue.in logratio uq reliability.in cantilever uq global interval.in textbook uq glob evidence.in rosen grad opt.in rosen syscall.in textbook opt conmin.in cantilever opt npsol.in container opt npsol.in rosen opt ea.in rosen opt patternsearch.in rosen opt ego.in herbie shubert opt ego.in textbook opt multiobj1.in mogatest1.in mogatest2.in mogatest3.in rosen opt scaled.in rosen opt nls.in textbook nls datafile.in textbook hybrid strat.in textbook pareto strat.in qsf multistart strat.in mogatest1 opt sbo.in rosen opt sbo.in textbook opt ouu1.in cantilever uq sop rel.in textbook uq surrogate.in Reference 2.4 3.3 20.22 2.9 5.2 5.14 5.17 5.8 5.10 5.20 5.23 2.7 2.12 20.2 20.16 13.1 6.3 6.1 6.5 20.7 6.6 6.8 20.17 20.19 6.10 20.4 14.1 14.4 14.2 14.10 14.8 15.5 15.2 46 CHAPTER 2. DAKOTA TUTORIAL Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 3 Parameter Study Capabilities 3.1 Overview Dakota parameter studies explore the effect of parametric changes within simulation models by computing response data sets at a selection of points in the parameter space, yielding one type of sensitivity analysis. (For a comparison with DACE-based sensitivity analysis, see Section 4.6.) The selection of points is deterministic and structured, or user-specified, in each of the four available parameter study methods: • Vector: Performs a parameter study along a line between any two points in an n-dimensional parameter space, where the user specifies the number of steps used in the study. • List: The user supplies a list of points in an n-dimensional space where Dakota will evaluate response data from the simulation code. • Centered: Given a point in an n-dimensional parameter space, this method evaluates nearby points along the coordinate axes of the parameter space. The user selects the number of steps and the step size. • Multidimensional: Forms a regular lattice or hypergrid in an n-dimensional parameter space, where the user specifies the number of intervals used for each parameter. More detail on these parameter studies is found in Sections 3.2 through 3.5 below. When used in parameter studies, the response data sets are not linked to any specific interpretation, so they may consist of any allowable specification from the responses keyword block, i.e., objective and constraint functions, least squares terms and constraints, or generic response functions. This allows the use of parameter studies in alternation with optimization, least squares, and uncertainty quantification studies with only minor modification to the input file. In addition, the response data sets may include gradients and Hessians of the response functions, which will be catalogued by the parameter study. This allows for several different approaches to “sensitivity analysis”: (1) the variation of function values over parameter ranges provides a global assessment as to the sensitivity of the functions to the parameters, (2) derivative information can be computed numerically, provided analytically by the simulator, or both (mixed gradients) in directly determining local sensitivity information at a point in parameter space, and (3) the global and local assessments can be combined to investigate the variation of derivative quantities through the parameter space by computing sensitivity information at multiple points. In addition to sensitivity analysis applications, parameter studies can be used for investigating nonsmoothness in simulation response variations (so that models can be refined or finite difference step sizes can be selected for 48 CHAPTER 3. PARAMETER STUDY CAPABILITIES computing numerical gradients), interrogating problem areas in the parameter space, or performing simulation code verification (verifying simulation robustness) through parameter ranges of interest. A parameter study can also be used in coordination with minimization methods as either a pre-processor (to identify a good starting point) or a post-processor (for post-optimality analysis). Parameter study methods will iterate any combination of design, uncertain, and state variables defined over continuous and discrete domains into any set of responses (any function, gradient, and Hessian definition). Parameter studies draw no distinction among the different types of continuous variables (design, uncertain, or state) or among the different types of response functions. They simply pass all of the variables defined in the variables specification into the interface, from which they expect to retrieve all of the responses defined in the responses specification. As described in Section 11.3, when gradient and/or Hessian information is being catalogued in the parameter study, it is assumed that derivative components will be computed with respect to all of the continuous variables (continuous design, continuous uncertain, and continuous state variables) specified, since derivatives with respect to discrete variables are assumed to be undefined. The specification of initial values or bounds is important for parameter studies. 3.1.1 Initial Values The vector and centered parameter studies use the initial values of the variables from the variables keyword block as the starting point and the central point of the parameter studies, respectively. In the case of design variables, the initial point is used, and in the case of state variables, the initial state is used (see the Dakota Reference Manual [3] for default values when initial point or initial state are unspecified). In the case of uncertain variables, initial values are inferred from the distribution specification: all uncertain initial values are set to their means, where mean values for bounded normal and bounded lognormal are repaired of needed to satisfy the specified distribution bounds, mean values for discrete integer range distributions are rounded down to the nearest integer, and mean values for discrete set distributions are rounded to the nearest set value. These parameter study starting values for design, uncertain, and state variables are referenced in the following sections using the identifier “Initial Values.” 3.1.2 Bounds The multidimensional parameter study uses the bounds of the variables from the variables keyword block to define the range of parameter values to study. In the case of design and state variables, the lower bounds and upper bounds specifications are used (see the Dakota Reference Manual [3] for default values when lower bounds or upper bounds are unspecified). In the case of uncertain variables, these values are either drawn or inferred from the distribution specification. Distribution lower and upper bounds can be drawn directly from required bounds specifications for uniform, loguniform, triangular, and beta distributions, as well as from optional bounds specifications for normal and lognormal. Distribution bounds are implicitly defined for histogram bin, histogram point, and interval variables (from the extreme values within the bin/point/interval specifications) as well as for binomial (0 to num trials) and hypergeometric (0 to min(num drawn,num selected)) variables. Finally, distribution bounds are inferred for normal and lognormal if optional bounds are unspecified, as well as for exponential, gamma, gumbel, frechet, weibull, poisson, negative binomial, and geometric (which have no bounds specifications); these bounds are [0, µ+3σ] for exponential, gamma, frechet, weibull, poisson, negative binomial, geometric, and unspecified lognormal, and [µ − 3σ, µ + 3σ] for gumbel and unspecified normal. Dakota Version 6.7 User’s Manual generated on November 13, 2017 3.2. VECTOR PARAMETER STUDY 3.2 49 Vector Parameter Study The vector parameter study computes response data sets at selected intervals along an n-dimensional vector in parameter space. This capability encompasses both single-coordinate parameter studies (to study the effect of a single variable on a response set) as well as multiple coordinate vector studies (to investigate the response variations along some arbitrary vector; e.g., to investigate a search direction failure). Dakota’s vector parameter study includes two possible specification formulations which are used in conjunction with the Initial Values (see Section 3.1.1) to define the vector and steps of the parameter study: final_point (vector of reals) and num_steps (integer) step_vector (vector of reals) and num_steps (integer) In both of these cases, the Initial Values are used as the parameter study starting point and the specification selection above defines the orientation of the vector and the increments to be evaluated along the vector. In the former case, the vector from initial to final point is partitioned by num steps, and in the latter case, the step vector is added num steps times. In the case of discrete range variables, both final point and step vector are specified in the actual values; and in the case of discrete sets (integer or real), final point is specified in the actual values but step vector must instead specify index offsets for the (ordered, unique) set. In all cases, the number of evaluations is num steps+1. Two examples are included below: Three continuous parameters with initial values of (1.0, 1.0, 1.0), num steps = 4, and either final point = (1.0, 2.0, 1.0) or step vector = (0, .25, 0): Parameters for function evaluation 1: 1.0000000000e+00 1.0000000000e+00 1.0000000000e+00 Parameters for function evaluation 2: 1.0000000000e+00 1.2500000000e+00 1.0000000000e+00 Parameters for function evaluation 3: 1.0000000000e+00 1.5000000000e+00 1.0000000000e+00 Parameters for function evaluation 4: 1.0000000000e+00 1.7500000000e+00 1.0000000000e+00 Parameters for function evaluation 5: 1.0000000000e+00 2.0000000000e+00 1.0000000000e+00 c1 c2 c3 c1 c2 c3 c1 c2 c3 c1 c2 c3 c1 c2 c3 Two continuous parameters with initial values of (1.0, 1.0), one discrete range parameter with initial value of 5, one discrete real set parameter with set values of (10., 12., 18., 30., 50.) and initial value of 10., num steps = 4, and either final point = (2.0, 1.4, 13, 50.) or step vector = (.25, .1, 2, 1): Parameters for function evaluation 1: 1.0000000000e+00 c1 1.0000000000e+00 c2 5 di1 Dakota Version 6.7 User’s Manual generated on November 13, 2017 50 CHAPTER 3. PARAMETER STUDY CAPABILITIES 1.0000000000e+01 Parameters for function evaluation 2: 1.2500000000e+00 1.1000000000e+00 7 1.2000000000e+01 Parameters for function evaluation 3: 1.5000000000e+00 1.2000000000e+00 9 1.8000000000e+01 Parameters for function evaluation 4: 1.7500000000e+00 1.3000000000e+00 11 3.0000000000e+01 Parameters for function evaluation 5: 2.0000000000e+00 1.4000000000e+00 13 5.0000000000e+01 dr1 c1 c2 di1 dr1 c1 c2 di1 dr1 c1 c2 di1 dr1 c1 c2 di1 dr1 An example using a vector parameter study is described in Section 3.7. 3.3 List Parameter Study The list parameter study computes response data sets at selected points in parameter space. These points are explicitly specified by the user and are not confined to lie on any line or surface. Thus, this parameter study provides a general facility that supports the case where the desired set of points to evaluate does not fit the prescribed structure of the vector, centered, or multidimensional parameter studies. The user input consists of a list of points specification which lists the requested parameter sets in succession. The list parameter study simply performs a simulation for the first parameter set (the first n entries in the list), followed by a simulation for the next parameter set (the next n entries), and so on, until the list of points has been exhausted. Since the Initial Values will not be used, they need not be specified. In the case of discrete range or discrete set variables, list values are specified using the actual values (not set indices). An example specification that would result in the same parameter sets as in the second example in Section 3.2 would be: list_of_points = 1.0 1.25 1.5 1.75 2.0 1.0 5 10. 1.1 7 12. 1.2 9 18. 1.3 11 30. 1.4 13 50. For convenience, the points for evaluation in a list parameter study may instead be specified via the import points file specification, e.g., ’import points file listpstudy.dat’, where the file listpstudy.dat may be in freeform or annotated format 12.1.1. The ordering of the points is in input specification order, with both active and inactive variables by default. Dakota Version 6.7 User’s Manual generated on November 13, 2017 3.4. CENTERED PARAMETER STUDY 3.4 51 Centered Parameter Study The centered parameter study executes multiple coordinate-based parameter studies, one per parameter, centered about the specified Initial Values. This is useful for investigation of function contours in the vicinity of a specific point. For example, after computing an optimum design, this capability could be used for post-optimality analysis in verifying that the computed solution is actually at a minimum or constraint boundary and in investigating the shape of this minimum or constraint boundary. This method requires step vector (list of reals) and steps per variable (list of integers) specifications, where the former specifies the size of the increments per variable (employed sequentially, not all at once as for the vector study in Section 3.2) and the latter specifies the number of increments per variable (employed sequentially, not all at once) for each of the positive and negative step directions. As for the vector study described in Section 3.2, step vector includes actual variable steps for continuous and discrete range variables, but employs index offsets for discrete set variables (integer or real). For example, with Initial Values of (1.0, 1.0), a step vector of (0.1, 0.1), and a steps per variable of (2, 2), the center point is evaluated followed by four function evaluations (two negative deltas and two positive deltas) per variable: Parameters for function evaluation 1: 1.0000000000e+00 1.0000000000e+00 Parameters for function evaluation 2: 8.0000000000e-01 1.0000000000e+00 Parameters for function evaluation 3: 9.0000000000e-01 1.0000000000e+00 Parameters for function evaluation 4: 1.1000000000e+00 1.0000000000e+00 Parameters for function evaluation 5: 1.2000000000e+00 1.0000000000e+00 Parameters for function evaluation 6: 1.0000000000e+00 8.0000000000e-01 Parameters for function evaluation 7: 1.0000000000e+00 9.0000000000e-01 Parameters for function evaluation 8: 1.0000000000e+00 1.1000000000e+00 Parameters for function evaluation 9: 1.0000000000e+00 1.2000000000e+00 d1 d2 d1 d2 d1 d2 d1 d2 d1 d2 d1 d2 d1 d2 d1 d2 d1 d2 This set of points in parameter space is depicted in Figure 3.1. Dakota Version 6.7 User’s Manual generated on November 13, 2017 52 CHAPTER 3. PARAMETER STUDY CAPABILITIES Figure 3.1: Example centered parameter study. 3.5 Multidimensional Parameter Study The multidimensional parameter study computes response data sets for an n-dimensional hypergrid of points. Each variable is partitioned into equally spaced intervals between its upper and lower bounds (see Section 3.1.2), and each combination of the values defined by these partitions is evaluated. As for the vector and centered studies described in Sections 3.2 and 3.4, partitioning occurs using the actual variable values for continuous and discrete range variables, but occurs within the space of valid indices for discrete set variables (integer or real). The number of function evaluations performed in the study is: n Y (partitionsi + 1) (3.1) i=1 The partitions information is specified using the partitions specification, which provides an integer list of the number of partitions for each variable (i.e., partitionsi ). Since the Initial Values will not be used, they need not be specified. In a two variable example problem with d1 ∈ [0,2] and d2 ∈ [0,3] (as defined by the upper and lower bounds from the variables specification) and with partitions = (2,3), the interval [0,2] is divided into two equal-sized partitions and the interval [0,3] is divided into three equal-sized partitions. This two-dimensional grid, shown in Figure 3.2, would result in the following twelve function evaluations: Parameters for function evaluation 1: 0.0000000000e+00 0.0000000000e+00 Parameters for function evaluation 2: 1.0000000000e+00 0.0000000000e+00 Parameters for function evaluation 3: 2.0000000000e+00 0.0000000000e+00 Parameters for function evaluation 4: 0.0000000000e+00 1.0000000000e+00 Parameters for function evaluation 5: d1 d2 d1 d2 d1 d2 d1 d2 Dakota Version 6.7 User’s Manual generated on November 13, 2017 3.5. MULTIDIMENSIONAL PARAMETER STUDY 53 Figure 3.2: Example multidimensional parameter study Parameters for Parameters for Parameters for Parameters for Parameters for Parameters for Parameters for 1.0000000000e+00 1.0000000000e+00 function evaluation 6: 2.0000000000e+00 1.0000000000e+00 function evaluation 7: 0.0000000000e+00 2.0000000000e+00 function evaluation 8: 1.0000000000e+00 2.0000000000e+00 function evaluation 9: 2.0000000000e+00 2.0000000000e+00 function evaluation 10: 0.0000000000e+00 3.0000000000e+00 function evaluation 11: 1.0000000000e+00 3.0000000000e+00 function evaluation 12: 2.0000000000e+00 3.0000000000e+00 d1 d2 d1 d2 d1 d2 d1 d2 d1 d2 d1 d2 d1 d2 d1 d2 The first example shown in this User’s Manual is a multi-dimensional parameter study. See Section 2.3.2. Dakota Version 6.7 User’s Manual generated on November 13, 2017 54 3.6 CHAPTER 3. PARAMETER STUDY CAPABILITIES Parameter Study Usage Guidelines Parameter studies, classical design of experiments (DOE), design/analysis of computer experiments (DACE), and sampling methods share the purpose of exploring the parameter space. Parameter Studies are recommended for simple studies with defined, repetitive structure. A local sensitivity analysis or an assessment of the smoothness of a response function is best addressed with a vector or centered parameter study. A multi-dimensional parameter study may be used to generate grid points for plotting response surfaces. For guidance on DACE and sampling methods, in contrast to parameter studies, see Section 4.7 and especially Table 4.4, which clarifies the different purposes of the method types. 3.7 Example: Vector Parameter Study with Rosenbrock This section demonstrates a vector parameter study on the Rosenbrock test function described in Section 2.3.1. An example of multidimensional parameter study is shown in Section 2.3.2. A vector parameter study is a study between any two design points in an n-dimensional parameter space. An input file for the vector parameter study is shown in Figure 3.3. The primary differences between this input file and the input file for the multidimensional parameter study are found in the variables and method sections. In the variables section, the keywords for the bounds are removed and replaced with the keyword initial point that specifies the starting point for the parameter study. In the method section, the vector parameter study keyword is used. The final point keyword indicates the stopping point for the parameter study, and num steps specifies the number of steps taken between the initial and final points in the parameter study. Figure 3.4(a) shows the graphics output created by Dakota. For this study, the simple Dakota graphics are more useful for visualizing the results. Figure 3.4(b) shows the locations of the 11 sample points generated in this study. It is evident from these figures that the parameter study starts within the banana-shaped valley, marches up the side of the hill, and then returns to the valley. Dakota Version 6.7 User’s Manual generated on November 13, 2017 3.7. EXAMPLE: VECTOR PARAMETER STUDY WITH ROSENBROCK 55 # Dakota Input File: rosen_ps_vector.in environment graphics tabular_data tabular_data_file = ’rosen_ps_vector.dat’ method vector_parameter_study final_point = 1.1 1.3 num_steps = 10 model single variables continuous_design = 2 initial_point -0.3 descriptors ’x1’ 0.2 "x2" interface analysis_drivers = ’rosenbrock’ direct responses objective_functions = 1 no_gradients no_hessians Figure 3.3: Rosenbrock vector parameter study Dakota/examples/users/rosen ps vector.in Dakota Version 6.7 User’s Manual generated on November 13, 2017 example: the Dakota input file – see 56 CHAPTER 3. PARAMETER STUDY CAPABILITIES (a) (b) Figure 3.4: Rosenbrock vector parameter study example: (a) screen capture of the Dakota graphics and (b) location of the design points (dots) evaluated. Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 4 Design of Experiments Capabilities 4.1 Overview Classical design of experiments (DoE) methods and the more modern design and analysis of computer experiments (DACE) methods are both techniques which seek to extract as much trend data from a parameter space as possible using a limited number of sample points. Classical DoE techniques arose from technical disciplines that assumed some randomness and nonrepeatability in field experiments (e.g., agricultural yield, experimental chemistry). DoE approaches such as central composite design, Box-Behnken design, and full and fractional factorial design generally put sample points at the extremes of the parameter space, since these designs offer more reliable trend extraction in the presence of nonrepeatability. DACE methods are distinguished from DoE methods in that the nonrepeatability component can be omitted since computer simulations are involved. In these cases, space filling designs such as orthogonal array sampling and Latin hypercube sampling are more commonly employed in order to accurately extract trend information. Quasi-Monte Carlo sampling techniques which are constructed to fill the unit hypercube with good uniformity of coverage can also be used for DACE. Dakota supports both DoE and DACE techniques. In common usage, only parameter bounds are used in selecting the samples within the parameter space. Thus, DoE and DACE can be viewed as special cases of the more general probabilistic sampling for uncertainty quantification (see following section), in which the DoE/DACE parameters are treated as having uniform probability distributions. The DoE/DACE techniques are commonly used for investigation of global response trends, identification of significant parameters (e.g., main effects), and as data generation methods for building response surface approximations. Dakota includes several approaches sampling and design of experiments, all implemented in included third-party software libraries. LHS (Latin hypercube sampling) [132] is a general-purpose sampling package developed at Sandia that has been used by the DOE national labs for several decades. DDACE (distributed design and analysis for computer experiments) is a more recent package for computer experiments developed at Sandia Labs [137]. DDACE provides the capability for generating orthogonal arrays, Box-Behnken designs, Central Composite designs, and random designs. The FSUDace (Florida State University’s Design and Analysis of Computer Experiments) package provides the following sampling techniques: quasi-Monte Carlo sampling based on Halton or Hammersley sequences, and Centroidal Voronoi Tessellation. Lawrence Livermore National Lab’s PSUADE (Problem Solving Environment for Uncertainty Analysis and Design Exploration) [136] includes several methods for model exploration, but only the Morris screening method is exposed in Dakota. This chapter describes DDACE, FSUDace, and PSUADE, with a focus on designing computer experiments. Latin Hypercube Sampling, also used in uncertainty quantification, is discussed in Section 5.2. 58 4.2 CHAPTER 4. DESIGN OF EXPERIMENTS CAPABILITIES Design of Computer Experiments What distinguishes design of computer experiments? Computer experiments are often different from physical experiments, such as those performed in agriculture, manufacturing, or biology. In physical experiments, one often applies the same treatment or factor level in an experiment several times to get an understanding of the variability of the output when that treatment is applied. For example, in an agricultural experiment, several fields (e.g., 8) may be subject to a low level of fertilizer and the same number of fields may be subject to a high level of fertilizer to see if the amount of fertilizer has a significant effect on crop output. In addition, one is often interested in the variability of the output within a treatment group: is the variability of the crop yields in the low fertilizer group much higher than that in the high fertilizer group, or not? In physical experiments, the process we are trying to examine is stochastic: that is, the same treatment may result in different outcomes. By contrast, in computer experiments, often we have a deterministic code. If we run the code with a particular set of input parameters, the code will always produce the same output. There certainly are stochastic codes, but the main focus of computer experimentation has been on deterministic codes. Thus, in computer experiments we often do not have the need to do replicates (running the code with the exact same input parameters several times to see differences in outputs). Instead, a major concern in computer experiments is to create an experimental design which can sample a high-dimensional space in a representative way with a minimum number of samples. The number of factors or parameters that we wish to explore in computer experiments is usually much higher than physical experiments. In physical experiments, one may be interested in varying a few parameters, usually five or less, while in computer experiments we often have dozens of parameters of interest. Choosing the levels of these parameters so that the samples adequately explore the input space is a challenging problem. There are many experimental designs and sampling methods which address the issue of adequate and representative sample selection. There are many goals of running a computer experiment: one may want to explore the input domain or the design space and get a better understanding of the range in the outputs for a particular domain. Another objective is to determine which inputs have the most influence on the output, or how changes in the inputs change the output. This is usually called sensitivity analysis. Another goal is to use the sampled input points and their corresponding output to create a response surface approximation for the computer code. The response surface approximation (e.g., a polynomial regression model, a Gaussian-process/Kriging model, a neural net) can then be used to emulate the computer code. Constructing a response surface approximation is particularly important for applications where running a computational model is extremely expensive: the computer model may take 10 or 20 hours to run on a high performance machine, whereas the response surface model may only take a few seconds. Thus, one often optimizes the response surface model or uses it within a framework such as surrogatebased optimization. Response surface models are also valuable in cases where the gradient (first derivative) and/or Hessian (second derivative) information required by optimization techniques are either not available, expensive to compute, or inaccurate because the derivatives are poorly approximated or the function evaluation is itself noisy due to roundoff errors. Furthermore, many optimization methods require a good initial point to ensure fast convergence or to converge to good solutions (e.g. for problems with multiple local minima). Under these circumstances, a good design of computer experiment framework coupled with response surface approximations can offer great advantages. In addition to the sensitivity analysis and response surface modeling mentioned above, we also may want to do uncertainty quantification on a computer model. Uncertainty quantification (UQ) refers to taking a particular set of distributions on the inputs, and propagating them through the model to obtain a distribution on the outputs. For example, if input parameter A follows a normal with mean 5 and variance 1, the computer produces a random draw from that distribution. If input parameter B follows a weibull distribution with alpha = 0.5 and beta = 1, the computer produces a random draw from that distribution. When all of the uncertain variables have samples drawn from their input distributions, we run the model with the sampled values as inputs. We do this repeatedly to build up a distribution of outputs. We can then use the cumulative distribution function of the output to ask questions Dakota Version 6.7 User’s Manual generated on November 13, 2017 4.3. DDACE 59 such as: what is the probability that the output is greater than 10? What is the 99th percentile of the output? Note that sampling-based uncertainty quantification and design of computer experiments are very similar. There is significant overlap in the purpose and methods used for UQ and for DACE. We have attempted to delineate the differences within Dakota as follows: we use the methods DDACE, FSUDACE, and PSUADE primarily for design of experiments, where we are interested in understanding the main effects of parameters and where we want to sample over an input domain to obtain values for constructing a response surface. We use the nondeterministic sampling methods (sampling) for uncertainty quantification, where we are propagating specific input distributions and interested in obtaining (for example) a cumulative distribution function on the output. If one has a problem with no distributional information, we recommend starting with a design of experiments approach. Note that DDACE, FSUDACE, and PSUADE currently do not support distributional information: they take an upper and lower bound for each uncertain input variable and sample within that. The uncertainty quantification methods in sampling (primarily Latin Hypercube sampling) offer the capability to sample from many distributional types. The distinction between UQ and DACE is somewhat arbitrary: both approaches often can yield insight about important parameters and both can determine sample points for response surface approximations. Three software packages are available in Dakota for design of computer experiments, DDACE (developed at Sandia Labs), FSUDACE (developed at Florida State University), and PSUADE (LLNL). 4.3 DDACE The Distributed Design and Analysis of Computer Experiments (DDACE) package includes both classical design of experiments methods [137] and stochastic sampling methods. The classical design of experiments methods in DDACE are central composite design (CCD) and Box-Behnken (BB) sampling. A grid-based sampling (fullfactorial) method is also available. The stochastic methods are orthogonal array sampling [89] (which permits main effects calculations), Monte Carlo (random) sampling, Latin hypercube sampling, and orthogonal arrayLatin hypercube sampling. While DDACE LHS supports variables with normal or uniform distributions, only uniform are supported through Dakota. Also DDACE does not allow enforcement of user-specified correlation structure among the variables. The sampling methods in DDACE can be used alone or in conjunction with other methods. For example, DDACE sampling can be used with both surrogate-based optimization and optimization under uncertainty advanced methods. See Figure 15.5 for an example of how the DDACE settings are used in Dakota. The following sections provide more detail about the sampling methods available for design of experiments in DDACE. 4.3.1 Central Composite Design A Box-Wilson Central Composite Design, commonly called a central composite design (CCD), contains an embedded factorial or fractional factorial design with center points that is augmented with a group of ’star points’ that allow estimation of curvature. If the distance from the center of the design space to a factorial point is ±1 unit for each factor, the distance from the center of the design space to a star point is ±α with | α |> 1. The precise value of α depends on certain properties desired for the design and on the number of factors involved. The CCD design is specified in Dakota with the method command dace central composite. As an example, with two input variables or factors, each having two levels, the factorial design is shown in Table 4.1 . With a CCD, the design in Table 4.1 would be augmented with the following points shown in Table 4.2 if α = 1.3. Dakota Version 6.7 User’s Manual generated on November 13, 2017 60 CHAPTER 4. DESIGN OF EXPERIMENTS CAPABILITIES Table 4.1: Simple Factorial Design Input 1 -1 -1 +1 +1 Input 2 -1 +1 -1 +1 These points define a circle around the original factorial design. Table 4.2: Additional Points to make the factorial design a CCD Input 1 0 0 1.3 -1.3 0 Input 2 +1.3 -1.3 0 0 0 Note that the number of sample points specified in a CCD,samples, is a function of the number of variables in the problem: samples = 1 + 2 ∗ N umV ar + 2N umV ar 4.3.2 Box-Behnken Design The Box-Behnken design is similar to a Central Composite design, with some differences. The Box-Behnken design is a quadratic design in that it does not contain an embedded factorial or fractional factorial design. In this design the treatment combinations are at the midpoints of edges of the process space and at the center, as compared with CCD designs where the extra points are placed at ’star points’ on a circle outside of the process space. BoxBehken designs are rotatable (or near rotatable) and require 3 levels of each factor. The designs have limited capability for orthogonal blocking compared to the central composite designs. Box-Behnken requires fewer runs than CCD for 3 factors, but this advantage goes away as the number of factors increases. The Box-Behnken design is specified in Dakota with the method command dace box behnken. Note that the number of sample points specified in a Box-Behnken design, samples, is a function of the number of variables in the problem: samples = 1 + 4 ∗ N umV ar + (N umV ar − 1)/2 Dakota Version 6.7 User’s Manual generated on November 13, 2017 4.3. DDACE 4.3.3 61 Orthogonal Array Designs Orthogonal array (OA) sampling is a widely used technique for running experiments and systematically testing factor effects [75]. An orthogonal array sample can be described as a 4-tuple (m, n, s, r), where m is the number of sample points, n is the number of input variables, s is the number of symbols, and r is the strength of the orthogonal array. The number of sample points, m, must be a multiple of the number of symbols, s. The number of symbols refers to the number of levels per input variable. The strength refers to the number of columns where we are guaranteed to see all the possibilities an equal number of times. For example, Table 4.3 shows an orthogonal array of strength 2 for m = 8, with 7 variables: Table 4.3: Orthogonal Array for Seven Variables Input 1 0 0 0 0 1 1 1 1 Input 2 0 0 1 1 0 0 1 1 Input 3 0 0 1 1 1 1 0 0 Input 4 0 1 0 1 0 1 0 1 Input 5 0 1 0 1 1 0 1 0 Input 6 0 1 1 0 0 1 1 0 Input 7 0 1 1 0 1 0 0 1 If one picks any two columns, say the first and the third, note that each of the four possible rows we might see there, 0 0, 0 1, 1 0, 1 1, appears exactly the same number of times, twice in this case. DDACE creates orthogonal arrays of strength 2. Further, the OAs generated by DDACE do not treat the factor levels as one fixed value (0 or 1 in the above example). Instead, once a level for a variable is determined in the array, DDACE samples a random variable from within that level. The orthogonal array design is specified in Dakota with the method command dace oas. The orthogonal array method in DDACE is the only method that allows for the calculation of main effects, specified with the command main effects. Main effects is a sensitivity analysis method which identifies the input variables that have the most influence on the output. In main effects, the idea is to look at the mean of the response function when variable A (for example) is at level 1 vs. when variable A is at level 2 or level 3. If these mean responses of the output are statistically significantly different at different levels of variable A, this is an indication that variable A has a significant effect on the response. The orthogonality of the columns is critical in performing main effects analysis, since the column orthogonality means that the effects of the other variables ’cancel out’ when looking at the overall effect from one variable at its different levels. There are ways of developing orthogonal arrays to calculate higher order interactions, such as two-way interactions (what is the influence of Variable A * Variable B on the output?), but this is not available in DDACE currently. At present, one way interactions are supported in the calculation of orthogonal array main effects within DDACE. The main effects are presented as a series of ANOVA tables. For each objective function and constraint, the decomposition of variance of that objective or constraint is presented as a function of the input variables. The p-value in the ANOVA table is used to indicate if the input factor is significant. The p-value is the probability that you would have obtained samples more extreme than you did if the input factor has no effect on the response. For example, if you set a level of significance at 0.05 for your p-value, and the actual p-value is 0.03, then the input factor has a significant effect on the response. Dakota Version 6.7 User’s Manual generated on November 13, 2017 62 4.3.4 CHAPTER 4. DESIGN OF EXPERIMENTS CAPABILITIES Grid Design In a grid design, a grid is placed over the input variable space. This is very similar to a multi-dimensional parameter study where the samples are taken over a set of partitions on each variable (see Section 3.5). The main difference is that in grid sampling, a small random perturbation is added to each sample value so that the grid points are not on a perfect grid. This is done to help capture certain features in the output such as periodic functions. A purely structured grid, with the samples exactly on the grid points, has the disadvantage of not being able to capture important features such as periodic functions with relatively high frequency (due to aliasing). Adding a random perturbation to the grid samples helps remedy this problem. Another disadvantage with grid sampling is that the number of sample points required depends exponentially on the input dimensions. In grid sampling, the number of samples is the number of symbols (grid partitions) raised to the number of variables. For example, if there are 2 variables, each with 5 partitions, the number of samples would be 52 . In this case, doubling the number of variables squares the sample size. The grid design is specified in Dakota with the method command dace grid. 4.3.5 Monte Carlo Design Monte Carlo designs simply involve pure Monte-Carlo random sampling from uniform distributions between the lower and upper bounds on each of the input variables. Monte Carlo designs, specified by dace random, are a way to generate a set of random samples over an input domain. 4.3.6 LHS Design DDACE offers the capability to generate Latin Hypercube designs. For more information on Latin Hypercube sampling, see Section 5.2. Note that the version of LHS in DDACE generates uniform samples (uniform between the variable bounds). The version of LHS offered with nondeterministic sampling can generate LHS samples according to a number of distribution types, including normal, lognormal, weibull, beta, etc. To specify the DDACE version of LHS, use the method command dace lhs. 4.3.7 OA-LHS Design DDACE offers a hybrid design which is combination of an orthogonal array and a Latin Hypercube sample. This design is specified with the method command dace oa lhs. This design has the advantages of both orthogonality of the inputs as well as stratification of the samples (see [111]). 4.4 FSUDace The Florida State University Design and Analysis of Computer Experiments (FSUDace) package provides quasiMonte Carlo sampling (Halton and Hammersley) and Centroidal Voronoi Tessellation (CVT) methods. All three methods natively generate sets of uniform random variables on the interval [0, 1] (or in Dakota, on user-specified uniform intervals). The quasi-Monte Carlo and CVT methods are designed with the goal of low discrepancy. Discrepancy refers to the nonuniformity of the sample points within the unit hypercube. Low discrepancy sequences tend to cover the unit hypercube reasonably uniformly. Quasi-Monte Carlo methods produce low discrepancy sequences, especially if Dakota Version 6.7 User’s Manual generated on November 13, 2017 4.5. PSUADE MOAT 63 one is interested in the uniformity of projections of the point sets onto lower dimensional faces of the hypercube (usually 1-D: how well do the marginal distributions approximate a uniform?) CVT does very well volumetrically: it spaces the points fairly equally throughout the space, so that the points cover the region and are isotropically distributed with no directional bias in the point placement. There are various measures of volumetric uniformity which take into account the distances between pairs of points, regularity measures, etc. Note that CVT does not produce low-discrepancy sequences in lower dimensions, however: the lower-dimension (such as 1-D) projections of CVT can have high discrepancy. The quasi-Monte Carlo sequences of Halton and Hammersley are deterministic sequences determined by a set of prime bases. A Halton design is specified in Dakota with the method command fsu quasi mc halton, and the Hammersley design is specified with the command fsu quasi mc hammersley. For more details about the input specification, see the Reference Manual. CVT points tend to arrange themselves in a pattern of cells that are roughly the same shape. To produce CVT points, an almost arbitrary set of initial points is chosen, and then an internal set of iterations is carried out. These iterations repeatedly replace the current set of sample points by an estimate of the centroids of the corresponding Voronoi subregions [30]. A CVT design is specified in Dakota with the method command fsu cvt. The methods in FSUDace are useful for design of experiments because they provide good coverage of the input space, thus allowing global sensitivity analysis. 4.5 PSUADE MOAT PSUADE (Problem Solving Environment for Uncertainty Analysis and Design Exploration) is a Lawrence Livermore National Laboratory tool for metamodeling, sensitivity analysis, uncertainty quantification, and optimization. Its features include non-intrusive and parallel function evaluations, sampling and analysis methods, an integrated design and analysis framework, global optimization, numerical integration, response surfaces (MARS and higher order regressions), graphical output with Pgplot or Matlab, and fault tolerance [136]. Dakota includes a prototype interface to its Morris One-At-A-Time (MOAT) screening method, a valuable tool for global sensitivity (including interaction) analysis. The Morris One-At-A-Time method, originally proposed by M. D. Morris [103], is a screening method, designed to explore a computational model to distinguish between input variables that have negligible, linear and additive, or nonlinear or interaction effects on the output. The computer experiments performed consist of individually randomized designs which vary one input factor at a time to create a sample of its elementary effects. With MOAT, each dimension of a k−dimensional input space is uniformly partitioned into p levels, creating a grid of pk points x ∈ Rk at which evaluations of the model y(x) might take place. An elementary effect corresponding to input i is computed by a forward difference di (x) = y(x + ∆ei ) − y(x) , ∆ (4.1) where ei is the ith coordinate vector, and the step ∆ is typically taken to be large (this is not intended to be a local derivative approximation). In the present implementation of MOAT, for an input variable scaled to [0, 1], p ∆ = 2(p−1) , so the step used to find elementary effects is slightly larger than half the input range. The distribution of elementary effects di over the input space characterizes the effect of input i on the output of interest. After generating r samples from this distribution, their mean, r µi = Dakota Version 6.7 User’s Manual generated on November 13, 2017 1 X (j) d , r j=1 i (4.2) 64 CHAPTER 4. DESIGN OF EXPERIMENTS CAPABILITIES modified mean r µ∗i = 1 X (j) |d |, r j=1 i (4.3) (using absolute value) and standard deviation v u X 2 u 1 r (j) σi = t di − µi r j=1 (4.4) are computed for each input i. The mean and modified mean give an indication of the overall effect of an input on the output. Standard deviation indicates nonlinear effects or interactions, since it is an indicator of elementary effects varying throughout the input space. The MOAT method is selected with method keyword psuade moat as shown in the sample Dakota input deck in Figure 4.1. The number of samples (samples) must be a positive integer multiple of (number of continuous design variables k + 1) and will be automatically adjusted if misspecified. The number of partitions (partitions) applies to each variable being studied and must be odd (the number of MOAT levels p per variable is partitions + 1, similar to Dakota multidimensional parameter studies). This will also be adjusted at runtime as necessary. Finite user-specified lower and upper bounds are required and will be scaled as needed by the method. For more information on use of MOAT sampling, see the Morris example in Section 20.9, or Saltelli, et al. [116]. 4.6 4.6.1 Sensitivity Analysis Sensitivity Analysis Overview In many engineering design applications, sensitivity analysis techniques and parameter study methods are useful in identifying which of the design parameters have the most influence on the response quantities. This information is helpful prior to an optimization study as it can be used to remove design parameters that do not strongly influence the responses. In addition, these techniques can provide assessments as to the behavior of the response functions (smooth or nonsmooth, unimodal or multimodal) which can be invaluable in algorithm selection for optimization, uncertainty quantification, and related methods. In a post-optimization role, sensitivity information is useful is determining whether or not the response functions are robust with respect to small changes in the optimum design point. In some instances, the term sensitivity analysis is used in a local sense to denote the computation of response derivatives at a point. These derivatives are then used in a simple analysis to make design decisions. Dakota supports this type of study through numerical finite-differencing or retrieval of analytic gradients computed within the analysis code. The desired gradient data is specified in the responses section of the Dakota input file and the collection of this data at a single point is accomplished through a parameter study method with no steps. This approach to sensitivity analysis should be distinguished from the activity of augmenting analysis codes to internally compute derivatives using techniques such as direct or adjoint differentiation, automatic differentiation (e.g., ADIFOR), or complex step modifications. These sensitivity augmentation activities are completely separate from Dakota and are outside the scope of this manual. However, once completed, Dakota can utilize these analytic gradients to perform optimization, uncertainty quantification, and related studies more reliably and efficiently. In other instances, the term sensitivity analysis is used in a more global sense to denote the investigation of variability in the response functions. Dakota supports this type of study through computation of response data sets (typically function values only, but all data sets are supported) at a series of points in the parameter space. The series of points is defined using either a vector, list, centered, or multidimensional parameter study method. Dakota Version 6.7 User’s Manual generated on November 13, 2017 4.6. SENSITIVITY ANALYSIS 65 # Dakota Input File: morris_ps_moat.in environment tabular_data tabular_data_file ’dakota_psuade.0.dat’ method psuade_moat samples = 84 partitions = 3 seed = 500 variables continuous_design = 20 lower_bounds = 0.0 0.0 0.0 0.0 upper_bounds = 1.0 1.0 1.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 1.0 1.0 0.0 0.0 1.0 1.0 interface analysis_drivers = ’morris’ fork asynchronous evaluation_concurrency = 5 responses objective_functions = 1 no_gradients no_hessians Figure 4.1: Dakota input file showing the Dakota/examples/users/morris ps moat.in Dakota Version 6.7 User’s Manual generated on November 13, 2017 Morris One-at-a-Time method – see 66 CHAPTER 4. DESIGN OF EXPERIMENTS CAPABILITIES For example, a set of closely-spaced points in a vector parameter study could be used to assess the smoothness of the response functions in order to select a finite difference step size, and a set of more widely-spaced points in a centered or multidimensional parameter study could be used to determine whether the response function variation is likely to be unimodal or multimodal. See Chapter 3 for additional information on these methods. These more global approaches to sensitivity analysis can be used to obtain trend data even in situations when gradients are unavailable or unreliable, and they are conceptually similar to the design of experiments methods and sampling approaches to uncertainty quantification described in the following sections. 4.6.2 Assessing Sensitivity with DACE Like parameter studies (see Chapter 3), the DACE techniques are useful for characterizing the behavior of the response functions of interest through the parameter ranges of interest. In addition to direct interrogation and visualization of the sampling results, a number of techniques have been developed for assessing the parameters which are most influential in the observed variability in the response functions. One example of this is the well-known technique of scatter plots, in which the set of samples is projected down and plotted against one parameter dimension, for each parameter in turn. Scatter plots with a uniformly distributed cloud of points indicate parameters with little influence on the results, whereas scatter plots with a defined shape to the cloud indicate parameters which are more significant. Related techniques include analysis of variance (ANOVA) [104] and main effects analysis, in which the parameters which have the greatest influence on the results are identified from sampling results. Scatter plots and ANOVA may be accessed through import of Dakota tabular results (see Section 13.3) into external statistical analysis programs such as S-plus, Minitab, etc. Running any of the design of experiments or sampling methods allows the user to save the results in a tabular data file, which then can be read into a spreadsheet or statistical package for further analysis. In addition, we have provided some functions to help determine the most important variables. We take the definition of uncertainty analysis from [116]: “The study of how uncertainty in the output of a model can be apportioned to different sources of uncertainty in the model input.” As a default, Dakota provides correlation analyses when running LHS. Correlation tables are printed with the simple, partial, and rank correlations between inputs and outputs. These can be useful to get a quick sense of how correlated the inputs are to each other, and how correlated various outputs are to inputs. The correlation analyses are explained further in Chapter 5.2. We also have the capability to calculate sensitivity indices through Variance-based Decomposition (VBD). Variancebased decomposition is a global sensitivity method that summarizes how the uncertainty in model output can be apportioned to uncertainty in individual input variables. VBD uses two primary measures, the main effect sensitivity index Si and the total effect index Ti . The main effect sensitivity index corresponds to the fraction of the uncertainty in the output, Y , that can be attributed to input xi alone. The total effects index corresponds to the fraction of the uncertainty in the output, Y , that can be attributed to input xi and its interactions with other variables. The main effect sensitivity index compares the variance of the conditional expectation V arxi [E(Y |xi )] against the total variance V ar(Y ). Formulas for the indices are: V arxi [E(Y |xi )] V ar(Y ) (4.5) E(V ar(Y |x−i )) V ar(Y ) − V ar(E[Y |x−i ]) = V ar(Y ) V ar(Y ) (4.6) Si = and Ti = where Y = f (x) and x−i = (x1 , ..., xi−1 , xi+1 , ..., xm ). Dakota Version 6.7 User’s Manual generated on November 13, 2017 4.7. DOE USAGE GUIDELINES 67 The calculation of Si and Ti requires the evaluation of m-dimensional integrals which are typically approximated by Monte-Carlo sampling. More details on the calculations and interpretation of the sensitivity indices can be found in [116]. In Dakota version 5.1, we have improved calculations for the calculation of the Si and Ti indices when using sampling. The implementation details of these calculatiosn are provided in [148]. VBD can be specified for any of the sampling or DACE methods using the command variance based decomposition. Note that VBD is extremely computationally intensive when using sampling since replicated sets of sample values are evaluated. If the user specified a number of samples, N , and a number of nondeterministic variables, M , variance-based decomposition requires the evaluation of N (M + 2) samples. To obtain sensitivity indices that are reasonably accurate, we recommend that N , the number of samples, be at least one hundred and preferably several hundred or thousands. Because of the computational cost, variance-based decomposition is turned off as a default for sampling or DACE. Another alternative, however, is to obtain these indices using one of the stochastic expansion methods described in Section 5.4. The calculation of the indices using expansion methods is much more efficient since the VBD indices are analytic functions of the coefficients in the stochastic expansion. The paper by Weirs et al. [148] compares different methods for calculating the sensitivity indices for nonlinear problems with significant interaction effects. In terms of interpretation of the sensitivity indices, a larger value of the sensitivity index, Si , means that the uncertainty in the input variable i has a larger effect on the variance of the output. Note that the sum of the main effect indices will be less than or equal to one. If the sum of the main effect indices is much less than one, it indicates that there are significant two-way, three-way, or higher order interactions that contribute significantly to the variance. There is no requirement that the sum of the total effect indices is one: in most cases, the sum of the total effect indices will be greater than one. An example of the Main and Total effects indices as calculated by Dakota using sampling is shown in Figure 4.2 Global sensitivity indices for each response function: response_fn_1 Sobol indices: Main Total 4.7508913283e-01 5.3242162037e-01 uuv_1 3.8112392892e-01 4.9912486515e-01 uuv_2 Figure 4.2: Dakota output for Variance-based Decomposition Finally, we have the capability to calculate a set of quality metrics for a particular input sample. These quality metrics measure various aspects relating to the volumetric spacing of the samples: are the points equally spaced, do they cover the region, are they isotropically distributed, do they have directional bias, etc.? The quality metrics are explained in more detail in the Reference Manual. 4.7 DOE Usage Guidelines Parameter studies, classical design of experiments (DOE), design/analysis of computer experiments (DACE), and sampling methods share the purpose of exploring the parameter space. When a global space-filling set of samples is desired, then the DOE, DACE, and sampling methods are recommended. These techniques are useful for scatter plot and variance analysis as well as surrogate model construction. The distinction between DOE and DACE methods is that the former are intended for physical experiments containing an element of nonrepeatability (and therefore tend to place samples at the extreme parameter vertices), whereas the latter are intended for repeatable computer experiments and are more space-filling in nature. The distinction between DOE/DACE and sampling is drawn based on the distributions of the parameters. DOE/Dakota Version 6.7 User’s Manual generated on November 13, 2017 68 CHAPTER 4. DESIGN OF EXPERIMENTS CAPABILITIES DACE methods typically assume uniform distributions, whereas the sampling approaches in Dakota support a broad range of probability distributions. To use sampling in design of experiments mode (as opposed to uncertainty quantification mode), an active view override (e.g., active all) can be included in the variables specification (see Section 9.5.1) of the Dakota input file. Design of experiments method selection recommendations are summarized in Table 4.4. Table 4.4: Guidelines for selection of parameter study, DOE, DACE, and sampling methods. Method Classification parameter study Applications Applicable Methods sensitivity analysis, directed parameter space investigations centered parameter study, list parameter study, multidim parameter study, vector parameter study classical design of experiments physical experiments (parameters are uniformly distributed) dace (box behnken, central composite) design of computer experiments variance analysis, space filling designs (parameters are uniformly distributed) dace (grid, random, oas, lhs, oa lhs), fsu quasi mc (halton, hammersley), fsu cvt, psuade moat sampling space filling designs (parameters have general probability distributions) sampling (Monte Carlo or LHS) with optional active view override Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 5 Uncertainty Quantification Capabilities 5.1 Overview At a high level, uncertainty quantification (UQ) or nondeterministic analysis is the process of characterizing input uncertainties, forward propagating these uncertainties through a computational model, and performing statistical or interval assessments on the resulting responses. This process determines the effect of uncertainties and assumptions on model outputs or results. In Dakota, uncertainty quantification methods specifically focus on the forward propagation part of the process, where probabilistic or interval information on parametric inputs are mapped through the computational model to assess statistics or intervals on outputs. For an overview of these approaches for engineering applications, consult [73]. UQ is related to sensitivity analysis in that the common goal is to gain an understanding of how variations in the parameters affect the response functions of the engineering design problem. However, for UQ, some or all of the components of the parameter vector, are considered to be uncertain as specified by particular probability distributions (e.g., normal, exponential, extreme value), or other uncertainty structures. By assigning specific distributional structure to the inputs, distributional structure for the outputs (i.e, response statistics) can be inferred. This migrates from an analysis that is more qualitative in nature, in the case of sensitivity analysis, to an analysis that is more rigorously quantitative. UQ methods are often distinguished by their ability to propagate aleatory or epistemic input uncertainty characterizations, where aleatory uncertainties are irreducible variabilities inherent in nature and epistemic uncertainties are reducible uncertainties resulting from a lack of knowledge. Since sufficient data is generally available for aleatory uncertainties, probabilistic methods are commonly used for computing response distribution statistics based on input probability distribution specifications. Conversely, for epistemic uncertainties, any use of probability distributions is based on subjective knowledge rather than objective data, and we may alternatively explore nonprobabilistic methods based on interval specifications. 5.1.1 Summary of Dakota UQ Methods Dakota contains capabilities for performing nondeterministic analysis with both types of input uncertainty. These UQ methods have been developed by Sandia Labs, in conjunction with collaborators in academia [53, 54, 35, 134]. The aleatory UQ methods in Dakota include various sampling-based approaches (e.g., Monte Carlo and Latin Hypercube sampling), local and global reliability methods, and stochastic expansion (polynomial chaos expansions 70 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES and stochastic collocation) approaches. The epistemic UQ methods include local and global interval analysis and Dempster-Shafer evidence theory. These are summarized below and then described in more depth in subsequent sections of this chapter. Dakota additionally supports mixed aleatory/epistemic UQ via interval-valued probability, second-order probability, and Dempster-Shafer theory of evidence. These involve advanced model recursions and are described in Section 15.1. LHS (Latin Hypercube Sampling): This package provides both Monte Carlo (random) sampling and Latin Hypercube sampling methods, which can be used with probabilistic variables in Dakota that have the following distributions: normal, lognormal, uniform, loguniform, triangular, exponential, beta, gamma, gumbel, frechet, weibull, poisson, binomial, negative binomial, geometric, hypergeometric, and user-supplied histograms. In addition, LHS accounts for correlations among the variables [82], which can be used to accommodate a user-supplied correlation matrix or to minimize correlation when a correlation matrix is not supplied. In addition to a standard sampling study, we support the capability to perform “incremental” LHS, where a user can specify an initial LHS study of N samples, and then re-run an additional incremental study which will double the number of samples (to 2N, with the first N being carried from the initial study). The full incremental sample of size 2N is also a Latin Hypercube, with proper stratification and correlation. Statistics for each increment are reported separately at the end of the study. Reliability Methods: This suite of methods includes both local and global reliability methods. Local methods include first- and second-order versions of the Mean Value method (MVFOSM and MVSOSM) and a variety of most probable point (MPP) search methods, including the Advanced Mean Value method (AMV and AMV2 ), the iterated Advanced Mean Value method (AMV+ and AMV2 +), the Two-point Adaptive Nonlinearity Approximation method (TANA-3), and the traditional First Order and Second Order Reliability Methods (FORM and SORM) [73]. The MPP search methods may be used in forward (Reliability Index Approach (RIA)) or inverse (Performance Measure Approach (PMA)) modes, as dictated by the type of level mappings. Each of the MPP search techniques solve local optimization problems in order to locate the MPP, which is then used as the point about which approximate probabilities are integrated (using first- or second-order integrations in combination with refinements based on importance sampling). Global reliability methods are designed to handle nonsmooth and multimodal failure surfaces, by creating global approximations based on Gaussian process models. They accurately resolve a particular contour of a response function and then estimate probabilities using multimodal adaptive importance sampling. Stochastic Expansion Methods: The development of these techniques mirrors that of deterministic finite element analysis utilizing the notions of projection, orthogonality, and weak convergence [53], [54]. Rather than estimating point probabilities, they form an approximation to the functional relationship between response functions and their random inputs, which provides a more complete uncertainty representation for use in multi-code simulations. Expansion methods include polynomial chaos expansions (PCE), which employ multivariate orthogonal polynomials that are tailored to representing particular input probability distributions, and stochastic collocation (SC), which employs multivariate interpolation polynomials. For PCE, expansion coefficients may be evaluated using a spectral projection approach (based on sampling, tensor-product quadrature, Smolyak sparse grid, or cubature methods for numerical integration) or a regression approach (least squares or compressive sensing). For SC, interpolants are formed over tensor-product or sparse grids and may be local or global, value-based or gradient-enhanced, and nodal or hierarchical. In global value-based cases (Lagrange polynomials), the barycentric formulation is used [11, 88, 79] to improve numerical efficiency and stability. Both sets of methods provide analytic response moments and variance-based metrics; however, CDF/CCDF probabilities are evaluated numerically by sampling on the expansion. Importance Sampling: Importance sampling is a method that allows one to estimate statistical quantities such as failure probabilities in a way that is more efficient than Monte Carlo sampling. The core idea in importance sampling is that one generates samples that are preferentially placed in important regions of the space (e.g. in or near the failure region or user-defined region of interest), then appropriately weights the samples to obtain an Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.1. OVERVIEW 71 unbiased estimate of the failure probability. Adaptive Sampling: The goal in performing adaptive sampling is to construct a surrogate model that can be used as an accurate predictor of an expensive simulation. The aim is to build a surrogate that minimizes the error over the entire domain of interest using as little data as possible from the expensive simulation. The adaptive sampling methods start with an initial LHS sample, and then adaptively choose samples that optimize a particular criteria. For example, if a set of additional possible sample points are generated, one criteria is to pick the next sample point as the point which maximizes the minimum distance to the existing points (maximin). Another criteria is to pick the sample point where the surrogate indicates the most uncertainty in its prediction. Recently, Dakota added a new method to assess failure probabilities based on ideas from computational geometry. Part of the idea underpinning this method is the idea of throwing “darts” which are higher dimensional objects than sample points (e.g. lines, planes, etc.) The POF (Probability-of-Failure) darts method uses these objects to estimate failure probabilities. Interval Analysis: Interval analysis is often used to model epistemic uncertainty. In interval analysis, one assumes that nothing is known about an epistemic uncertain variable except that its value lies somewhere within an interval. In this situation, it is NOT assumed that the value has a uniform probability of occurring within the interval. Instead, the interpretation is that any value within the interval is a possible value or a potential realization of that variable. In interval analysis, the uncertainty quantification problem is one of determining the resulting bounds on the output (defining the output interval) given interval bounds on the inputs. Again, any output response that falls within the output interval is a possible output with no frequency information assigned to it. We have the capability to perform interval analysis using either global or local methods. In the global approach, one uses either a global optimization method (based on a Gaussian process surrogate model) or a sampling method to assess the bounds. The local method uses gradient information in a derivative-based optimization approach, using either SQP (sequential quadratic programming) or a NIP (nonlinear interior point) method to obtain bounds. Dempster-Shafer Theory of Evidence: The objective of evidence theory is to model the effects of epistemic uncertainties. Epistemic uncertainty refers to the situation where one does not know enough to specify a probability distribution on a variable. Sometimes epistemic uncertainty is referred to as subjective, reducible, or lack of knowledge uncertainty. In contrast, aleatory uncertainty refers to the situation where one does have enough information to specify a probability distribution. In Dempster-Shafer theory of evidence, the uncertain input variables are modeled as sets of intervals. The user assigns a basic probability assignment (BPA) to each interval, indicating how likely it is that the uncertain input falls within the interval. The intervals may be overlapping, contiguous, or have gaps. The intervals and their associated BPAs are then propagated through the simulation to obtain cumulative distribution functions on belief and plausibility. Belief is the lower bound on a probability estimate that is consistent with the evidence, and plausibility is the upper bound on a probability estimate that is consistent with the evidence. In addition to the full evidence theory structure, we have a simplified capability for users wanting to perform pure interval analysis (e.g. what is the interval on the output given intervals on the input) using either global or local optimization methods. Interval analysis is often used to model epistemic variables in nested analyses, where probability theory is used to model aleatory variables. Bayesian Calibration: In Bayesian calibration, uncertain input parameters are described by a “prior” distribution. The priors are updated with experimental data, in a Bayesian framework that involves the experimental data and a likelihood function which describes how well each parameter value is supported by the data. After the process of Bayesian calibration, the prior distribution becomes a posterior distribution. The posterior distribution of the parameters represents the parameter values that have been updated with the data (e.g. the model, when run at these parameters, should yield results that are consistent with the observational data). Dakota Version 6.7 User’s Manual generated on November 13, 2017 72 5.1.2 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES Variables and Responses for UQ All the UQ methods perform a forward uncertainty propagation in which probability or interval information for input parameters is mapped to probability or interval information for output response functions. The m functions in the Dakota response data set are interpreted as m general response functions by the Dakota methods (with no specific interpretation of the functions as for optimization and least squares). Within the variables specification, uncertain variable descriptions are employed to define the parameter probability distributions (see Section 9.3). The continuous aleatory distribution types include: normal (Gaussian), lognormal, uniform, loguniform, triangular, exponential, beta, gamma, gumbel, frechet, weibull, and histogram bin. The discrete aleatory distribution types include: poisson, binomial, negative binomial, geometric, hypergeometric, and histogram point. The epistemic distribution type is interval for continuous variables. For epistemic discrete variables, there are three types: integer range, integer set, and real set. When gradient and/or Hessian information is used in an uncertainty assessment, derivative components are normally computed with respect to the active continuous variables, which could be aleatory uncertain, epistemic uncertain, aleatory and epistemic uncertain, or all continuous variables, depending on the active view (see Section 9.5). 5.2 Sampling Methods Sampling techniques are selected using the sampling method selection. This method generates sets of samples according to the probability distributions of the uncertain variables and maps them into corresponding sets of response functions, where the number of samples is specified by the samples integer specification. Means, standard deviations, coefficients of variation (COVs), and 95% confidence intervals are computed for the response functions. Probabilities and reliabilities may be computed for response levels specifications, and response levels may be computed for either probability levels or reliability levels specifications (refer to the Method keywords section in the Dakota Reference Manual [3] for additional information). Currently, traditional Monte Carlo (MC) and Latin hypercube sampling (LHS) are supported by Dakota and are chosen by specifying sample type as random or lhs. In Monte Carlo sampling, the samples are selected randomly according to the user-specified probability distributions. Latin hypercube sampling is a stratified sampling technique for which the range of each uncertain variable is divided into Ns segments of equal probability, where Ns is the number of samples requested. The relative lengths of the segments are determined by the nature of the specified probability distribution (e.g., uniform has segments of equal width, normal has small segments near the mean and larger segments in the tails). For each of the uncertain variables, a sample is selected randomly from each of these equal probability segments. These Ns values for each of the individual parameters are then combined in a shuffling operation to create a set of Ns parameter vectors with a specified correlation structure. A feature of the resulting sample set is that every row and column in the hypercube of partitions has exactly one sample. Since the total number of samples is exactly equal to the number of partitions used for each uncertain variable, an arbitrary number of desired samples is easily accommodated (as compared to less flexible approaches in which the total number of samples is a product or exponential function of the number of intervals for each variable, i.e., many classical design of experiments methods). Advantages of sampling-based methods include their relatively simple implementation and their independence from the scientific disciplines involved in the analysis. The main drawback of these techniques is the large number of function evaluations needed to generate converged statistics, which can render such an analysis computationally very expensive, if not intractable, for real-world engineering applications. LHS techniques, in general, require fewer samples than traditional Monte Carlo for the same accuracy in statistics, but they still can be prohibitively expensive. For further information on the method and its relationship to other sampling techniques, one is referred to the works by McKay, et al. [101], Iman and Shortencarier [82], and Helton and Davis [76]. Note that under Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.2. SAMPLING METHODS 73 certain separability conditions associated with the function to be sampled, Latin hypercube sampling provides a more accurate estimate of the mean value than does random sampling. That is, given an equal number of samples, the LHS estimate of the mean will have less variance than the mean value obtained through random sampling. Figure 5.1 demonstrates Latin hypercube sampling on a two-variable parameter space. Here, the range of both parameters, x1 and x2 , is [0, 1]. Also, for this example both x1 and x2 have uniform statistical distributions. For Latin hypercube sampling, the range of each parameter is divided into p “bins” of equal probability. For parameters with uniform distributions, this corresponds to partitions of equal size. For n design parameters, this partitioning yields a total of pn bins in the parameter space. Next, p samples are randomly selected in the parameter space, with the following restrictions: (a) each sample is randomly placed inside a bin, and (b) for all one-dimensional projections of the p samples and bins, there will be one and only one sample in each bin. In a two-dimensional example such as that shown in Figure 5.1, these LHS rules guarantee that only one bin can be selected in each row and column. For p = 4, there are four partitions in both x1 and x2 . This gives a total of 16 bins, of which four will be chosen according to the criteria described above. Note that there is more than one possible arrangement of bins that meet the LHS criteria. The dots in Figure 5.1 represent the four sample sites in this example, where each sample is randomly located in its bin. There is no restriction on the number of bins in the range of each parameter, however, all parameters must have the same number of bins. Figure 5.1: An example of Latin hypercube sampling with four bins in design parameters x1 and x2 . The dots are the sample sites. The actual algorithm for generating Latin hypercube samples is more complex than indicated by the description given above. For example, the Latin hypercube sampling method implemented in the LHS code [132] takes into account a user-specified correlation structure when selecting the sample sites. For more details on the implementation of the LHS algorithm, see Reference [132]. In addition to Monte Carlo vs. LHS design choices, Dakota sampling methods support options for incrementallyrefined designs, generation of approximately determinant-optimal (D-optimal) designs, and selection of sample sizes to satisfy Wilks’ criteria. Dakota Version 6.7 User’s Manual generated on November 13, 2017 74 5.2.1 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES Uncertainty Quantification Example using Sampling Methods The input file in Figure 5.2 demonstrates the use of Latin hypercube Monte Carlo sampling for assessing probability of failure as measured by specified response levels. The two-variable Textbook example problem (see Equation 20.1) will be used to demonstrate the application of sampling methods for uncertainty quantification where it is assumed that x1 and x2 are uniform uncertain variables on the interval [0, 1]. The number of samples to perform is controlled with the samples specification, the type of sampling algorithm to use is controlled with the sample type specification, the levels used for computing statistics on the response functions is specified with the response levels input, and the seed specification controls the sequence of the pseudo-random numbers generated by the sampling algorithms. The input samples generated are shown in Figure 5.3 for the case where samples = 5 and samples = 10 for both random (◦) and lhs (+) sample types. # Dakota Input File: textbook_uq_sampling.in environment tabular_data tabular_data_file = ’textbook_uq_sampling.dat’ top_method_pointer = ’UQ’ method id_method = ’UQ’ sampling sample_type lhs samples = 10 seed = 98765 response_levels = 0.1 0.2 0.6 0.1 0.2 0.6 0.1 0.2 0.6 distribution cumulative variables uniform_uncertain = 2 lower_bounds = 0. 0. upper_bounds = 1. 1. descriptors = ’x1’ ’x2’ interface id_interface = ’I1’ analysis_drivers = ’text_book’ fork asynchronous evaluation_concurrency = 5 responses response_functions = 3 no_gradients no_hessians Figure 5.2: Dakota input file for UQ Dakota/examples/users/textbook uq sampling.in example using LHS – see Latin hypercube sampling ensures full coverage of the range of the input variables, which is often a problem with Monte Carlo sampling when the number of samples is small. In the case of samples = 5, poor stratification Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.2. SAMPLING METHODS 75 Figure 5.3: Distribution of input sample points for random (4) and lhs (+) sampling for samples=5 and 10. is evident in x1 as four out of the five Monte Carlo samples are clustered in the range 0.35 < x1 < 0.55, and the regions x1 < 0.3 and 0.6 < x1 < 0.9 are completely missed. For the case where samples = 10, some clustering in the Monte Carlo samples is again evident with 4 samples in the range 0.5 < x1 < 0.55. In both cases, the stratification with LHS is superior. The response function statistics returned by Dakota are shown in Figure 5.4. The first block of output specifies the response sample means, sample standard deviations, and skewness and kurtosis. The second block of output displays confidence intervals on the means and standard deviations of the responses. The third block defines Probability Density Function (PDF) histograms of the samples: the histogram bins are defined by the lower and upper values of the bin and the corresponding density for that bin. Note that these bin endpoints correspond to the response levels and/or probability levels defined by the user in the Dakota input file. If there are just a few levels, these histograms may be coarse. Dakota does not do anything to optimize the bin size or spacing. Finally, the last section of the output defines the Cumulative Distribution Function (CDF) pairs. In this case, distribution cumulative was specified for the response functions, and Dakota presents the probability levels corresponding to the specified response levels (response levels) that were set. The default compute probabilities was used. Alternatively, Dakota could have provided CCDF pairings, reliability levels corresponding to prescribed response levels, or response levels corresponding to prescribed probability or reliability levels. In addition to obtaining statistical summary information of the type shown in Figure 5.4, the results of LHS sampling also include correlations. Four types of correlations are returned in the output: simple and partial “raw” correlations, and simple and partial “rank” correlations. The raw correlations refer to correlations performed on the actual input and output data. Rank correlations refer to correlations performed on the ranks of the data. Ranks are obtained by replacing the actual data by the ranked values, which are obtained by ordering the data in ascending order. For example, the smallest value in a set of input samples would be given a rank 1, the next smallest value a rank 2, etc. Rank correlations are useful when some of the inputs and outputs differ greatly in magnitude: then it is easier to compare if the smallest ranked input sample is correlated with the smallest ranked output, for example. Dakota Version 6.7 User’s Manual generated on November 13, 2017 76 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES Statistics based on 10 samples: Sample moment statistics for each Mean response_fn_1 3.8383990322e-01 response_fn_2 7.4798705803e-02 response_fn_3 7.0946176558e-02 response function: Std Dev Skewness Kurtosis 4.0281539886e-01 1.2404952971e+00 6.5529797327e-01 3.4686110941e-01 4.5716015887e-01 -5.8418924529e-01 3.4153246532e-01 5.2851897926e-01 -8.2527332042e-01 95% confidence intervals for each LowerCI_Mean response_fn_1 9.5683125821e-02 response_fn_2 -1.7333078422e-01 response_fn_3 -1.7337143113e-01 response function: UpperCI_Mean LowerCI_StdDev 6.7199668063e-01 2.7707061315e-01 3.2292819583e-01 2.3858328290e-01 3.1526378424e-01 2.3491805390e-01 UpperCI_StdDev 7.3538389383e-01 6.3323317325e-01 6.2350514636e-01 Probability Density Function (PDF) histograms for each response function: PDF for response_fn_1: Bin Lower Bin Upper Density Value ----------------------------2.3066424677e-02 1.0000000000e-01 3.8994678038e+00 1.0000000000e-01 2.0000000000e-01 2.0000000000e+00 2.0000000000e-01 6.0000000000e-01 5.0000000000e-01 6.0000000000e-01 1.2250968624e+00 4.7992562123e-01 PDF for response_fn_2: Bin Lower Bin Upper Density Value -----------------------------3.5261164651e-01 1.0000000000e-01 1.1046998102e+00 1.0000000000e-01 2.0000000000e-01 2.0000000000e+00 2.0000000000e-01 6.0000000000e-01 5.0000000000e-01 6.0000000000e-01 6.9844576220e-01 1.0157877573e+00 PDF for response_fn_3: Bin Lower Bin Upper Density Value -----------------------------3.8118095128e-01 1.0000000000e-01 1.2469321539e+00 1.0000000000e-01 2.0000000000e-01 0.0000000000e+00 2.0000000000e-01 6.0000000000e-01 7.5000000000e-01 6.0000000000e-01 6.4526450977e-01 2.2092363423e+00 Level mappings for each response function: Cumulative Distribution Function (CDF) for response_fn_1: Response Level Probability Level Reliability Index General Rel Index -------------- ----------------- ----------------- ----------------1.0000000000e-01 3.0000000000e-01 2.0000000000e-01 5.0000000000e-01 6.0000000000e-01 7.0000000000e-01 Cumulative Distribution Function (CDF) for response_fn_2: Response Level Probability Level Reliability Index General Rel Index -------------- ----------------- ----------------- ----------------1.0000000000e-01 5.0000000000e-01 2.0000000000e-01 7.0000000000e-01 6.0000000000e-01 9.0000000000e-01 Cumulative Distribution Function (CDF) for response_fn_3: Response Level Probability Level Reliability Index General Rel Index -------------- ----------------- ----------------- ----------------1.0000000000e-01 6.0000000000e-01 2.0000000000e-01 6.0000000000e-01 6.0000000000e-01 9.0000000000e-01 Dakota Version 6.7 User’s Manual generated on November 13, 2017 Figure 5.4: Dakota response function statistics from UQ sampling example. 5.2. SAMPLING METHODS 77 Correlations are always calculated between two sets of sample data. One can calculate correlation coefficients between two input variables, between an input and an output variable (probably the most useful), or between two output variables. The simple correlation coefficients presented in the output P tables are Pearson’s correlation co(x −x̄)(y −ȳ) . Partial correlation efficient, which is defined for two variables x and y as: Corr(x, y) = √P i i 2 Pi 2 i (xi −x̄) i (yi −ȳ) coefficients are similar to simple correlations, but a partial correlation coefficient between two variables measures their correlation while adjusting for the effects of the other variables. For example, say one has a problem with two inputs and one output; and the two inputs are highly correlated. Then the correlation of the second input and the output may be very low after accounting for the effect of the first input. The rank correlations in Dakota are obtained using Spearman’s rank correlation. Spearman’s rank is the same as the Pearson correlation coefficient except that it is calculated on the rank data. Figure 5.5 shows an example of the correlation output provided by Dakota for the input file in Figure 5.2. Note that these correlations are presently only available when one specifies lhs as the sampling method under sampling. Also note that the simple and partial correlations should be similar in most cases (in terms of values of correlation coefficients). This is because we use a default “restricted pairing” method in the LHS routine which forces nearzero correlation amongst uncorrelated inputs. Simple Correlation Matrix between input and output: x1 x2 response_fn_1 response_fn_2 response_fn_3 x1 1.00000e+00 x2 -7.22482e-02 1.00000e+00 response_fn_1 -7.04965e-01 -6.27351e-01 1.00000e+00 response_fn_2 8.61628e-01 -5.31298e-01 -2.60486e-01 1.00000e+00 response_fn_3 -5.83075e-01 8.33989e-01 -1.23374e-01 -8.92771e-01 1.00000e+00 Partial Correlation Matrix between input and output: response_fn_1 response_fn_2 response_fn_3 x1 -9.65994e-01 9.74285e-01 -9.49997e-01 x2 -9.58854e-01 -9.26578e-01 9.77252e-01 Simple Rank Correlation Matrix between input and output: x1 x2 response_fn_1 response_fn_2 response_fn_3 x1 1.00000e+00 x2 -6.66667e-02 1.00000e+00 response_fn_1 -6.60606e-01 -5.27273e-01 1.00000e+00 response_fn_2 8.18182e-01 -6.00000e-01 -2.36364e-01 1.00000e+00 response_fn_3 -6.24242e-01 7.93939e-01 -5.45455e-02 -9.27273e-01 1.00000e+00 Partial Rank Correlation Matrix between input and output: response_fn_1 response_fn_2 response_fn_3 x1 -8.20657e-01 9.74896e-01 -9.41760e-01 x2 -7.62704e-01 -9.50799e-01 9.65145e-01 Figure 5.5: Correlation results using LHS Sampling. Finally, note that the LHS package can be used for design of experiments over design and state variables by including an active view override in the variables specification section of the Dakota input file (see Section 9.5.1). Then, instead of iterating on only the uncertain variables, the LHS package will sample over all of the active variables. In the active all view, continuous design and continuous state variables are treated as having uniform probability distributions within their upper and lower bounds, discrete design and state variables are sampled uniformly from within their sets or ranges, and any uncertain variables are sampled within their specified Dakota Version 6.7 User’s Manual generated on November 13, 2017 78 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES probability distributions. 5.2.2 Incremental Sampling In many situations, one may run an initial sample set and then need to perform further sampling to get better estimates of the mean, variance, and percentiles, and to obtain more comprehensive sample coverage. We call this capability incremental sampling. Typically, a Dakota restart file (dakota.rst) would be available from the original sample, so only the newly generated samples would need to be evaluated. Incremental sampling supports continuous uncertain variables and discrete uncertain variables such as discrete distributions (e.g. binomial, Poisson, etc.) as well as histogram variables and uncertain set types. There are two cases, incremental random and incremental Latin hypercube sampling, with incremental LHS being the most common. One major advantage of LHS incremental sampling is that it maintains the stratification and correlation structure of the original LHS sample. That is, if one generated two independent LHS samples and simply merged them, the calculation of the accuracy of statistical measures such as the mean and the variance would be slightly incorrect. However, in the incremental case, the full sample (double the original size) is a Latin Hypercube sample itself and statistical measures and their accuracy can be properly calculated. The incremental sampling capability is most useful when one is starting off with very small samples. Once the sample size is more than a few hundred, the benefit of incremental sampling diminishes. 1. Incremental random sampling: With incremental random sampling, the original sample set with N 1 samples must be generated using sample type = random and samples = N1. Then, the user can duplicate the Dakota input file and add refinement samples = N2 with the number of new samples N 2 to be added. Random incremental sampling does not require a doubling of samples each time. Thus, the user can specify any number of refinement samples (from an additional one sample to a large integer). For example, if the first sample has 50 samples, and 10 more samples are desired, the second Dakota run should specify samples = 50, refinement samples = 10. In this situation, only 10 new samples will be generated, and the final statistics will be reported at the end of the study both for the initial 50 samples and for the full sample of 60. The command line syntax for running the second sample is dakota -i input60.in -r dakota.50.rst where input60.in is the input file with the refinement samples specification and dakota.50.rst is the restart file containing the initial 50 samples. Note that if the restart file has a different name, that is fine; the correct restart file name should be used. This process can be repeated if desired,arbitrarily extending the total sample size each time, e.g, samples = 50, refinement samples = 10 3 73 102. 2. Incremental Latin hypercube sampling: With incremental LHS sampling, the original sample set with N 1 samples must be generated using sample type = lhs and samples = N1. Then, the user can duplicate the Dakota input file and add refinement samples = N1. The sample size must double each time, so the first set of refinement samples must be the same size as the initial set. That is, if one starts with a very small sample size of 10, then one can use the incremental sampling capability to generate sample sizes of 20, 40, 80, etc. For example, if the first sample has 50 samples, in the second Dakota run, the number of refinement samples should be set to 50 for a total of 100. In this situation, only 50 new samples will be generated, and at the end of the study final statistics will be reported both for the initial 50 samples and for the full sample of 100. The command line syntax for running the second sample is dakota -i input100.in -r dakota.50.rst, where input100.in is the input file with the incremental sampling specification and dakota.50.rst is the restart file containing the initial 50 samples. Note that if the restart file has a different name, that is fine; the correct restart file name should be used. Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.3. RELIABILITY METHODS 79 This process can be repeated if desired, doubling the total sample size each time, e.g, samples = 50, refinement samples = 50 100 200 400. 5.2.3 Principal Component Analysis As of Dakota 6.3, we added a capability to perform Principal Component Analysis on field response data when using LHS sampling. Principal components analysis (PCA) is a data reduction method and allows one to express an ensemble of field data with a set of principal components responsible for the spread of that data. Dakota can calculate the principal components of the response matrix of N samples * L responses (the field response of length L) using the keyword principal components. The Dakota implementation is under active development: the PCA capability may ultimately be specified elsewhere or used in different ways. For now, it is performed as a post-processing analysis based on a set of Latin Hypercube samples. If the user specifies LHS sampling with field data responses and also specifies principal components, Dakota will calculate the principal components by calculating the eigenvalues and eigenvectors of a centered data matrix. Further, if the user specifies percent variance explained = 0.99, the number of components that accounts for at least 99 percent of the variance in the responses will be retained. The default for this percentage is 0.95. In many applications, only a few principal components explain the majority of the variance, resulting in significant data reduction. The principal components are written to a file, princ comp.txt. Dakota also uses the principal components to create a surrogate model by representing the overall response as weighted sum of M principal components, where the weights will be determined by Gaussian processes which are a function of the input uncertain variables. This reduced form then can be used for sensitivity analysis, calibration, etc. 5.2.4 Wilks-based Sample Sizes Most of the sampling methods require the user to specify the number of samples in advance. However, if one specifies random sampling, one can use an approach developed by Wilks[149] to determine the number of samples that ensures a particular confidence level in a percentile of interest. The Wilks method of computing the number of samples to execute for a random sampling study is based on order statistics, eg considering the outputs ordered from smallest to largest [149, 108]. Given a probability level, α, and confidence level, β, the Wilks calculation determines the minimum number of samples required such that there is (β ∗ 100)% confidence that the (α ∗ 100)%-ile of the uncertain distribution on model output will fall below the actual (α ∗ 100)%-ile given by the sample. To be more specific, if we wish to calculate the 95% confidence limit on the 95th percentile, Wilks indicates that 59 samples are needed. If we order the responses and take the largest one, that value defines a tolerance limit on the 95th percentile: we have a situation where 95% of the time, the 95th percentile will fall at or below that sampled value. This represents a one sided upper treatment applicable to the largest output value. This treatment can be reversed to apply to the lowest output value by using the one sided lower option, and further expansion to include an interval containing both the smallest and the largest output values in the statistical statement can be specified via the two sided option. Additional generalization to higher order statistics, eg a statement applied to the N largest outputs (one sided upper) or the N smallest and N largest outputs (two sided), can be specified using the order option along with value N. 5.3 Reliability Methods Reliability methods provide an alternative approach to uncertainty quantification which can be less computationally demanding than sampling techniques. Reliability methods for uncertainty quantification are based on Dakota Version 6.7 User’s Manual generated on November 13, 2017 80 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES probabilistic approaches that compute approximate response function distribution statistics based on specified uncertain variable distributions. These response statistics include response mean, response standard deviation, and cumulative or complementary cumulative distribution functions (CDF/CCDF). These methods are often more efficient at computing statistics in the tails of the response distributions (events with low probability) than sampling based approaches since the number of samples required to resolve a low probability can be prohibitive. The methods all answer the fundamental question: “Given a set of uncertain input variables, X, and a scalar response function, g, what is the probability that the response function is below or above a certain level, z̄?” The former can be written as P [g(X) ≤ z̄] = Fg (z̄) where Fg (z̄) is the cumulative distribution function (CDF) of the uncertain response g(X) over a set of response levels. The latter can be written as P [g(X) > z̄] and defines the complementary cumulative distribution function (CCDF). This probability calculation involves a multi-dimensional integral over an irregularly shaped domain of interest, D, where g(X) < z as displayed in Figure 5.6 for the case of two variables. The reliability methods all involve the transformation of the user-specified uncertain variables, X, with probability density function, p(x1 , x2 ), which can be non-normal and correlated, to a space of independent Gaussian random variables, u, possessing a mean value of zero and unit variance (i.e., standard normal variables). The region of interest, D, is also mapped to the transformed space to yield, Du , where g(U) < z as shown in Figure 5.7. The Nataf transformation [28], which is identical to the Rosenblatt transformation [114] in the case of independent random variables, is used in Dakota to accomplish this mapping. This transformation is performed to make the probability calculation more tractable. In the transformed space, probability contours are circular in nature as shown in Figure 5.7 unlike in the original uncertain variable space, Figure 5.6. Also, the multi-dimensional integrals can be approximated by simple functions of a single parameter, β, called the reliability index. β is the minimum Euclidean distance from the origin in the transformed space to the response surface. This point is also known as the most probable point (MPP) of failure. Note, however, the methodology is equally applicable for generic functions, not simply those corresponding to failure criteria; this nomenclature is due to the origin of these methods within the disciplines of structural safety and reliability. Note that there are local and global reliability methods. The majority of the methods available are local, meaning that a local optimization formulation is used to locate one MPP. In contrast, global methods can find multiple MPPs if they exist. Figure 5.6: Graphical depiction of calculation of cumulative distribution function in the original uncertain variable space. Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.3. RELIABILITY METHODS 81 Table 5.1: Mapping from Dakota options to standard reliability methods. Order of approximation and integration MPP search First order Second order none MVFOSM MVSOSM x taylor mean AMV AMV2 u taylor mean u-space AMV u-space AMV2 x taylor mpp AMV+ AMV2 + u taylor mpp u-space AMV+ u-space AMV2 + x two point TANA u two point u-space TANA no approx FORM SORM Figure 5.7: Graphical depiction of integration for the calculation of cumulative distribution function in the transformed uncertain variable space. 5.3.1 Local Reliability Methods The Dakota Theory Manual [4] provides the algorithmic details for the local reliability methods, including the Mean Value method and the family of most probable point (MPP) search methods. 5.3.1.1 Method mapping Given settings for limit state approximation, approximation order, integration approach, and other details presented to this point, it is evident that the number of algorithmic combinations is high. Table 5.1 provides a succinct mapping for some of these combinations to common method names from the reliability literature, where blue indicates the most well-known combinations and gray indicates other supported combinations. Within the Dakota specification (refer to local reliability in the keywords section of the Reference Manual [3]) within the Reference Manual), the MPP search and integration order selections are explicit in the method specification, but the order of the approximation is inferred from the associated response specification (as is done Dakota Version 6.7 User’s Manual generated on November 13, 2017 82 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES with local taylor series approximations described in Section 8.4.3.2). Thus, reliability methods do not have to be synchronized in approximation and integration order as shown in the table; however, it is often desirable to do so. 5.3.2 Global Reliability Methods Global reliability methods are designed to handle nonsmooth and multimodal failure surfaces, by creating global approximations based on Gaussian process models. They accurately resolve a particular contour of a response function and then estimate probabilities using multimodal adaptive importance sampling. The global reliability method in Dakota is called Efficient Global Reliability Analysis (EGRA) [12]. The name is due to its roots in efficient global optimization (EGO) [84, 81]. The main idea in EGO-type optimization methods is that a global approximation is made of the underlying function. This approximation, which is a Gaussian process model, is used to guide the search by finding points which maximize the expected improvement function (EIF). The EIF is used to select the location at which a new training point should be added to the Gaussian process model by maximizing the amount of improvement in the objective function that can be expected by adding that point. A point could be expected to produce an improvement in the objective function if its predicted value is better than the current best solution, or if the uncertainty in its prediction is such that the probability of it producing a better solution is high. Because the uncertainty is higher in regions of the design space with fewer observations, this provides a balance between exploiting areas of the design space that predict good solutions, and exploring areas where more information is needed. The general procedure of these EGO-type methods is: 1. Build an initial Gaussian process model of the objective function. 2. Find the point that maximizes the EIF. If the EIF value at this point is sufficiently small, stop. 3. Evaluate the objective function at the point where the EIF is maximized. Update the Gaussian process model using this new point. Go to Step 2. Gaussian process (GP) models are used because they provide not just a predicted value at an unsampled point, but also an estimate of the prediction variance. This variance gives an indication of the uncertainty in the GP model, which results from the construction of the covariance function. This function is based on the idea that when input points are near one another, the correlation between their corresponding outputs will be high. As a result, the uncertainty associated with the model’s predictions will be small for input points which are near the points used to train the model, and will increase as one moves further from the training points. The expected improvement function is used in EGO algorithms to select the location at which a new training point should be added. The EIF is defined as the expectation that any point in the search space will provide a better solution than the current best solution based on the expected values and variances predicted by the GP model. It is important to understand how the use of this EIF leads to optimal solutions. The EIF indicates how much the objective function value at a new potential location is expected to be less than the predicted value at the current best solution. Because the GP model provides a Gaussian distribution at each predicted point, expectations can be calculated. Points with good expected values and even a small variance will have a significant expectation of producing a better solution (exploitation), but so will points that have relatively poor expected values and greater variance (exploration). The application of EGO to reliability analysis, however, is made more complicated due to the inclusion of equality constraints. In forward reliability analysis, the response function appears as a constraint rather than the objective. That is, we want to satisfy the constraint that the response equals a threshold value and is on the limit state: G(u) = z̄. Therefore, the EIF function was modified to focus on feasibility, and instead of using an expected Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.3. RELIABILITY METHODS 83 improvement function, we use an expected feasibility function (EFF) [12]. The EFF provides an indication of how well the response is expected to satisfy the equality constraint. Points where the expected value is close to the threshold (µG ≈ z̄) and points with a large uncertainty in the prediction will have large expected feasibility values. The general outline of the EGRA algorithm is as follows: LHS sampling is used to generate a small number of samples from the true response function. Then, an initial Gaussian process model is constructed. Based on the EFF, the point with maximum EFF is found using the global optimizer DIRECT. The true response function is then evaluated at this new point, and this point is added to the sample set and the process of building a new GP model and maximizing the EFF is repeated until the maximum EFF is small. At this stage, the GP model is accurate in the vicinity of the limit state. The GP model is then used to calculate the probability of failure using multimodal importance sampling, which is explained below. One method to calculate the probability of failure is to directly perform the probability integration numerically by sampling the response function. Sampling methods can be prohibitively expensive because they generally require a large number of response function evaluations. Importance sampling methods reduce this expense by focusing the samples in the important regions of the uncertain space. They do this by centering the sampling density function at the MPP rather than at the mean. This ensures the samples will lie the region of interest, thus increasing the efficiency of the sampling method. Adaptive importance sampling (AIS) further improves the efficiency by adaptively updating the sampling density function. Multimodal adaptive importance sampling [29] is a variation of AIS that allows for the use of multiple sampling densities making it better suited for cases where multiple sections of the limit state are highly probable. Note that importance sampling methods require that the location of at least one MPP be known because it is used to center the initial sampling density. However, current gradient-based, local search methods used in MPP search may fail to converge or may converge to poor solutions for highly nonlinear problems, possibly making these methods inapplicable. The EGRA algorithm described above does not depend on the availability of accurate gradient information, making convergence more reliable for nonsmooth response functions. Moreover, EGRA has the ability to locate multiple failure points, which can provide multiple starting points and thus a good multimodal sampling density for the initial steps of multimodal AIS. The probability assessment using multimodal AIS thus incorporates probability of failure at multiple points. 5.3.3 Uncertainty Quantification Examples using Reliability Analysis In summary, the user can choose to perform either forward (RIA) or inverse (PMA) mappings when performing a reliability analysis. With either approach, there are a variety of methods from which to choose in terms of limit state approximations (MVFOSM, MVSOSM, x-/u-space AMV, x-/u-space AMV2 , x-/u-space AMV+, x-/u-space AMV2 +, x-/u-space TANA, and FORM/SORM), probability integrations (first-order or second-order), limit state Hessian selection (analytic, finite difference, BFGS, or SR1), and MPP optimization algorithm (SQP or NIP) selections. All reliability methods output approximate values of the CDF/CCDF response-probability-reliability levels for prescribed response levels (RIA) or prescribed probability or reliability levels (PMA). In addition, mean value methods output estimates of the response means and standard deviations as well as importance factors that attribute variance among the set of uncertain variables (provided a nonzero response variance estimate). 5.3.3.1 Mean-value Reliability with Textbook Figure 5.8 shows the Dakota input file for an example problem that demonstrates the simplest reliability method, called the mean value method (also referred to as the Mean Value First Order Second Moment method). It is Dakota Version 6.7 User’s Manual generated on November 13, 2017 84 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES specified with method keyword local reliability. This method calculates the mean and variance of the response function based on information about the mean and variance of the inputs and gradient information at the mean of the inputs. The mean value method is extremely cheap computationally (only five runs were required for the textbook function), but can be quite inaccurate, especially for nonlinear problems and/or problems with uncertain inputs that are significantly non-normal. More detail on the mean value method can be found in the Local Reliability Methods section of the Dakota Theory Manual [4], and more detail on reliability methods in general (including the more advanced methods) is found in Section 5.3. Example output from the mean value method is displayed in Figure 5.9. Note that since the mean of both inputs is 1, the mean value of the output for response 1 is zero. However, the mean values of the constraints are both 0.5. The mean value results indicate that variable x1 is more important in constraint 1 while x2 is more important in constraint 2, which is the case based on Equation 20.1. The importance factors are not available for the first response as the standard deviation is zero. # Dakota Input File: textbook_uq_meanvalue.in environment #graphics method local_reliability interface analysis_drivers = ’text_book’ fork asynchronous variables lognormal_uncertain = 2 means = std_deviations = descriptors = 1. 1. 0.5 0.5 ’TF1ln’ ’TF2ln’ responses response_functions = 3 numerical_gradients method_source dakota interval_type central fd_gradient_step_size = 1.e-4 no_hessians Figure 5.8: Mean Value Reliability Method: Dakota/examples/users/textbook uq meanvalue.in 5.3.3.2 the Dakota input file – see FORM Reliability with Lognormal Ratio This example quantifies the uncertainty in the “log ratio” response function: g(x1 , x2 ) = x1 x2 (5.1) Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.3. RELIABILITY METHODS 85 MV Statistics for response_fn_1: Approximate Mean Response = Approximate Standard Deviation of Response = Importance Factors not available. MV Statistics for response_fn_2: Approximate Mean Response = Approximate Standard Deviation of Response = Importance Factor for TF1ln = Importance Factor for TF2ln = MV Statistics for response_fn_3: Approximate Mean Response = Approximate Standard Deviation of Response = Importance Factor for TF1ln = Importance Factor for TF2ln = 0.0000000000e+00 0.0000000000e+00 5.0000000000e-01 1.0307764064e+00 9.4117647059e-01 5.8823529412e-02 5.0000000000e-01 1.0307764064e+00 5.8823529412e-02 9.4117647059e-01 Figure 5.9: Results of the Mean Value Method on the Textbook Function by computing approximate response statistics using reliability analysis to determine the response cumulative distribution function: P [g(x1 , x2 ) < z̄] (5.2) where X1 and X2 are identically distributed lognormal random variables with means of 1, standard deviations of 0.5, and correlation coefficient of 0.3. A Dakota input file showing RIA using FORM (option 7 in limit state approximations combined with first-order integration) is listed in Figure 5.10. The user first specifies the local reliability method, followed by the MPP search approach and integration order. In this example, we specify mpp search no approx and utilize the default first-order integration to select FORM. Finally, the user specifies response levels or probability/reliability levels to determine if the problem will be solved using an RIA approach or a PMA approach. In the example figure of 5.10, we use RIA by specifying a range of response levels for the problem. The resulting output for this input is shown in Figure 5.11, with probability and reliability levels listed for each response level. Figure 5.12 shows that FORM compares favorably to an exact analytic solution for this problem. Also note that FORM does have some error in the calculation of CDF values for this problem, but it is a very small error (on the order of e-11), much smaller than the error obtained when using a Mean Value method, which will be discussed next. If the user specifies local reliability as a method with no additional specification on how to do the MPP search (for example, by commenting out mpp search no approx in Figure 5.10), then no MPP search is done: the Mean Value method is used. The mean value results are shown in Figure 5.13 and consist of approximate mean and standard deviation of the response, the importance factors for each uncertain variable, and approximate probability/reliability levels for the prescribed response levels that have been inferred from the approximate mean and standard deviation (see Mean Value section in Reliability Methods Chapter of Dakota Theory Manual [4]). It is evident that the statistics are considerably different from the fully converged FORM results; however, these rough approximations are also much less expensive to calculate. The importance factors are a measure of the sensitivity of the response function(s) to the uncertain input variables. A comparison of the mean value results with the FORM results is shown in Figure 5.12. The mean value results are not accurate near the tail values of the CDF, and can differ from the exact solution by as much as 0.11 in CDF estimates. A comprehensive comparison of various reliability methods applied to the logratio problem is provided in [36]. Additional reliability analysis and design results are provided in Sections 20.10.1-20.10.5. Dakota Version 6.7 User’s Manual generated on November 13, 2017 86 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES # Dakota Input File: logratio_uq_reliability.in environment graphics method local_reliability mpp_search no_approx response_levels = .4 .5 .55 .6 .65 .7 .75 .8 .85 .9 1. 1.05 1.15 1.2 1.25 1.3 1.35 1.4 1.5 1.55 1.6 1.65 1.7 1.75 variables lognormal_uncertain = 2 means = 1. 1 std_deviations = 0.5 0.5 initial_point = 0.6 1.4 descriptors = ’TF1ln’ uncertain_correlation_matrix = ’TF2ln’ 1 0.3 0.3 1 interface analysis_drivers = ’log_ratio’ direct # fork asynch responses response_functions = 1 numerical_gradients method_source dakota interval_type central fd_step_size = 1.e-4 no_hessians Figure 5.10: Dakota input file for Reliability UQ Dakota/examples/users/logratio uq reliability.in example using FORM – see Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.3. RELIABILITY METHODS 87 Cumulative Distribution Function (CDF) for response_fn_1: Response Level Probability Level Reliability Index -------------- ----------------- ----------------4.0000000000e-01 4.7624085962e-02 1.6683404020e+00 5.0000000000e-01 1.0346525475e-01 1.2620507942e+00 5.5000000000e-01 1.3818404972e-01 1.0885143628e+00 6.0000000000e-01 1.7616275822e-01 9.3008801339e-01 6.5000000000e-01 2.1641741368e-01 7.8434989943e-01 7.0000000000e-01 2.5803428381e-01 6.4941748143e-01 7.5000000000e-01 3.0020938124e-01 5.2379840558e-01 8.0000000000e-01 3.4226491013e-01 4.0628960782e-01 8.5000000000e-01 3.8365052982e-01 2.9590705956e-01 9.0000000000e-01 4.2393548232e-01 1.9183562480e-01 1.0000000000e+00 5.0000000000e-01 6.8682233460e-12 1.0500000000e+00 5.3539344228e-01 -8.8834907167e-02 1.1500000000e+00 6.0043460094e-01 -2.5447217462e-01 1.2000000000e+00 6.3004131827e-01 -3.3196278078e-01 1.2500000000e+00 6.5773508987e-01 -4.0628960782e-01 1.3000000000e+00 6.8356844630e-01 -4.7770089473e-01 1.3500000000e+00 7.0761025532e-01 -5.4641676380e-01 1.4000000000e+00 7.2994058691e-01 -6.1263331274e-01 1.5000000000e+00 7.6981945355e-01 -7.3825238860e-01 1.5500000000e+00 7.8755158269e-01 -7.9795460350e-01 1.6000000000e+00 8.0393505584e-01 -8.5576118635e-01 1.6500000000e+00 8.1906005158e-01 -9.1178881995e-01 1.7000000000e+00 8.3301386860e-01 -9.6614373461e-01 1.7500000000e+00 8.4588021938e-01 -1.0189229206e+00 Figure 5.11: Output from Reliability UQ example using FORM. Figure 5.12: Comparison of the cumulative distribution function (CDF) computed by FORM, the Mean Value method, and the exact CDF for g(x1 , x2 ) = xx12 Dakota Version 6.7 User’s Manual generated on November 13, 2017 88 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES MV Statistics for response_fn_1: Approximate Mean Response = 1.0000000000e+00 Approximate Standard Deviation of Response = 5.9160798127e-01 Importance Factor for TF1ln = 7.1428570714e-01 Importance Factor for TF2ln = 7.1428572143e-01 Importance Factor for TF1ln TF2ln = -4.2857142857e-01 Cumulative Distribution Function (CDF) for response_fn_1: Response Level Probability Level Reliability Index General Rel Index -------------- ----------------- ----------------- ----------------4.0000000000e-01 1.5524721837e-01 1.0141851006e+00 1.0141851006e+00 5.0000000000e-01 1.9901236093e-01 8.4515425050e-01 8.4515425050e-01 5.5000000000e-01 2.2343641149e-01 7.6063882545e-01 7.6063882545e-01 6.0000000000e-01 2.4948115037e-01 6.7612340040e-01 6.7612340040e-01 6.5000000000e-01 2.7705656603e-01 5.9160797535e-01 5.9160797535e-01 7.0000000000e-01 3.0604494093e-01 5.0709255030e-01 5.0709255030e-01 7.5000000000e-01 3.3630190949e-01 4.2257712525e-01 4.2257712525e-01 8.0000000000e-01 3.6765834596e-01 3.3806170020e-01 3.3806170020e-01 8.5000000000e-01 3.9992305332e-01 2.5354627515e-01 2.5354627515e-01 9.0000000000e-01 4.3288618783e-01 1.6903085010e-01 1.6903085010e-01 1.0000000000e+00 5.0000000000e-01 0.0000000000e+00 0.0000000000e+00 1.0500000000e+00 5.3367668035e-01 -8.4515425050e-02 -8.4515425050e-02 1.1500000000e+00 6.0007694668e-01 -2.5354627515e-01 -2.5354627515e-01 1.2000000000e+00 6.3234165404e-01 -3.3806170020e-01 -3.3806170020e-01 1.2500000000e+00 6.6369809051e-01 -4.2257712525e-01 -4.2257712525e-01 1.3000000000e+00 6.9395505907e-01 -5.0709255030e-01 -5.0709255030e-01 1.3500000000e+00 7.2294343397e-01 -5.9160797535e-01 -5.9160797535e-01 1.4000000000e+00 7.5051884963e-01 -6.7612340040e-01 -6.7612340040e-01 1.5000000000e+00 8.0098763907e-01 -8.4515425050e-01 -8.4515425050e-01 1.5500000000e+00 8.2372893005e-01 -9.2966967555e-01 -9.2966967555e-01 1.6000000000e+00 8.4475278163e-01 -1.0141851006e+00 -1.0141851006e+00 1.6500000000e+00 8.6405064339e-01 -1.0987005257e+00 -1.0987005257e+00 1.7000000000e+00 8.8163821351e-01 -1.1832159507e+00 -1.1832159507e+00 1.7500000000e+00 8.9755305196e-01 -1.2677313758e+00 -1.2677313758e+00 Figure 5.13: Output from Reliability UQ example using mean value. Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.4. STOCHASTIC EXPANSION METHODS 5.4 89 Stochastic Expansion Methods The development of these techniques mirrors that of deterministic finite element analysis through the utilization of the concepts of projection, orthogonality, and weak convergence. The polynomial chaos expansion is based on a multidimensional orthogonal polynomial approximation and the stochastic collocation approach is based on a multidimensional interpolation polynomial approximation, both formed in terms of standardized random variables. A distinguishing feature of these two methodologies is that the final solution is expressed as a functional mapping, and not merely as a set of statistics as is the case for many nondeterministic methodologies. This makes these techniques particularly attractive for use in multi-physics applications which link different analysis packages. The first stochastic expansion method is the polynomial chaos expansion (PCE) [53, 54]. For smooth functions (i.e., analytic, infinitely-differentiable) in L2 (i.e., possessing finite variance), exponential convergence rates can be obtained under order refinement for integrated statistical quantities of interest such as mean, variance, and probability. Dakota implements the generalized PCE approach using the Wiener-Askey scheme [152], in which Hermite, Legendre, Laguerre, Jacobi, and generalized Laguerre orthogonal polynomials are used for modeling the effect of continuous random variables described by normal, uniform, exponential, beta, and gamma probability distributions, respectively1 . These orthogonal polynomial selections are optimal for these distribution types since the inner product weighting function corresponds2 to the probability density functions for these continuous distributions. Orthogonal polynomials can be computed for any positive weight function, so these five classical orthogonal polynomials may be augmented with numerically-generated polynomials for other probability distributions (e.g., for lognormal, extreme value, and histogram distributions). When independent standard random variables are used (or computed through transformation), the variable expansions are uncoupled, allowing the polynomial orthogonality properties to be applied on a per-dimension basis. This allows one to mix and match the polynomial basis used for each variable without interference with the spectral projection scheme for the response. In non-intrusive PCE, simulations are used as black boxes and the calculation of chaos expansion coefficients for response metrics of interest is based on a set of simulation response evaluations. To calculate these response PCE coefficients, two primary classes of approaches have been proposed: spectral projection and regression. The spectral projection approach projects the response against each basis function using inner products and employs the polynomial orthogonality properties to extract each coefficient. Each inner product involves a multidimensional integral over the support range of the weighting function, which can be evaluated numerically using sampling, tensor-product quadrature, Smolyak sparse grid [124], or cubature [128] approaches. The regression approach finds a set of PCE coefficients which best match a set of response values obtained from either a design of computer experiments (“point collocation” [145]) or from a randomly selected subset of tensor Gauss points (“probabilistic collocation” [135]). Various methods can be used to solve the resulting linear system, including least squares methods for over-determined systems and compressed sensing methods for under-determined systems. Details of these methods are documented in the Linear regression section of the Dakota Theory Manual [4] and the necessary specifications needed to activate these techniques are listed in the keyword section of the Dakota Reference Manual [3]. Stochastic collocation (SC) is another stochastic expansion technique for UQ that is closely related to PCE. As for PCE, exponential convergence rates can be obtained under order refinement for integrated statistical quantities of interest, provided that the response functions are smooth with finite variance. The primary distinction is that, whereas PCE estimates coefficients for known multivariate orthogonal polynomial basis functions, SC forms multivariate interpolation polynomial bases for known coefficients. The interpolation polynomials may be either local or global and either value-based or gradient-enhanced (four combinations: Lagrange, Hermite, piecewise linear spline, and piecewise cubic spline), and may be used within nodal or hierarchical interpolation formulations. Interpolation is performed on structured grids such as tensor-product or sparse grids. Starting from a tensor1 Orthogonal 2 Identical polynomial selections also exist for discrete probability distributions, but are not yet supported in Dakota. support range; weight differs by at most a constant factor. Dakota Version 6.7 User’s Manual generated on November 13, 2017 90 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES product multidimensional interpolation polynomial in the value-based case (Lagrange or piecewise linear spline), we have the feature that the ith interpolation polynomial has a value of 1 at collocation point i and a value of 0 for all other collocation points, leading to the use of expansion coefficients that are just the response values at each of the collocation points. In the gradient-enhanced case (Hermite or piecewise cubic spline), SC includes both type 1 and type 2 interpolation polynomials, where the former interpolate the values while producing zero gradients and the latter interpolate the gradients while producing zero values (refer to [4] for additional details). Sparse interpolants are weighted sums of these tensor interpolants; however, they are only interpolatory for sparse grids based on fully nested rules and will exhibit some interpolation error at the collocation points for sparse grids based on non-nested rules. A key to maximizing performance with SC is performing collocation using the Gauss points and weights from the same optimal orthogonal polynomials used in PCE. For use of standard Gauss integration rules (not nested variants such as Gauss-Patterson or Genz-Keister) within tensor-product quadrature, tensor PCE expansions and tensor SC interpolants are equivalent in that identical polynomial approximations are generated [22]. Moreover, this equivalence can be extended to sparse grids based on standard Gauss rules, provided that a sparse PCE is formed based on a weighted sum of tensor expansions [20]. The Dakota Theory Manual [4] provides full algorithmic details for the PCE and SC methods. 5.4.1 Uncertainty Quantification Examples using Stochastic Expansions 5.4.1.1 Polynomial Chaos Expansion for Rosenbrock A typical Dakota input file for performing an uncertainty quantification using PCE is shown in Figure 5.14. In this example, we compute CDF probabilities for six response levels of Rosenbrock’s function. Since Rosenbrock is a fourth order polynomial and we employ a fourth-order expansion using an optimal basis (Legendre for uniform random variables), we can readily obtain a polynomial expansion which exactly matches the Rosenbrock function. In this example, we select Gaussian quadratures using an anisotropic approach (fifth-order quadrature in x1 and third-order quadrature in x2 ), resulting in a total of 15 function evaluations to compute the PCE coefficients. The tensor product quadature points upon which the expansion is calculated are shown in Figure 5.15. The tensor product generates all combinations of values from each individual dimension: it is an all-way pairing of points. Once the expansion coefficients have been calculated, some statistics are available analytically and others must be evaluated numerically. For the numerical portion, the input file specifies the use of 10000 samples, which will be evaluated on the expansion to compute the CDF probabilities. In Figure 5.16, excerpts from the results summary are presented, where we first see a summary of the PCE coefficients which exactly reproduce Rosenbrock for a Legendre polynomial basis. The analytic statistics for mean, standard deviation, and COV are then presented. For example, the mean is 455.66 and the standard deviation is 606.56. The moments are followed by global sensitivity indices (Sobol’ indices).This example shows that variable x1 has the largest main effect (0.497) as compared with variable x2 (0.296) or the interaction between x1 and x2 (0.206). After the global sensitivity indices, the local sensitivities are presented, evaluated at the mean values. Finally, we see the numerical results for the CDF probabilities based on 10000 samples performed on the expansion. For example, the probability that the Rosenbrock function is less than 100 over these two uncertain variables is 0.342. Note that this is a very similar estimate to what was obtained using 200 Monte Carlo samples, with fewer true function evaluations. 5.4.1.2 Uncertainty Quantification Example using Stochastic Collocation Compared to the previous PCE example, this section presents a more sophisticated example, where we use stochastic collocation built on an anisotropic sparse grid defined from numerically-generated orthogonal polynomials. The uncertain variables are lognormal in this example and the orthogonal polynomials are generated Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.4. STOCHASTIC EXPANSION METHODS 91 # Dakota Input File: rosen_uq_pce.in environment #graphics method polynomial_chaos quadrature_order = 5 dimension_preference = 5 3 samples_on_emulator = 10000 seed = 12347 response_levels = .1 1. 50. 100. 500. 1000. variance_based_decomp #interaction_order = 1 variables uniform_uncertain = 2 lower_bounds = -2. -2. upper_bounds = 2. 2. descriptors = ’x1’ ’x2’ interface analysis_drivers = ’rosenbrock’ direct responses response_functions = 1 no_gradients no_hessians Figure 5.14: Dakota input file for performing UQ using polynomial chaos expansions – see Dakota/examples/users/rosen uq pce.in Figure 5.15: Rosenbrock polynomial chaos example: tensor product quadrature points. Dakota Version 6.7 User’s Manual generated on November 13, 2017 92 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES Polynomial Chaos coefficients for response_fn_1: coefficient u1 u2 ----------- ---- ---4.5566666667e+02 P0 P0 -4.0000000000e+00 P1 P0 9.1695238095e+02 P2 P0 -9.9475983006e-14 P3 P0 3.6571428571e+02 P4 P0 -5.3333333333e+02 P0 P1 -3.9968028887e-14 P1 P1 -1.0666666667e+03 P2 P1 -3.3573144265e-13 P3 P1 1.2829737273e-12 P4 P1 2.6666666667e+02 P0 P2 2.2648549702e-13 P1 P2 4.8849813084e-13 P2 P2 2.8754776338e-13 P3 P2 -2.8477220582e-13 P4 P2 ------------------------------------------------------------------Statistics derived analytically from polynomial expansion: Moment-based statistics for each response function: Mean Std Dev Skewness response_fn_1 expansion: 4.5566666667e+02 6.0656024184e+02 numerical: 4.5566666667e+02 6.0656024184e+02 1.9633285271e+00 Kurtosis 3.3633861456e+00 Covariance among response functions: [[ 3.6791532698e+05 ]] Local sensitivities for each response function evaluated at uncertain variable means: response_fn_1: [ -2.0000000000e+00 2.4055757386e-13 ] Global sensitivity indices for each response function: response_fn_1 Sobol indices: Main Total 4.9746891383e-01 7.0363551328e-01 x1 2.9636448672e-01 5.0253108617e-01 x2 Interaction 2.0616659946e-01 x1 x2 Statistics based on 10000 samples performed on polynomial expansion: Probability Density Function (PDF) histograms for each response function: PDF for response_fn_1: Bin Lower Bin Upper Density Value ----------------------------6.8311107124e-03 1.0000000000e-01 2.0393073423e-02 1.0000000000e-01 1.0000000000e+00 1.3000000000e-02 1.0000000000e+00 5.0000000000e+01 4.7000000000e-03 5.0000000000e+01 1.0000000000e+02 1.9680000000e-03 1.0000000000e+02 5.0000000000e+02 9.2150000000e-04 5.0000000000e+02 1.0000000000e+03 2.8300000000e-04 1.0000000000e+03 3.5755437782e+03 5.7308286215e-05 Level mappings for each response function: Cumulative Distribution Function (CDF) for response_fn_1: Response Level Probability Level Reliability Index General Rel Index -------------- ----------------- ----------------- ----------------1.0000000000e-01 1.9000000000e-03 1.0000000000e+00 1.3600000000e-02 5.0000000000e+01 2.4390000000e-01 1.0000000000e+02 3.4230000000e-01 5.0000000000e+02 7.1090000000e-01 1.0000000000e+03 8.5240000000e-01 ------------------------------------------------------------------- Figure 5.16: Excerpt of UQ output for polynomial chaos example. Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.4. STOCHASTIC EXPANSION METHODS 93 from Gauss-Wigert recursion coefficients [121] in combination with the Golub-Welsch procedure [65]. The input file is shown in Figure 5.17. Note that the dimension preference of (2, 1) is inverted to define a γ weighting vector of (0.5, 1) (and γ of 0.5) for use in the anisotropic Smolyak index set constraint (see Smolyak sparse grids section in Stochastic Expansion Methods chapter in Dakota Theory Manual [4]). In this example, we compute CDF probabilities for six response levels of Rosenbrock’s function. This example requires 19 function evaluations to calculate the interpolating polynomials in stochastic collocation and the resulting expansion exactly reproduces Rosenbrock’s function. The placement of the points generated by the sparse grid is shown in Figure 5.18. # Dakota Input File: rosen_uq_sc.in environment #graphics method stoch_collocation sparse_grid_level = 3 dimension_preference = 2 1 samples_on_emulator = 10000 seed = 12347 response_levels = .1 1. 50. 100. 500. 1000. variance_based_decomp #interaction_order = 1 output silent variables lognormal_uncertain means std_deviations descriptors = 2 = 1. 1. = 0.5 0.5 = ’x1’ ’x2’ interface analysis_drivers = ’rosenbrock’ direct responses response_functions = 1 no_gradients no_hessians Figure 5.17: Dakota input file for performing Dakota/examples/users/rosen uq sc.in UQ using stochastic collocation – see Once the expansion coefficients have been calculated, some statistics are available analytically and others must be evaluated numerically. For the numerical portion, the input file specifies the use of 10000 samples, which will be evaluated on the expansion to compute the CDF probabilities. In Figure 5.19, excerpts from the results summary are presented. We first see the moment statistics for mean, standard deviation, skewness, and kurtosis computed by numerical integration (see Analytic moments section in Stochastic Expansion Methods chapter in Dakota Theory Manual [4]), where the numerical row corresponds to integration using the original response values and the expansion row corresponds to integration using values from the interpolant. The response covariance (collapsing to a single variance value for one response function) and global sensitivity indices (Sobol’ indices) are presented next. This example shows that variable x1 has the largest main effect (0.99) as compared with variable x2 (0.0007) or the interaction between x1 and x2 (0.005). Finally, we see the numerical results for the CDF probabilities based on 10000 samples performed on the expansion. For example, the probability that the Rosenbrock function is less Dakota Version 6.7 User’s Manual generated on November 13, 2017 94 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES Figure 5.18: Rosenbrock stochastic collocation example: sparse grid points. than 100 is 0.7233. Note that these results are significantly different than the ones presented in Section 5.4.1.1 because of the different assumptions about the inputs: uniform[-2,2] versus lognormals with means of 1.0 and standard deviations of 0.5. 5.5 Importance Sampling Methods Importance sampling is a method that allows one to estimate statistical quantities such as failure probabilities (e.g. the probability that a response quantity will exceed a threshold or fall below a threshold value) in a way that is more efficient than Monte Carlo sampling. The core idea in importance sampling is that one generates samples that preferentially samples important regions in the space (e.g. in or near the failure region or user-defined region of interest), and then appropriately weights the samples to obtain an unbiased estimate of the failure probability [126]. In importance sampling, the samples are generated from a density which is called the importance density: it is not the original probability density of the input distributions. The importance density should be centered near the failure region of interest. For black-box simulations such as those commonly interfaced with Dakota, it is difficult to specify the importance density a priori: the user often does not know where the failure region lies, especially in a high-dimensional space. [131] More formally, we define the objective of importance sampling as calculating the probability, P , that the output will exceed a threshold level. This is a failure probability, where the failure probability is defined as some scalar function, y (X), exceeding a threshold, T , where the inputs, X, are randomly distributed with density, ρ (X). When evaluating y (X) is sufficiently expensive or P is sufficiently small, Monte Carlo (MC) sampling methods to estimate P will be infeasible due to the large number of function evaluations required for a specified accuracy. The probability of failure can be thought of as the mean rate of occurrence of failure. The Monte Carlo (MC) estimate of P is therefore the sample mean of the indicator function, I (X), PM C = N 1 X I (Xi ) X ∼ ρ (X) , N i=1 (5.3) where N samples, Xi , are drawn from ρ (X), and the indicator function I (X) is 1 if failure occurs and zero otherwise. Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.5. IMPORTANCE SAMPLING METHODS 95 Statistics derived analytically from polynomial expansion: Moment-based statistics for each response function: Mean Std Dev Skewness response_fn_1 expansion: 2.5671972656e+02 2.0484189184e+03 2.7419241630e+02 numerical: 2.5671972656e+02 2.0484189184e+03 2.7419241630e+02 Kurtosis 1.9594567379e+06 1.9594567379e+06 Covariance among response functions: [[ 4.1960200651e+06 ]] Global sensitivity indices for each response function: response_fn_1 Sobol indices: Main Total 9.9391978710e-01 9.9928724777e-01 x1 7.1275222945e-04 6.0802128961e-03 x2 Interaction 5.3674606667e-03 x1 x2 Statistics based on 10000 samples performed on polynomial expansion: Level mappings for each response function: Cumulative Distribution Function (CDF) for response_fn_1: Response Level Probability Level Reliability Index -------------- ----------------- ----------------1.0000000000e-01 1.8100000000e-02 1.0000000000e+00 8.7800000000e-02 5.0000000000e+01 5.8410000000e-01 1.0000000000e+02 7.2330000000e-01 5.0000000000e+02 9.2010000000e-01 1.0000000000e+03 9.5660000000e-01 General Rel Index ----------------- Figure 5.19: Excerpt of UQ output for stochastic collocation example. Dakota Version 6.7 User’s Manual generated on November 13, 2017 96 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES Importance sampling draws samples from the importance density ρ0 (X) and scales the sample mean by the importance density: N 1 X ρ (Xi ) PIS = I (Xi ) 0 X ∼ ρ0 (X) . (5.4) N i=1 ρ (Xi ) This reduces the asymptotic error variance from: 2 = σerr MC to E 2 = σerr IS 2 Inspection of Eq. 5.6 reveals σerr IS h i 2 E (I (X) − P ) (5.5) N I (X) ρρ(X) 0 (X) − P 2 . N = 0 if ρ0 (X) equals the ideal importance density ρ∗ (X), ρ∗ (X) = I (X) ρ (X) . P (5.6) (5.7) However, ρ∗ (X) is unknown a priori because I (X) is only known where it has been evaluated. Therefore, the required P in the denominator is also unknown: this is what we are trying to estimate. If importance sampling is to be effective, the practitioner must be able to choose a good ρ0 (X) without already knowing I (X) everywhere. There is a danger: a poor choice for ρ0 (X) can put most of the samples in unimportant 2 2 regions and make σerr much greater than σerr . In particular, importance sampling can be challenging for IS MC very low probability events in high-dimensional spaces where the output y is calculated by a simulation. In these cases, usually one does not know anything a priori about where the failure region exists in input space. We have developed two importance sampling approaches which do not rely on the user explicitly specifying an importance density. 5.5.1 Importance Sampling Method based on Reliability Approach The first method is based on ideas in reliability modeling 5.3.1. An initial Latin Hypercube sampling is performed to generate an initial set of samples. These initial samples are augmented with samples from an importance density as follows: The variables are transformed to standard normal space. In the transformed space, the importance density is a set of normal densities centered around points which are in the failure region. Note that this is similar in spirit to the reliability methods, in which importance sampling is centered around a Most Probable Point (MPP). In the case of the LHS samples, the importance sampling density will simply by a mixture of normal distributions centered around points in the failure region. This method is specified by the keyword importance sampling. The options for importance sampling are as follows: import centers a sampling density at one of the initial LHS samples identified in the failure region. It then generates the importance samples, weights them by their probability of occurence given the original density, and calculates the required probability (CDF or CCDF level). adapt import is the same as import but is performed iteratively until the failure probability estimate converges. mm adapt import starts with all of the samples located in the failure region to build a multimodal sampling density. First, it uses a small number of samples around each of the initial samples in the failure region. Note that these samples are allocated to the different points based on their relative probabilities of occurrence: more probable points get more samples. This early part of the approach is done to search for “representative” points. Once these are located, the multimodal sampling density is set and then the multi-modal adaptive method proceeds similarly to the adaptive method (sample until convergence). Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.6. ADAPTIVE SAMPLING METHODS 5.5.2 97 Gaussian Process Adaptive Importance Sampling Method The second importance sampling method in Dakota is the one we recommend, at least for problems that have a relatively small number of input variables (e.g. less than 10). This method, Gaussian Process Adaptive Importance Sampling, is outlined in the paper [24]. This method starts with an initial set of LHS samples and adds samples one at a time, with the goal of adaptively improving the estimate of the ideal importance density during the process. The approach uses a mixture of component densities. An iterative process is used to construct the sequence of improving component densities. At each iteration, a Gaussian process (GP) surrogate is used to help identify areas in the space where failure is likely to occur. The GPs are not used to directly calculate the failure probability; they are only used to approximate the importance density. Thus, the Gaussian process adaptive importance sampling algorithm overcomes limitations involving using a potentially inaccurate surrogate model directly in importance sampling calculations. This method is specified with the keyword gpais. There are three main controls which govern the behavior of the algorithm. samples specifies the initial number of Latin Hypercube samples which are used to create the initial Gaussian process surrogate. emulator samples specifies the number of samples taken on the latest Gaussian process model each iteration of the algorithm. These samples are used in the construction of the next importance sampling density. The default is 10,000 samples. The third control is max iterations, which controls the number of iterations of the algorithm. Each iteration, one additional sample of the “true” simulation is taken. Thus, if samples were set at 100 and max iterations were set to 200, there would be a total of 300 function evaluations of the simulator model taken. 5.6 Adaptive Sampling Methods The goal in performing adaptive sampling is to construct a surrogate model that can be used as an accurate predictor to some expensive simulation, thus it is to one’s advantage to build a surrogate that minimizes the error over the entire domain of interest using as little data as possible from the expensive simulation. The adaptive part alludes to the fact that the surrogate will be refined by focusing samples of the expensive simulation on particular areas of interest rather than rely on random selection or standard space-filling techniques. 5.6.1 Adaptive sampling based on surrogates At a high-level, the adaptive sampling pipeline is a four-step process: 1. Evaluate the expensive simulation (referred to as the true model) at initial sample points 2. Fit/refit a surrogate model 3. Create a candidate set and score based on information from surrogate 4. Select a candidate point to evaluate the true model and Repeat 2-4 In terms of the Dakota implementation, the adaptive sampling method currently uses Latin Hypercube sampling (LHS) to generate the initial points in Step 1 above. For Step 2, we use a Gaussian process model. The user can specify the scoring metric used to select the next point (or points) to evaluate and add to the set. We have investigated several scoring metrics with which to evaluate candidate points for Step 3. There are some classical ones such as distance (e.g. add a point which maximizes the minimum distance to all of the existing points). This distance metric tends to generate points that are space-filling. We have investigated several methods that involve Dakota Version 6.7 User’s Manual generated on November 13, 2017 98 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES interesting topological features of the space (e.g. points that are near saddle points). These are an area of active investigation but are not currently included in Dakota. The fitness metrics for scoring candidate points currently include: Predicted Variance First introduced in [97] and later used in [120], this method uses the predicted variance of the Gaussian process surrogate as the score of a candidate point. Thus, the adaptively chosen points will be in areas of highest uncertainty according to the Gaussian process model. Distance A candidate’s score is the Euclidean distance in domain space between the candidate and its nearest neighbor in the set of points already evaluated on the true model. Therefore, the most undersampled area of the domain will always be selected. The adaptivity of this method could be brought to question as it would chose the exact same points regardless of the surrogate model used. However, it is useful to use to compare other adaptive metrics to one that relies purely on space-filling in an equivalent context. Gradient Similar to the above metric, a candidate’s nearest neighbor is determined as in the distance metric, only now the score is the absolute value of the difference in range space of the two points. The range space values used are predicted from the surrogate model. Though this method is called the gradient metric, it actually does not take into account how close the candidate and its neighbor are in domain space. This method attempts to evenly fill the range space of the surrogate. Note that in our approach, a Latin Hypercube sample is generated (a new one, different from the initial sample) and the surrogate model is evaluated at this points. These are the “candidate points” that are then evaluated according to the fitness metric outlined above. The number of candidates used in practice should be high enough to fill most of the input domain: we recommend at least hundreds of points for a low- dimensional problem. All of the candidates (samples on the emulator) are given a score and then the highest-scoring candidate is selected to be evaluated on the true model. The adaptive sampling method also can generate batches of points to add at a time. With batch or multi-point selection, the true model can be evaluated in parallel and thus increase throughput before refitting our surrogate model. This proposes a new challenge as the problem of choosing a single point and choosing multiple points off a surrogate are fundamentally different. Selecting the n best scoring candidates is more than likely to generate a set of points clustered in one area which will not be conducive to adapting the surrogate. We have implemented several strategies for batch selection of points: Naive Selection This strategy will select the n highest scoring candidates regardless of their position. This tends to group an entire round of points in the same area. Distance Penalized Re-weighted Scoring In this strategy, the highest scoring candidate is selected and then all remaining candidates are re-scored with a distance penalization factor added in to the score. Only points selected within a round are used for the distance penalization. The factor is the same as used in the distance penalization scoring metrics from [98]. First, compute all of the minimum distances from each remaining candidate to the selected candidates. Then, determine the median value of these distances. If the smallest distance, d, between a point and the selected set is less than the computed median distance its score is unaltered, otherwise the score is multiplied by a value ρ determined by the following equation: ρ = 1.5 ∗ d − 0.5 ∗ d3 (5.8) Topological Maxima of Scoring Function In this strategy we look at the topology of the scoring function and select the n highest maxima in the topology. To determine local maxima, we construct the approximate Morse-Smale complex. If the number of local maxima is less than n, we revert to the distance strategy above. As a further extension, one may want to filter low-persistence maxima, but to keep the framework Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.6. ADAPTIVE SAMPLING METHODS 99 general, we chose to omit this feature as defining a threshold for what deems a critical point as ”low persistence” can vary drastically from problem to problem. Constant Liar We adapt the constant liar strategy presented in [59] with the scoring metrics. The strategy first selects the highest scoring candidate, and then refits the surrogate using a “lie” value at the point selected and repeating until n points have been selected whereupon the lie values are removed from the surrogate and the selected points are evaluated on the true model and the surrogate is refit with these values. The adaptive sampling method is specified by the method keyword adaptive sampling. There are many controls, including the number of candidate samples to investigate each iteration (emulator samples), the fitness metric used in scoring candidates (fitness metric), and the number of iterations to perform the adaptive sampling (max iterations). For batch selection of points, one specifies a batch selection strategy and a batch size. The details of the specification are provided in the Dakota reference manual. 5.6.2 Adaptive sampling based on dart throwing pof darts is a novel method for estimating the tail probability (Probability of Failure) based on random spherepacking in the uncertain parameter space. Random points are sequentially sampled from the domain and consequently surrounded by protecting spheres, with the constraint that each new sphere center has to be outside all prior spheres [31]. The radius of each sphere is chosen such that the entire sphere lies either in the failure or the non-failure region. This radius depends of the function evaluation at the disk center, the failure threshold and an estimate of the function gradient at the disk center. After exhausting the sampling budget specified by build samples, which is the number of spheres per failure threshold, the domain is decomposed into two regions. These regions correspond to failure and non-failure categories, each represented by the union of the spheres of each type. The volume of the union of failure spheres gives a lower bound on the required estimate of the probability of failure, while the volume of the union of the non-failure spheres subtracted from the volume of the domain gives an upper estimate. After all the spheres are constructed, we construct a surrogate model, specified via a model pointer, and sample the surrogate model extensively to estimate the probability of failure for each threshold. pof darts handles multiple response functions and allows each to have multiple failure thresholds. For each failure threshold pof darts will insert a number of spheres specified by the user-input parameter ”samples”. However, estimating the probability of failure for each failure threshold would utilize the total number of disks sampled for all failure thresholds. For each failure threshold, the sphere radii changes to generate the right spatial decomposition. The POF-Darts method is specified by the method keyword pof darts. The sample budget is specified by build samples. By default, the method employs a local approach to estimate the Lipschitz constant per sphere. The surrogate model used by the pof darts method for extensive sampling is specified using a model pointer, and its parameters are therefore defined in that model. It can typically be any global surrogate in Dakota (e.g., Gaussian process, polynomial chaos expansion, polynomial regression, etc). POF-Darts can also use piecewisedecomposed surrogates which build local pieces of the surrogate over different domain patches. The piecewise decomposition option is a new capability added to Dakota to help construct surrogates in high-dimensional spaces, using known function evaluations as well as gradient and Hessian information, if available. The piecewise decomposition option is declared using the keyword domain decomp and currently supports polynomial, Gaussian Process (GP), and Radial Basis Functions (RBF) surroagte models only. For example: a polynomial regression global surrogate is specified with model polynomial, its order is selected using surrogate order, and the piecewise decomposition option is specified with domain decomp. The domain decomp option is parametrized by a cell type set by default to Voronoi cells, an optional number of support layers, and an optional discontinuity detection capability. See 8.4.3.10 for more details. Dakota Version 6.7 User’s Manual generated on November 13, 2017 100 5.7 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES Epistemic Nondeterministic Methods Uncertainty quantification is often used as part of the risk assessment of performance, reliability, and safety of engineered systems. Increasingly, uncertainty is separated into two categories for analysis purposes: aleatory and epistemic uncertainty [109, 77]. Aleatory uncertainty is also referred to as variability, irreducible or inherent uncertainty, or uncertainty due to chance. Examples of aleatory uncertainty include the height of individuals in a population, or the temperature in a processing environment. Aleatory uncertainty is usually modeled with probability distributions, and sampling methods such as Latin Hypercube sampling in Dakota can be used to model aleatory uncertainty. In contrast, epistemic uncertainty refers to lack of knowledge or lack of information about a particular aspect of the simulation model, including the system and environment being modeled. An increase in knowledge or information relating to epistemic uncertainty will lead to a reduction in the predicted uncertainty of the system response or performance. For epistemic uncertain variables, typically one does not know enough to specify a probability distribution on a variable. Epistemic uncertainty is referred to as subjective, reducible, or lack of knowledge uncertainty. Examples of epistemic uncertainty include little or no experimental data for a fixed but unknown physical parameter, incomplete understanding of complex physical phenomena, uncertainty about the correct model form to use, etc. There are many approaches which have been developed to model epistemic uncertainty, including fuzzy set theory, possibility theory, and evidence theory. It is also possible to use simple interval analysis in an epistemic context. Interval analysis and evidence theory are described in more detail below. 5.7.1 Interval Methods for Epistemic Analysis In interval analysis, one assumes that nothing is known about an epistemic uncertain variable except that its value lies somewhere within an interval. In this situation, it is NOT assumed that the value has a uniform probability of occuring within the interval. Instead, the interpretation is that any value within the interval is a possible value or a potential realization of that variable. In interval analysis, the uncertainty quantification problem is one of determining the resulting bounds on the output (defining the output interval) given interval bounds on the inputs. Again, any output response that falls within the output interval is a possible output with no frequency information assigned to it. We have the capability to perform interval analysis using either global interval est or local interval est. In the global approach, one uses either a global optimization method or a sampling method to assess the bounds. global interval est allows the user to specify either lhs, which performs Latin Hypercube Sampling and takes the minimum and maximum of the samples as the bounds (no optimization is performed) or ego. In the case of ego, the efficient global optimization method is used to calculate bounds. The ego method is described in Section 6.2.3. If the problem is amenable to local optimization methods (e.g. can provide derivatives or use finite difference method to calculate derivatives), then one can use local methods to calculate these bounds. local interval est allows the user to specify either sqp which is sequential quadratic programming, or nip which is a nonlinear interior point method. Note that when performing interval analysis, it is necessary to define interval uncertain variables as described in Section 9.3. For interval analysis, one must define only one interval per input variable, in contrast with DempsterShafer evidence theory, where an input can have several possible intervals. Interval analysis can be considered a special case of Dempster-Shafer evidence theory where each input is defined by one input interval with a basic probability assignment of one. In Dakota, however, the methods are separate and semantic differences exist in the output presentation. If you are performing a pure interval analysis, we recommend using either global interval est or local interval est instead of global evidence or local evidence, for reasons of simplicity. Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.7. EPISTEMIC NONDETERMINISTIC METHODS 101 These interval methods can also be used as the outer loop within an interval-valued probability analysis for propagating mixed aleatory and epistemic uncertainty – refer to Section 15.1.1 for additional details. An example of interval estimation is shown in Figure 5.20, with example results in Figure 5.21. This example is a demonstration of calculating interval bounds for three outputs of the cantilever beam problem. The cantilever beam problem is described in detail in Section 20.7. Given input intervals of [1,10] on beam width and beam thickness, we can see that the interval estimate of beam weight is approximately [1,100]. # Dakota Input File: cantilever_uq_global_interval.in environment tabular_data # tabular_data_file = ’cantilever_uq_global_interval.dat’ method global_interval_est ego seed = 1234567 output verbose variables continuous_interval_uncertain = 2 num_intervals = 1 1 interval_probabilities = 1.0 1.0 lower_bounds = 1.0 1.0 upper_bounds = 10.0 10.0 descriptors ’w’ ’t’ continuous_state = 4 initial_state = 40000. 29.E+6 500. 1000. descriptors = ’R’ ’E’ ’X’ ’Y’ interface analysis_drivers = ’cantilever’ direct responses response_functions = 3 descriptors = ’weight’ ’stress’ ’displ’ no_gradients no_hessians Figure 5.20: Dakota input file for performing UQ using Dakota/examples/users/cantilever uq global interval.in 5.7.2 interval analysis – see Dempster-Shafer Theory of Evidence We have chosen to pursue evidence theory at Sandia as a way to model epistemic uncertainty, in part because evidence theory is a generalization of probability theory. Evidence theory is also referred to as Dempster-Shafer theory or the theory of random sets [109]. This section focuses on the use of Dempster-Shafer evidence theory for propagating epistemic uncertainties. When aleatory uncertainties are also present, we may choose either to discretize the aleatory probability distributions into sets of intervals and treat them as well-characterized episDakota Version 6.7 User’s Manual generated on November 13, 2017 102 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES -----------------------------------------------------------------Min and Max estimated values for each response function: weight: Min = 1.0000169352e+00 Max = 9.9999491948e+01 stress: Min = -9.7749994284e-01 Max = 2.1499428450e+01 displ: Min = -9.9315672724e-01 Max = 6.7429714485e+01 ----------------------------------------------------------------- Figure 5.21: Excerpt of UQ output for interval example. temic variables, or we may choose to segregate the aleatory uncertainties and treat them within an inner loop. A nested Dempster-Shafer approach for propagating mixed aleatory and epistemic uncertainty is described in Section 15.1.3. In evidence theory, there are two complementary measures of uncertainty: belief and plausibility. Together, belief and plausibility can be thought of as defining lower and upper bounds, respectively, on probabilities. Belief and plausibility define the lower and upper limits or intervals on probability values. Typical plots of cumulative and complementary cumulative belief and plausibility functions are shown in Figure 5.22 [77]. In evidence theory, Figure 5.22: Example cumulative belief and plausibility distribution functions on left; complementary cumulative belief and plausibility distribution functions on right it is not possible to specify one probability value. Instead, there is a range of values that is consistent with the evidence. The range of values is defined by belief and plausibility. Note that no statement or claim is made about one value within an interval being more or less likely than any other value. In Dempster-Shafer evidence theory, the uncertain input variables are modeled as sets of intervals. The user assigns a basic probability assignment (BPA) to each interval, indicating how likely it is that the uncertain input falls within the interval. The BPAs for a particular uncertain input variable must sum to one. The intervals may be overlapping, contiguous, or have gaps. In Dakota, an interval uncertain variable is specified as interval uncertain. When one defines an interval type variable in Dakota, it is also necessary to specify the number of intervals defined for each variable with iuv num intervals as well the basic probability assignments per interval, iuv interval probs, and the associated bounds per each interval, iuv interval bounds. Figure 5.23 shows the input specification for interval uncertain variables. The example has two epistemic uncerDakota Version 6.7 User’s Manual generated on November 13, 2017 5.7. EPISTEMIC NONDETERMINISTIC METHODS 103 tain interval variables. The first uncertain variable has three intervals and the second has two. The basic probability assignments for the first variable are 0.5, 0.1, and 0.4, while the BPAs for the second variable are 0.7 and 0.3. Note that it is possible (and often the case) to define an interval uncertain variable with only ONE interval. This means that you only know that the possible value of that variable falls within the interval, and the BPA for that interval would be 1.0. In the case we have shown, the interval bounds on the first interval for the first variable are 0.6 and 0.9, and the bounds for the second interval for the first variable are 0.1 to 0.5, etc. # Dakota Input File: textbook_uq_glob_evidence.in environment tabular_data tabular_data_file = ’textbook_uq_glob_evidence.dat’ method global_evidence lhs samples = 1000 seed = 59334 response_levels = 0.001 0.03 0.2 0.8 0.001 0.2 0.6 0.8 probability_levels = 0.25 0.5 0.75 0.25 0.5 0.75 distribution cumulative output verbose variables continuous_interval_uncertain = num_intervals = 3 2 interval_probabilities = 0.5 lower_bounds = 0.6 0.1 0.5 upper_bounds = 0.9 0.5 1.0 2 0.1 0.4 0.7 0.3 0.3 0.6 0.5 0.8 interface analysis_drivers = ’text_book’ direct responses response_functions = 2 no_gradients no_hessians Figure 5.23: Dakota input file for UQ example using Dakota/examples/users/textbook uq glob evidence.in Evidence Theory – see Once the intervals, the BPAs, and the interval bounds are defined, the user can run an epistemic analysis by specifying the method as either global evidence or local evidence in the Dakota input file. Both of these methods perform Dempster-Shafer calculations: the difference is that the local method uses a local optimization algorithm to calculate the interval bounds and the global method uses either sampling or a global optimization approach to calculate an interval bound. These differences are discussed in more detail below. The intervals and their associated BPAs are then propagated through the simulation to obtain cumulative distribution functions on belief and plausibility. As mentioned above, belief is the lower bound on a probability estimate that is consistent with the evidence, and plausibility is the upper bound on a probability estimate that is consistent with the evidence. Dakota Version 6.7 User’s Manual generated on November 13, 2017 104 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES Figure 5.24 shows results for the first response function obtained when running the example in Figure 5.23. In this example, there are 6 output intervals (as a result of the 2 interval input variables with 3 and 2 intervals, respectively). The output intervals are ordered to obtain cumulative bound functions for both belief and plausibility. The cumulative distribution function is presented for both belief (CBF) and plausibility (CPF). The CBF value is the cumulative belief corresponding to a certain output value. For example, the belief that the output value is less than or equal to 0.2 for response 1 is 0.27, and the plausibility that the output is less than or equal to 0.2 is 1 for response 1. The belief that the output value is less than 0.6217 is 0.75, while the plausbility that the output is less than 0.0806 is 0.75. The CBF and CPF may be plotted on a graph and interpreted as bounding the cumulative distribution function (CDF), which is the probability that the output is less than or equal to a certain value. The interval bounds on probability values show the value of epistemic uncertainty analysis: the intervals are usually much larger than expected, giving one a truer picture of the total output uncertainty caused by lack of knowledge or information about the epistemic input quantities. Belief and Plausibility for each response function: Cumulative Belief/Plausibility Functions (CBF/CPF) for response_fn_1: Response Level Belief Prob Level Plaus Prob Level -------------- -------------------------------1.0000000000e-03 0.0000000000e+00 0.0000000000e+00 3.0000000000e-02 0.0000000000e+00 2.7000000000e-01 2.0000000000e-01 2.7000000000e-01 1.0000000000e+00 8.0000000000e-01 9.3000000000e-01 1.0000000000e+00 Probability Level Belief Resp Level Plaus Resp Level ----------------- -------------------------------2.5000000000e-01 2.6187288772e-01 6.2609206069e-02 5.0000000000e-01 2.9829775860e-01 6.3736734971e-02 7.5000000000e-01 6.2173551556e-01 8.0596931719e-02 Figure 5.24: Results of an Epistemic Uncertainty Quantification using Evidence Theory. As in other nondeterministic methods, with local evidence or global evidence, one can specify probability levels and response levels. If response levels are specified, the belief and plausibility function values corresponding to those response levels are calculated (see Belief Prob Level and Plaus Prob Level in the tables shown in Figure 5.24). Similarly, if probability levels are specified, these are first interpreted to be belief values, and the corresponding response levels are calculated (see Belief Resp Level); then they are interpreted to be plausibility values and the corresponding response levels are calculated (see Plaus Resp Level in the table in Figure 5.24). We have recently added the capability to support generalized reliability mappings in the evidence methods. If the user specifies a generalized reliability level, it will be first converted to a probability, then interpreted as a belief and plausibility and the corresponding response levels will be calculated. Likewise, if response levels are specified, the corresponding belief and plausibility values will be mapped to bounds on the generalized reliability levels. To elaborate on the differences between global evidence and local evidence: both of these methods take the Dempster-Shafer structures specified on the inputs and calculate a resulting Dempster-Shafer structure on the outputs (e.g. a cumulative belief and plausibility function). To calculate the belief and plausibility measures, it is necessary to calculate the minimum and maximum of the response function in each “interval cell combination.” For example, in a two variable problem, if the first variable had three intervals and associated BPAs assigned and the second variable had two intervals and associated BPAs assigned, there would be 6 interval cells in total. In each of these six cells, one needs to identify a minimum and maximum value of the response function. This is easy to do if the function is monotonic in both variables, but in general it is not. We offer the capability to use local optimization methods to calculate these bounds: local evidence allows the user to specify either sqp Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.8. BAYESIAN CALIBRATION METHODS 105 which is sequential quadratic programming, or nip which is a nonlinear interior point method. We also offer the capability to use global methods to assess these interval cell bounds. global evidence allows the user to specify either lhs, which performs Latin Hypercube Sampling and takes the minimum and maximum of the samples within each cell as the bounds (no optimization is performed) or ego. In the case of ego, the efficient global optimization method is used to calculate bounds. The ego method is described in Section 6.2.3. Note that for a situation with many uncertain variables, each with a fairly complicated Dempster-Shafer structure described by many intervals, there will be a huge number of interval calls, and the overall process of performing DempsterShafer analysis will be extremely expensive. Reference [133] provides more details about the implementation of the optimization methods to perform Dempster-Shafer calculations, as well as comparisons on test problems. 5.8 Bayesian Calibration Methods In Bayesian calibration a “prior distribution” on a parameter is updated through a Bayesian framework involving experimental data and a likelihood function. Bayesian inference theory is best left to other sources [86] and only a brief summary is given here. In Bayesian methods, uncertain parameters are characterized by probability density functions. These probability density functions define the permissible parameter values - the support, as well as the relative plausibility of each permissible parameter value. In the context of calibration or any inference step, the probability density function that describes knowledge before the incorporation of data is called the prior, fΘ (θ). When data are available, the likelihood function describes how well each parameter value is supported by the data. Bayes Theorem [83], shown in Equation 5.9, is used for inference: to derive the plausible parameter values, based on the prior probability density and the data d. The result is the posterior probability density function of the parameters fΘ|D (θ|d). It is interpreted the same way as the prior, but includes the information derived from the data. fΘ|D (θ|d) = fΘ (θ) L (θ; d) . fD (d) (5.9) The likelihood function is used to describe how well a model’s predictions are supported by the data. The likelihood function can be written generally as: L (θ; d) = F(q(θ) − d), where θ are the parameters of model quantity of interest q. The form of the function F can greatly influence the results. The specific likelihood function used in Dakota is based on Gaussian probability density functions. This means that we assume the difference between the model quantity (e.g. quantity of interest returned from a computer simulation) and the experimental observations are Gaussian: di = qi (θ) + i , (5.10) where i is a random variable that can encompass both measurement errors on di and modeling errors associated with the simulation quantity of interest qi , for each of n observations. If we assume that all experiments and observations are independent, then the probabilistic model defined by Eq. (5.10) results in a likelihood function for θ that is the product of n normal probability density functions: " # n 2 Y 1 (di − qi (θ)) √ exp − L(θ; d) = , (5.11) 2σd 2 σ 2π i=1 d Dakota Version 6.7 User’s Manual generated on November 13, 2017 106 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES where σd 2 refers to the measurement error of the data, assumed constant across all data observations in this case. We also support the more general case of a full covariance matrix, Σd , that specifies the covariance between each observation i and j. In this case, the likelihood is commonly written in log form, where the log-likelihood is: 1 log L(θ; d) ∝ − r T Σd −1 r, 2 (5.12) where r is the vector of residuals between the data points and the model quantity of interest, q(θ) − d. Dakota admits four experiment variance type options to specify the measurement error covariance: none for no measurement error specified (in this case, the variance is assumed to be one), scalar where a constant value σd 2 is given for all observations, diagonal where a value is specified for the diagonal elements of the covariance matrix Σd meaning that each observation has its own measurement error but there are no cross terms, and matrix where the full covariance matrix Σd is specified. The diagonal and matrix terms are only available for field response data. In contrast to earlier versions of Dakota, all measurement error variance should be specified with units of variance/covariance, not standard deviation. Markov Chain Monte Carlo (MCMC) is the prototypical method used to estimate posterior parameter densities, given the observational data and the priors. There are many references that describe the basic algorithm [55] and this is an active research area. MCMC algorithms may require hundreds of thousands of steps to converge, depending on dimensionality, response nonlinearity, and the desired set of posterior statistics. Since each iteration involves an evaluation of the model to obtain q(θ), surrogate models of expensive simulations are often employed to make the MCMC process tractable. Dakota offers four approaches for Bayesian calibration: QUESO, DREAM, GPMSA, and WASABI. They are specified with the bayes calibration keyword in combination with the queso, dream, gpmsa, or wasabi selections, respectively. The QUESO and GPMSA methods use components from the QUESO library (Quantification of Uncertainty for Estimation, Simulation, and Optimization) developed at The University of Texas at Austin. It implements the Delayed Rejection and Adaptive Metropolis [71] (DRAM) algorithm, among others. Algorithm variants selectively combine the delayed rejection and adaptive elements. The QUESO/GPMSA capability is based on the GPMSA Matlab toolbox developed at Los Alamos National Laboratory and uses tightly integrated Gaussian process models during calibration. It’s Dakota implementation is in an early prototype stage. DREAM uses an implementation of DiffeRential Evolution Adaptive Metropolis developed by John Burkardt. The DREAM approach runs concurrent chains for global exploration, and automatically tunes the proposal covariance during the process by a self-adaptive randomized subspace sampling [142]. The prototype WASABI method is an MCMC-free Bayesian calibration approach. QUESO/DRAM and variants are the most well-developed within Dakota. 5.8.1 QUESO The QUESO library includes several sampling algorithm variants. One can use a standard Metropolis-Hastings algorithm (metropolis hastings), adaptive Metropolis (adaptive metropolis) for adapting the proposal covariance during the sampling, delayed rejection (delayed rejection) for backtracking from sample rejections, the full DRAM (dram) which involves both delayed rejection and adaptive Metropolis, or a multi-level algorithm (multilevel). This last option is not yet production-ready in Dakota. With any choice of sampling algorithm, one may manually set the burn in period for the MCMC chain with burn in samples. If a sub sampling period is specified, the MCMC chain is further filtered such that only the sample at the beginning of each period is in the final MCMC chain. The sub sampling period should therefore be greater than or equal to the correlation length of the samples. With the QUESO method, one may run the MCMC sampling on the simulation model directly. However, if the Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.8. BAYESIAN CALIBRATION METHODS 107 model is expensive, use of a surrogate (emulator) model is recommended. Options include a Gaussian process, a polynomial chaos expansion, or a stochastic collocation expansion. The proposal covariance refers to the covariance structure of a multivariate normal distribution, which governs sample generation in the chain. One may specify proposal covariance, followed by prior (the default), values, filename, or derivatives. With the prior setting, the proposal covariance will be set to the variance of the prior distributions of the parameters being calibrated. When specifying the proposal covariance with input file values or from a separate file, the user may specify only the diagonals of the covariance matrix or the full covariance matrix. The derivatives option will use derivatives from the simulation or emulator model to form or approximate the Hessian of the misfit function (the negative log likelihood). Especially when derivative information is available inexpensively (e.g. from an emulator), the derivative-based proposal covariance forms a more accurate proposal distribution, resulting in lower rejection rates and faster chain mixing. When using an emulator, the derivativebased proposal covariance should be updated periodically using the posterior adaptive specification. This will add simulation truth evaluations in areas of high-likelihood to the emulator training data, thus refining the Hessian. For more detail about derivative-based formulations involving the misfit Hessian, refer to the Theory Manual. An additional control for QUESO is to perform a logit transformation (logit transform) which performs an internal variable transformation from bounded domains to unbounded domains. This option can be helpful when regions of high posterior density exist in the corners of a multi-dimensional bounded domain. In these cases, it may be difficult to generate feasible samples from the proposal density, and transformation to unbounded domains may greatly reduce sample rejection rates. The pre solve option will perform a deterministic gradient-based optimization before the MCMC sampling to get a good starting point for the chain. This pre-solve seeks to maximize the log-posterior (the log-likelihood minus the log-prior) to identify the maximum a posteriori probability point, also called the MAP point. The Markov Chain will then start at the MAP point, which can circumvent a lot of initial searching for high posterior probability points. The pre-solve option can be used with an emulator or with no emulator. Credible and prediction intervals will be calculated if probability levels is specified. Credible intervals propagate uncertainties in parameter density information to the quantity of interest and quantify how well the model fits the provided data, while prediction intervals propagate both parameter and experimental measurement uncertainties and contain the next experimental or simulated observation with the specified probability. Further details can be found in [123]. If probability levels is specified, credible intervals will always be calculated. Prediction intervals will only be calculated if experiment variance type is also specified in the responses block. By specifying posterior stats, information-theoretic metrics may be calculated using the posterior distribution of parameters. If the kl divergence option is selected, the Kullback-Leibler Divergence will be calculated between the posterior and the prior distributions such that Z fΘ|D (θ|d) dθ. (5.13) DKL = fΘ|D (θ|d) log fΘ (θ) This quantity represents the amount of information gained about the parameters during the Bayesian update. Further details regarding the calculation and use of DKL can be found in [4]. 5.8.2 DREAM For the DREAM method, one can define the number of chains used with the chains specification. The total number of generations per chain in DREAM is the number of samples divided by the number of chains. The number of chains randomly selected to be used in the crossover each time a crossover occurs is crossover chain pairs. Dakota Version 6.7 User’s Manual generated on November 13, 2017 108 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES There is an extra adaptation during burn-in, in which DREAM estimates a distribution of crossover probabilities that favors large jumps over smaller ones in each of the chains. Normalization is required to ensure that all of the input dimensions contribute equally. In this process, a discrete number of candidate points for each crossover value is generated, which can be specified with num cr. The gr threshold control is the convergence tolerance for the Gelman-Rubin statistic, which governs the convergence of the multiple chain process. The integer jump step forces a long jump every jump step generations. For more details about these parameters, refer to [142]. Credible and prediction intervals can be calculated by specifying probability levels, and statistics regarding the posterior may be calculated by specifying posterior stats, as described in Section 5.8.1. 5.8.3 GPMSA Core to GPMSA is the construction of a Gaussian process emulator from simulation runs collected at various settings of input parameters. The emulator is a statistical model of the system response, and it is used to incorporate the observational data to improve system predictions and constrain or calibrate the unknown parameters. The GPMSA code draws heavily on the theory developed in the seminal Bayesian calibration paper by Kennedy and O’Hagan [86]. The particular approach developed by the Los Alamos group, and implemented in QUESO and therefore Dakota, is provided in [78]. It includes an embedded discrepancy model and the ability to estimate various hyper-parameters of the Gaussian process, observation error model, and discrepancy model. Dakota’s GPMSA capability is an experimental prototype with a number of limitations. See the Dakota Reference Manual [3] for more information. 5.8.4 WASABI WASABI differs from the other Bayesian approaches in that it is not an MCMC-based approach. Instead, it is based on the idea of “consistent Bayes” which is outlined in [14]. This approach to stochastic inference uses measure-theoretic principles to construct a probability measure or density on model parameters that is consistent with the model and the data. The idea is that the probability measure on the parameters, when “pushed-foward” through the computational model, will give results that match the probability measure on the observational data. We use a similar notation as with the Bayesian methods, but the interpretation is different here. The goal is to identify the posterior density on the parameters, πpost (θ), which is equal to the prior density on the parameters obs , describes the relative likelihood that the output of the model times a ratio. The numerator of the ratio, πD corresponds to the observed data D: this is the density of the data evaluated at the model output. q(θ) refers q to the model output. πDprior refers to the push-forward of the prior through the model and represents a forward propagation of uncertainty. πpost (θ) = πprior (θ) obs πD (q(θ)) . qprior πD (q(θ)) (5.14) The Theory Manual [4] has more detail about the assumptions and mathematical foundations for this method. Note a major difference in interpretation of the posterior results with respect to a standard Bayesian approach: In a standard Bayesian approach, the posterior reflects an updated state of information about the prior distribution on parameter values induced by the observational data. In consistent Bayes, the posterior reflects a stochastic mapping of parameter values such that the posterior parameters, when pushed-forward through the model, give results that are consistent with the density of the observational data. WASABI is a prototype capability. See the Dakota Reference Manual [3] for more information. Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.8. BAYESIAN CALIBRATION METHODS Table 5.2: Capabilities of Bayesian methods in Dakota QUESO GPMSA DREAM Any continuous Any continuous Uniform only variable type variable type Inference Type MCMC with DR, MCMC with DR, MCMC with AM, DRAM, or MH AM, DRAM, or MH Differential Evolution Adaptive Metropolis Can use PCE/SC Emulator Yes No Yes Can use GP Emulator Yes Yes (required) Yes Likelihood-based adaptive Yes No No emulator update Initialize with MAP pre-solve Yes No No Proposal covariance prior, user, prior, user n/a options derivative-based Can calibrate error Yes Yes (internal) Yes covariance multipliers Supports standardized space Yes Yes Yes Logit transform Yes Yes No Posterior export samples samples samples Capability Prior Distributions 5.8.5 109 WASABI Uniform only MCMC-free interval analysis Yes Yes No No n/a No Yes No samples, density Feature Comparison Table 5.2 compares the options available with the QUESO and DREAM implementations in Dakota. 5.8.6 Bayesian Calibration Example To run a QUESO-based Bayesian calibration in Dakota, create a Dakota input file such as the one shown in Figure 5.25. Here, the QUESO DRAM (delayed rejection adaptive metropolis) solver is selected. The number of samples = 1000 indicates how many points to generate in the acceptance chain3 . This example uses the mod cantilever algebraic model, so an emulator is not warranted. The proposal covariance used has diagonal element values of 1.e6 and 0.1. Two credible and prediction intervals will be calculated for each model output: a 5/95 interval and a 10/90 interval. The calibration terms in the responses section refers to the number of outputs that will be used in the calibration process: in this case, it is just two. The calibration data file has the observational data: in this case, it is a freeform file (e.g. no header or annotation) with ten experiments. For each experiment, there are two experiment values, one for stress and one for displacement, followed by two variance values for the error associated with that experiment for each quantity of interest. When the input file shown in 5.25 is run, Dakota will run the MCMC algorithm and generate a posterior sample of θ in accordance with Bayes Theorem 5.9 and the likelihood function 5.11. Dakota’s final output summary reports an evaluation count summary and the best sample point visited during the MCMC process as measured by maximum posterior probability (an estimate of the MAP point). The final output also summarizes the moments of the posterior samples from the chain (e.g. mean of the chain, standard deviation of the chain samples, etc.), as well as the credible and prediction intervals for each model output. Auxilliary output is also generated to a directory called QuesoDiagnostics/ in the directory from which Dakota is run. The file display sub0.txt contains diagnostic information regarding the MCMC process. 3 If delayed rejection is active, the number of simulation evaluations will typically be higher due to backtracking. Dakota Version 6.7 User’s Manual generated on November 13, 2017 110 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES method, bayes_calibration queso chain_samples = 1000 seed = 348 dram # | delayed_rejection | adaptive_metropolis | metropolis_hastings proposal_covariance values 1.0e6 1.0e-1 diagonal logit_transform # default is off probability_levels 0.05 0.1 0.05 0.1 output debug variables, uniform_uncertain 2 upper_bounds 1.e8 10.0 lower_bounds 1.e6 0.1 initial_point 2.85e7 2.5 descriptors ’E’ ’w’ continuous_state 4 initial_state 3 40000 500 1000 descriptors ’t’ ’R’ ’X’ ’Y’ interface, analysis_drivers = ’mod_cantilever’ direct responses, calibration_terms = 2 calibration_data_file = ’dakota_cantilever_queso.withsigma.dat’ freeform num_experiments = 10 variance_type = ’scalar’ # read 2 scalar sigmas in each row descriptors = ’stress’ ’displacement’ no_gradients no_hessians Figure 5.25: Dakota input file for UQ example using Bayesian Calibration Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.8. BAYESIAN CALIBRATION METHODS 111 The Matlab files contained in the QuesoDiagnostics directory contain the chain files. The files to load in Matlab are raw chain.m or filtered chain.m, containing the full chain or the filtered chain values of the parameters4 . In addition, the accepted chain values that Dakota generates are written to a file in the run directory called (by default) dakota mcmc tabular.dat. The first columns of this file are the posterior values of the input variables. If burn in or sub sampling period are specified, the filtered acceptance chain is instead written to the file dakota mcmc tabular.dat. This file contains the posterior values of the filtered MCMC chain, as well as the values of state variables and the resulting model responses. Finally, if one wants to see the likelihood of each point, specifying verbose output in the method section will result in the likelihoods being printed. 5.8.7 Calibrating the Observation Error Model As discussed in Section 12.2.3, Dakota accepts user information on the covariance Σd among observation errors and includes this in the likelihood formulation: 1 log L(θ; d) ∝ − r T Σd −1 r. 2 In some cases, it can be helpful to fine tune the assumptions in this covariance during the calibration process. Dakota supports calibrating one or more multipliers on the blocks of the error covariance. So if Σd is block diagonal such that Σd = diag(Σd1 , ..., ΣdN ), we instead formulate as Σd = diag(m1 Σd1 , ..., mP ΣdP ) and calibrate the multipliers mi as hyper-parameters in the Bayesian inference process. The supported modes for calibrating observation error multipliers are shown in Figure 5.26: one, per experiment, per response, and both. Here, the two major blocks denote two experiments, while the inner blocks denote five response groups (two scalar, three field). The priors on the hyper-parameters mi are taken to be inverse gamma distributions, with mean and mode approximately 1.0 and standard deviation approximately 0.1. 5.8.8 Model Discrepancy Whether in a Bayesian setting or otherwise, the goal of model calibration is to minimize the difference between the observational data di and the corresponding model response qi (θ). In the presence of scenario or configuration variables x, Eq. 5.10 can be modified, di (x) = qi (θ, x) + i , (5.15) with the ensuing equations of the likelihood and Bayes Theorem updated likewise. The configuration variables represent experimental settings, such as temperature or pressure, that may vary between experiments. However, it is often the case that the agreement between the data and the model after calibration is not sufficiently close. This is generally attributed to model form or structural error, and can be corrected to some extent with the use of model discrepancy. The Kennedy and O’Hagan [86] formulation takes the form di (x) = qi (θ, x) + δi (x) + i , (5.16) where δi (x) represents the model discrepancy. It should be noted that the discrepancy depends only on the configuration variables. Furthermore, one discrepancy model is calculated for each observable di , i = 1, . . . , n, yielding δ1 , . . . , δn . Future work includes developing a discrepancy model that accounts for field data so that a single, global δ(x) will be approximated. 4 The full chain will be output in cases of adaptive posterior refinement or proposal updating, since these use cases access the entire acceptance chain to identify refinement data or restarting points, respectively. Dakota Version 6.7 User’s Manual generated on November 13, 2017 112 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES (a) (b) (c) (d) Figure 5.26: Calibrating observational error covariance multipliers: (a) one multiplier on whole user-provided covariance structure, (b) multiplier per-experiment, (c) multiplier per-response, and (d) both.. Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.8. BAYESIAN CALIBRATION METHODS 113 The current implementation of the model discrepancy capability in Dakota serves as a post-processing mechanism after the completion of a Bayesian update. If model discrepancy is specified in the input file, Dakota will perform the Bayesian update as detailed in the section above, and then begin the process of approximating δi . For each observable di and for each configuration xj , δi (xj ) = di (xj ) − qi (θ ∗ , xj ) , (5.17) where θ ∗ is the average of the calibrated posterior distribution of the model parameters. The ith discrepancy function will be built over the computed δi (xj ), j = 1, . . . , m. Within the Dakota input file, the user may specify the discrepancy type to be either a Gaussian process or polynomial regression model with the gaussian process or polynomial commands, respectively. Additionally, the order of the trend function may be selected using the correction order command and choosing one of constant, linear, or quadratic. Any specifications using these keywords will apply to all δi . By default, Dakota will build a Gaussian process discrepancy model with a quadratic trend function. Information regarding how polynomial and Gaussian process models are built can be found in Sections 8.4.3.4 and 8.4.3.5, respectively. The user may specify new “prediction” configurations at which the corrected model should be calculated. For each response and for each new configuration, qi (θ, xk,new ) + δi (xk,new ) will be computed. The prediction configurations can be specified in one of three ways. If num prediction configs is included, Dakota will uniformly distribute the indicated number of prediction configurations throughout the domain of the configuration variable that is given in the variables block of the input file. Alternatively, the user may explicitly list desired prediction configuration locations within the input file following the prediction configs keyword, or in an external file to be read in with the import prediction configs option. If none of these three options is selected, Dakota will automatically calculate corrected model predictions at ten configurations, spread uniformly in the configuration variable domain. Calculations corresponding to each prediction configuration and to each observable will be output to tabular files. The responses from the discrepancy model itself is output to dakota discrepancy tabular.dat. Those from the corrected model are output to dakota corrected tabular.dat. The user may specify the output file names for the discrepancy and corrected model tabular files using the export discrepancy file and export corrected model file keywords, respectively. Variance information corresponding to each specified configuration location and for each observable is also computed. In a prediction setting, the variance calculated from the discrepancy model is additively combined with the variance information provided with the experimental data, such that 2 Σtotal,i (x) = Σδ,i (x) + σexp,i I (5.18) for each observable i. Further details of how the variance Σδ,i (x) is computed for Gaussian process and polynomial regression models can be found in the Dakota Theory Manual [4]. The experimental variance provided for 2 parameter calibration may vary for the same observable from experiment to experiment, thus σexp,i is taken to be the maximum variance given for each observable. That is, 2 σexp,i = max σi2 (xj ), j (5.19) where σi2 (xj ) is the variance provided for the ith observable di , computed or measured with the configuration variable xj . When each corrected model value qi (θ ∗ , xk,new ) + δi (xk,new ) is considered, the variance calculated via 5.18 2 provides a prediction interval, similar to those described in Section 5.8.1. Including σexp,i in the variance calculation accounts for the uncertainty in the model predictions that arise due to uncertainties in the calibration data. These prediction variances are output to the file dakota discrepancy variance tabular.dat by Dakota Version 6.7 User’s Manual generated on November 13, 2017 114 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES default. The name of this file can be modified using the export corrected variance file keyword in the input script. Additional details and an illustrative example of these calculations are given in the Dakota Theory Manual [4]. 5.8.9 Bayesian Experimental Design The goal of experimental design is to add observational data to the Bayesian update that informs model parameters and reduces their uncertainties. In Bayesian approaches, data from physical experiments is typically used to calibrate a model. However, another common practice is to use responses or output from a high-fidelity model as “truth data” in place of experimental data in a low-fidelity model calibration. This can be done in with a single Bayesian calibration, or it can be done iteratively with the use of experimental design, where an initial set of high-fidelity runs is augmented sequentially to find the next “best” high-fidelity design point at which to run the high-fidelity model to add to the calibration data. The low-fidelity posterior parameter distribution is then updated again using Bayesian calibration. The mutual information is used as the selection criterion to guide the process of high-fidelity data acquisition. In Dakota, design conditions, such as temperature or spatial location, can be specified using so-called configuration variables. The design selection algorithm implemented in Dakota uses a user-specified high-fidelity code to produce the “experimental” or observational data that is used in the calibration of the desired low-fidelity model. The high-fidelity model is dependent only upon the design or configuration variables while the low-fidelity model depends on both the design variables and uncertain model parameters. An example Dakota input file that implements this Bayesian experimental design algorithm is shown in Figures 5.27-5.28. Note that there are three model blocks, one describing the model hierarchy and one each for the high-fidelity and low-fidelity models. There are two variables, interface, and responses blocks such that each model has its own specifications. The low-fidelity variables block contains information about both the design variables, which are specified with continuous state, and the parameters to be updated via Bayes’ Theorem 5.9, which are specified using one of the aleatory uncertain variable types discussed in Section 9.3.1.1. In the high-fidelity variables block, only the continuous state parameters are included. The specifications of the design variables should be consistent in both blocks. Each interface block should point to the appropriate high- or low-fidelity code, and the responses blocks should contain consistent details about the responses from each code. For example, both of the models should return the same number of calibration terms. The mutual information experimental design algorithm is selected by specifying bayesian calibration, queso, and experimental design within the method block of the input file, and the first model block should contain the hierarchical specification of the surrogate keyword. The algorithm starts by performing a Bayesian calibration using a number of data points, specified in Dakota by initial samples. These initial data points can be pulled from external data using the calibration data file keyword in the high-fidelity response block. In this case, num config variables should be specified and set to the number of configuration variables captured in the variables blocks. Furthermore, for use in Bayesian experimental design, calibration data file should contain the configuration variables and the corresponding high-fidelity model responses. Scalar variance information may be included for the calibration data through the use of the experimental variance type or simulation variance command within the high-fidelity responses block. The former is applied to any user-provided data, such as through the calibration data file keyword, while the latter applies only to those high-fidelity model responses produced by the high-fidelity code run by Dakota. Further information can be found in the Dakota Reference Manual [3]. If the number of points taken from this file is less than initial samples, or if no such file is provided, Latin Hypercube Sampling is used to draw samples of the design space, and the high-fidelity model is run at these points to supplement the user-specified data. After this initial calibration, a set of design conditions (i.e. conDakota Version 6.7 User’s Manual generated on November 13, 2017 5.8. BAYESIAN CALIBRATION METHODS 115 environment tabular data method , b a y e s c a l i b r a t i o n queso dram s e e d = 34785 c h a i n s a m p l e s = 500 experimental design initial samples = 2 num candidates = 2 i m p o r t c a n d i d a t e p o i n t s f i l e = ’ d a k o t a b a y e s e x p d e s i g n . cand . dat ’ freeform # ksg2 max hifi evaluations = 1 m o d e l p o i n t e r = ’HIERARCH’ model , i d m o d e l = ’HIERARCH’ v a r i a b l e s p o i n t e r = ’ALL VARS’ surrogate hierarchical o r d e r e d m o d e l f i d e l i t i e s = ’LF ’ ’HF’ model , i d m o d e l = ’LF ’ single interface pointer = ’ lofi IF ’ v a r i a b l e s p o i n t e r = ’ALL VARS’ responses pointer = ’ lofi resp ’ model , i d m o d e l = ’HF’ single interface pointer = ’ hifi IF ’ v a r i a b l e s p o i n t e r = ’CONFIG VARS ’ responses pointer = ’ hifi resp ’ variables , i d v a r i a b l e s = ’ALL VARS’ continuous state = 1 u p p e r b o u n d s = 70 l o w e r b o u n d s = 10 uniform uncertain = 3 upper bounds 0.06 0 lower bounds 0 −8 260 0 variables , i d v a r i a b l e s = ’CONFIG VARS ’ active state continuous state = 1 u p p e r b o u n d s = 70 l o w e r b o u n d s = 10 Figure 5.27: Dakota input file for Bayesian Experimental Design. Dakota Version 6.7 User’s Manual generated on November 13, 2017 116 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES interface , id interface = ’ hifi IF ’ a n a l y s i s d r i v e r s = ’ expdesign high ’ fork interface , id interface = ’ lofi IF ’ a n a l y s i s d r i v e r s = ’ expdesign low ’ fork responses , id responses = ’ lofi resp ’ calibration terms = 1 no gradients no hessians responses , id responses = ’ hifi resp ’ calibration terms = 1 c a l i b r a t i o n d a t a f i l e = ’ d a k o t a b a y e s e x p d e s i g n . dat ’ freeform num config variables = 1 num experiments = 2 e x p e r i m e n t v a r i a n c e t y p e = ’ none ’ # experiment variance type = ’ scalar ’ no gradients no hessians Figure 5.28: Dakota input file for Bayesian Experimental Design. Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.8. BAYESIAN CALIBRATION METHODS 117 figuration variables) of size num candidates is proposed. Users may specify these candidate points through the import candidate points file command. Again, if the number of points in this file is less than num candidates, or if no such file is provided, Latin Hypercube Sampling is used to draw samples of the design space. From these candidate designs, that which maximizes the mutual information with respect to the low-fidelity model parameters is deemed “optimal.” The mutual information is approximated using the low-fidelity model and a knearest neighbor algorithm, as detailed in [95]. This optimal design is used in the high-fidelity model to create a new observation, which is appended to the initial data. This updated data is used to recalculate the Bayesian posterior, and the process repeats until one of three stopping criteria are met. There are two algorithms that may be used to calculate the mutual information, both of which are derived in [90]. The first algorithm discussed therein is used as the default algorithm within Dakota; the second may be selected by including the keyword ksg2 in the Dakota input script. Furthermore, the user may choose to include, during the computation of the mutual information, a stochastic error term on the low-fidelity model responses. This is done by specifying simulation variance in the responses block corresponding to the low-fidelity model. See the Dakota Theory Manual [4] for more information regarding the implementation of the mutual information calculations. There are three criteria by which this algorithm is considered complete. The user may specify max hifi evaluations, which limits the number of high-fidelity model simulations Dakota will run. Note that this does not include any simulations needed to perform the initial Bayesian calibration of the low-fidelity model parameters. Alternatively, if the change in the mutual information from one iteration to the next is sufficiently small or if all candidate points have been exhausted, the algorithm will terminate. Progress of the algorithm will be reported to the screen with the rest of the Dakota output. Furthermore, a summary of the algorithm’s results, including, for each iteration, the optimal design, the mutual information, and the corresponding high-fidelity model response, can be found in the file experimental design output.txt. 5.8.9.1 One-at-a-time Implementation There may be some applications for which the high-fidelity model must be run independently of Dakota. This algorithm may still be implemented in this case, however, it requires some extra work by the user to ensure that the high-fidelity model information is properly communicated to Dakota, as a ”dummy” high-fidelity model code must be supplied to Dakota. The data to be used in the initial Bayesian calibration should be gathered from the high-fidelity model or physical experiment and imported via the calibration data file in the high-fidelity responses block, and extra care should be taken to ensure that initial samples matches the number of experiments in this file. It is also best, for this use-case, to use import candidate points file, with num candidates exactly matching the number of candidate points in the file. By setting max hifi evaluations to zero, Dakota will run the initial calibration of the low-fidelity model, select the “optimal” design from those provided in import candidate points file, and exit without running the “dummy” high-fidelity model code. The selected design will be output to the screen, as well as to experimental design output.txt, as detailed above. The high-fidelity model may then be run offline with the newly selected design point. The user must update calibration data file with the new high-fidelity data when it becomes available, as well as remove the previously selected design point from import candidate points file. Within the Dakota input file, initial samples, num experiments, and num candidates should be correspondingly updated. Dakota may then be run again to yield the next optimal experimental design. It should be noted that the stopping criteria will not be automatically evaluated by Dakota when one-at-a-time implementation is used. The user must determine when the algorithm should be terminated. Dakota Version 6.7 User’s Manual generated on November 13, 2017 118 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES 5.9 Uncertainty Quantification Usage Guidelines The choice of uncertainty quantification method depends on how the input uncertainty is characterized, the computational budget, and the desired output accuracy. The recommendations for UQ methods are summarized in Table 5.3 and are discussed in the remainder of the section. Table 5.3: Guidelines for UQ method selection. Method Classification Sampling Epistemic Desired Problem Characteristics nonsmooth, multimodal response functions; response evaluations are relatively inexpensive smooth, unimodal response functions; larger sets of random variables; estimation of tail probabilities smooth or limited nonsmooth response; multimodal response; low dimensional; estimation of tail probabilities smooth or limited nonsmooth response; multimodal response; low dimensional; estimation of tail probabilities smooth or limited nonsmooth response; multimodal response; low dimensional; estimation of moments or moment-based metrics uncertainties are poorly characterized Mixed UQ some uncertainties are poorly characterized Local reliability Global reliability Adaptive Sampling Stochastic expansions Applicable Methods sampling (Monte Carlo or LHS) local reliability (MV, AMV/AMV2 , AMV+/AMV2 +, TANA, FORM/SORM) global reliability importance sampling, gpais, adaptive sampling, pof darts polynomial chaos, stoch collocation interval: local interval est, global interval est, sampling; BPA: local evidence, global evidence nested UQ (IVP, SOP, DSTE) with epistemic outer loop and aleatory inner loop, sampling Sampling Methods Sampling-based methods are the most robust uncertainty techniques available, are applicable to almost all simulations, and possess rigorous error bounds; consequently, they should be used whenever the function is relatively inexpensive to compute and adequate sampling can be performed. In the case of terascale computational simulations, however, the number of function evaluations required by traditional techniques such as Monte Carlo and Latin hypercube sampling (LHS) quickly becomes prohibitive, especially if tail statistics are needed. Alternatively, one can apply the traditional sampling techniques to a surrogate function approximating the expensive computational simulation (see Section 15.3). However, if this approach is selected, the user should be aware that it is very difficult to assess the accuracy of the results obtained. Unlike the case of surrogate-based local minimization (see Section 14.6.1), there is no simple pointwise calculation to verify the accuracy of the approximate results. This is due to the functional nature of uncertainty quantification, i.e. the accuracy of the surrogate over the entire parameter space needs to be considered, not just around a candidate optimum as in the case of surrogate-based local. This issue especially manifests itself when trying to estimate low probability events such as the catastrophic failure of a system. Reliability Methods Local reliability methods (e.g., MV, AMV/AMV2 , AMV+/AMV2 +, TANA, and FORM/SORM) are more computationally efficient in general than the sampling methods and are effective when applied to reasonably wellbehaved response functions; i.e., functions that are smooth, unimodal, and only mildly nonlinear. They can be Dakota Version 6.7 User’s Manual generated on November 13, 2017 5.9. UNCERTAINTY QUANTIFICATION USAGE GUIDELINES 119 used to provide qualitative sensitivity information concerning which uncertain variables are important (with relatively few function evaluations), or compute full cumulative or complementary cumulative response functions (with additional computational effort). Since they rely on gradient calculations to compute local optima (most probable points of failure), they scale well for increasing numbers of random variables, but issues with nonsmooth, discontinuous, and multimodal response functions are relevant concerns. In addition, even if there is a single MPP and it is calculated accurately, first-order and second-order integrations may fail to accurately capture the shape of the failure domain. In these cases, adaptive importance sampling around the MPP can be helpful. Overall, local reliability methods should be used with some care and their accuracy should be verified whenever possible. An effective alternative to local reliability analysis when confronted with nonsmooth, multimodal, and/or highly nonlinear response functions is efficient global reliability analysis (EGRA). This technique employs Gaussian process global surrogate models to accurately resolve the failure domain and then employs multimodal adaptive importance sampling to resolve the probabilities. For relatively low dimensional problems (i.e, on the order of 10 variables), this method displays the efficiency of local reliability analysis with the accuracy of exhaustive sampling. While extremely promising, this method is still relatively new and is the subject of ongoing refinements as we deploy it to additional applications. Adaptive Sampling Methods There are now a number of methods in Dakota which are tailored to estimating tail probabilities. These methods include both standard importance sampling and Gaussian Process Adaptive Importance Sampling, as well as adaptive sampling and the POF-darts method. These methods are suitable for smooth or limited non-smooth responses, and work well in low dimensions. GPAIS and POF-darts utilize a Gaussian process surrogate model. Stochastic Expansions Methods The next class of UQ methods available in Dakota is comprised of stochastic expansion methods (polynomial chaos and stochastic collocation), which are general purpose techniques provided that the response functions possess finite second order moments. Further, these methods capture the underlying functional relationship between a key response metric and its random variables, rather than just approximating statistics such as mean and standard deviation. This class of methods parallels traditional variational methods in mechanics; in that vein, efforts are underway to compute rigorous error bounds of the approximations produced by the methods. Another strength of these methods is their potential use in a multiphysics environment as a means to propagate the uncertainty through a series of simulations while retaining as much information as possible at each stage of the analysis. The current challenge in the development of these methods, as for other global surrogate-based methods, is effective scaling for large numbers of random variables. Recent advances in adaptive collocation and sparsity detection methods address some of the scaling issues for stochastic expansions. Epistemic Uncertainty Quantification Methods The final class of UQ methods available in Dakota are focused on epistemic uncertainties, or uncertainties resulting from a lack of knowledge. In these problems, the assignment of input probability distributions when data is sparse can be somewhat suspect. One approach to handling epistemic uncertainties is interval analysis (local interval est and global interval est), where a set of intervals on inputs, one interval for each input variable, is mapped to a set of intervals on outputs. To perform this process efficiently, optimization methods can be used. Another related technique is Dempster-Shafer theory of evidence (Dakota methods local evidence and global evidence), where multiple intervals per input variable (which can be overlapping, contiguous, or disjoint) are propagated, again potentially using optimization methods. The choice between local or global optimization methods for interval computation is governed by the same issues described in Section 6.4. Mixed Aleatoric and Epistemic Methods For problems with a mixture of epistemic and aleatoric uncertainties, it is desirable to segregate the two uncertainty types within a nested analysis, allowing stronger probabilistic inferences for the portion of the problem Dakota Version 6.7 User’s Manual generated on November 13, 2017 120 CHAPTER 5. UNCERTAINTY QUANTIFICATION CAPABILITIES where they are appropriate. In this nested approach, an outer epistemic level selects realizations of epistemic parameters (augmented variables) and/or realizations of random variable distribution parameters (inserted variables). These realizations define the objective probabilistic analysis to be performed on the inner aleatoric level. In the case where the outer loop involves propagation of subjective probability, the nested approach is known as second-order probability and the study generates a family of CDF/CCDF respresentations known as a “horse tail” plot. In the case where the outer loop is an interval propagation approach (local interval est or global interval est), the nested approach is known as interval-valued probability (see also Section 8.5) . In the case where the outer loop is an evidence-based approach (local evidence or global evidence), the approach generates epistemic belief and plausibility bounds on aleatory statistics. Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 6 Optimization Capabilities Optimization algorithms work to minimize (or maximize) an objective function, typically calculated by the user simulation code, subject to constraints on design variables and responses. Available approaches in Dakota include well-tested and proven gradient-based, derivative-free local, and global methods for use in science and engineering design applications. Dakota also offers more advanced algorithms, e.g., to manage multi-objective optimization or perform surrogate-based minimization. This chapter summarizes optimization problem formulation, standard algorithms available in Dakota (mostly through included third-party libraries, see 6.5), some advanced capabilities, and offers usage guidelines. 6.1 Optimization Formulations This section provides a basic introduction to the mathematical formulation of optimization, problems. The primary goal of this section is to introduce terms relating to these topics, and is not intended to be a description of theory or numerical algorithms. For further details, consult [7], [57], [72], [107], and [140]. A general optimization problem is formulated as follows: minimize: f (x) x ∈ . will be written and Dakota will exit if the active subspace model is used with a non-compatible method. The set of compatible methods will be expanded in future releases. The active subspace implementation in Dakota 6.4 first transforms uncertain variables to standard normal distributions using a Nataf transformm before forming the subspace. This is a nonlinear transformation for non-normally distributed uncertain variables and may potentially decrease sparse structure in a fullspace model. Future Dakota releases will not use this transformation and should perform better in the general case. The only required keyword when using a subspace model is the actual model pointer which points to the underlying model (specified by its id model) on which to build the subspace. The subspace model requires either analytical (preferred) or numerical gradients of the response functions. The active subspace model first samples the gradient of the fullspace model. The number of gradient samples can be specified with initial samples. The gradient samples are compiled into the columns of a matrix. A singular value decomposition is performed of the derivative matrix and the resulting singular values and vectors are used to determine the basis vectors and size of the active subspace. Constantine [18] recommends choosing initial samples such that: initial samples = αk log(m), where α is an oversampling factor between 2 and 10, k is the number of singular values to approximate, and m is the number of fullspace variables. To ensure accurate results, k should be greater than the estimated subspace size determined by one of the truncation methods described below. Dakota has everal metrics to estimate the size of an active subspace: Dakota Version 6.7 User’s Manual generated on November 13, 2017 170 CHAPTER 8. MODELS • constantine (default) • bing li • energy • cross validation Additionally, if the desired subspace size is known it can be explicitly selected using the input parameter dimension. The constantine and bing li truncation methods both use bootstrap sampling of the compiled derivative matrix to estimate an active subspace size. The number of bootstrap samples used with these methods can be specified with the keyword bootstrap samples, but typically the default value works well. The energy method computes the number of bases so that the subspace representation accounts for all but a maximum percentage (specified as a decimal) of the total eigenvalue energy. This value is specified using the truncation tolerance keyword. For more information on active subspaces please consult the Theory Manual [4] and/or references [21, 19, 18]. Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 9 Variables 9.1 Overview The variables specification in a Dakota input file specifies the parameter set to be iterated by a particular method. In the case of an optimization study, these variables are adjusted in order to locate an optimal design; in the case of parameter studies/sensitivity analysis/design of experiments, these parameters are perturbed to explore the parameter space; and in the case of uncertainty analysis, the variables are associated with distribution/interval characterizations which are used to compute corresponding distribution/interval characterizations for response functions. To accommodate these and other types of studies, Dakota supports design, uncertain, and state variable types for continuous and discrete variable domains. Uncertain types can be further categorized as either aleatory or epistemic, and discrete domains can include discrete range, discrete integer set, discrete string set, and discrete real set. This chapter will present a brief overview of the main types of variables and their uses, as well as cover some user issues relating to file formats and the active set vector. For a detailed description of variables section syntax and example specifications, refer to the variables keywords in the Dakota Reference Manual [3]. 9.2 Design Variables Design variables are those variables which are modified in the course of determining an optimal design. These variables may be continuous (real-valued between bounds), discrete range (integer-valued between bounds), discrete set of integers (integer-valued from finite set), discrete set of strings (string-valued from finite set), and discrete set of reals (real-valued from finite set). 9.2.1 Continuous Design Variables The most common type of design variables encountered in engineering applications are of the continuous type. These variables may assume any real value (e.g., 12.34, -1.735e+07) within their bounds. All but a handful of the optimization algorithms in Dakota support continuous design variables exclusively. 172 CHAPTER 9. VARIABLES 9.2.2 Discrete Design Variables Engineering design problems may contain discrete variables such as material types, feature counts, stock gauge selections, etc. These variables may assume only a fixed number of values, as compared to a continuous variable which has an uncountable number of possible values within its range. Discrete variables may involve a range of consecutive integers (x can be any integer between 1 and 10), a set of integer values (x can be 101, 212, or 355), a set of string values (x can be ’direct’, ’gmres’, or ’jacobi’), or a set of real values (e.g., x can be identically 4.2, 6.4, or 8.5). Discrete variables may be classified as either “categorical” or “noncategorical.” In the latter noncategorical case, the discrete requirement can be relaxed during the solution process since the model can still compute meaningful response functions for values outside the allowed discrete range or set. For example, a discrete variable representing the thickness of a structure is generally a noncategorical variable since it can assume a continuous range of values during the algorithm iterations, even if it is desired to have a stock gauge thickness in the end. In the former categorical case, the discrete requirement cannot be relaxed since the model cannot obtain a solution for values outside the range or set. For example, feature counts are generally categorical discrete variables, since most computational models will not support a non-integer value for the number of instances of some feature (e.g., number of support brackets). Dakota supports a categorical specification to indicate which discrete real and discrete integer variables are restricted vs. relaxable. String variables cannot be relaxed. Gradient-based optimization methods cannot be directly applied to problems with discrete variables since derivatives only exist for a variable continuum. For problems with noncategorical variables, the experimental branch and bound capability (branch and bound) can be used to relax the discrete requirements and apply gradientbased methods to a series of generated subproblems. For problems with categorical variables, nongradient-based methods (e.g., coliny ea) are commonly used; however, most of those methods do not take advantage of any structure that may be associated with the categorical variables. The exception is mesh adaptive search. If it is possible to define a subjective relationship between the different values a given categorical variable can take on, the user can communicate that relationship in the form of an adjacency matrix. The mesh adaptive search will take that relationship into consideration. Further documentation can be found in [3] under the keywords adjacency matrix and neighbor order. Branch and bound techniques are discussed in Section 14.5 and nongradient-based methods are further described in Chapter 6. In addition to engineering applications, many non-engineering applications in the fields of scheduling, logistics, and resource allocation contain discrete design parameters. Within the Department of Energy, solution techniques for these problems impact programs in stockpile evaluation and management, production planning, nonproliferation, transportation (routing, packing, logistics), infrastructure analysis and design, energy production, environmental remediation, and tools for massively parallel computing such as domain decomposition and meshing. 9.2.2.1 Discrete Design Integer Variables There are two types of discrete design integer variables supported by Dakota. • The discrete design range specification supports a range of consecutive integers between specified lower bounds and upper bounds. • The discrete design set integer specification supports a set of enumerated integer values through the elements specification. The set of values specified is stored internally as an STL set container, which enforces an ordered, unique representation of the integer data. Underlying this set of ordered, unique integers is a set of indices that run from 0 to one less than the number of set values. These indices are used by some iterative algorithms (e.g., parameter studies, SCOLIB iterators) for simplicity in discrete value Dakota Version 6.7 User’s Manual generated on November 13, 2017 9.3. UNCERTAIN VARIABLES 173 enumeration when the actual corresponding set values are immaterial. In the case of parameter studies, this index representation is exposed through certain step and partition control specifications (see Chapter 3). 9.2.2.2 Discrete Design String Variables There is one type of discrete design string variable supported by Dakota. • The discrete design set string specification supports a set of enumerated string values through the elements specification. As for the discrete integer set variables described in Section 9.2.2.1, internal storage of the set values is ordered and unique and an underlying index representation is exposed for the specification of some iterative algorithms. Each string element value must be quoted in the Dakota input file and may contain alphanumeric, dash, underscore, and colon. White space, quote characters, and backslash/meta-characters are not permitted. 9.2.2.3 Discrete Design Real Variables There is one type of discrete design real variable supported by Dakota. • The discrete design set real specification specification supports a set of enumerated real values through the set values specification. As for the discrete integer set variables described in Section 9.2.2.1, internal storage of the set values is ordered and unique and an underlying index representation is exposed for the specification of some iterative algorithms. 9.3 Uncertain Variables Deterministic variables (i.e., those with a single known value) do not capture the behavior of the input variables in all situations. In many cases, the exact value of a model parameter is not precisely known. An example of such an input variable is the thickness of a heat treatment coating on a structural steel I-beam used in building construction. Due to variability and tolerances in the coating process, the thickness of the layer is known to follow a normal distribution with a certain mean and standard deviation as determined from experimental data. The inclusion of the uncertainty in the coating thickness is essential to accurately represent the resulting uncertainty in the response of the building. 9.3.1 Aleatory Uncertain Variables Aleatory uncertainties are irreducible variabilities inherent in nature. They are characterized by having a sufficiently rich set of data as to allow modeling using probability distributions, and probabilistic methods are commonly used for propagating input aleatory uncertainties described by probability distribution specifications. The two following sections describe the continuous and discrete aleatory uncertain variables supported by Dakota. For aleatory random variables, Dakota supports a user-supplied correlation matrix to provide correlations among the input variables. By default, the correlation matrix is set to the identity matrix, i.e., no correlation among the uncertain variables. Dakota Version 6.7 User’s Manual generated on November 13, 2017 174 CHAPTER 9. VARIABLES For additional information on random variable probability distributions, refer to [73] and [132]. Refer to the Dakota Reference Manual [3] for more detail on the uncertain variable specifications and to Chapter 5 for a description of methods available to quantify the uncertainty in the response. 9.3.1.1 Continuous Aleatory Uncertain Variables • Normal: a probability distribution characterized by a mean and standard deviation. Also referred to as Gaussian. Bounded normal is also supported by some methods with an additional specification of lower and upper bounds. • Lognormal: a probability distribution characterized by a mean and either a standard deviation or an error factor. The natural logarithm of a lognormal variable has a normal distribution. Bounded lognormal is also supported by some methods with an additional specification of lower and upper bounds. • Uniform: a probability distribution characterized by a lower bound and an upper bound. Probability is constant between the bounds. • Loguniform: a probability distribution characterized by a lower bound and an upper bound. The natural logarithm of a loguniform variable has a uniform distribution. • Triangular: a probability distribution characterized by a mode, a lower bound, and an upper bound. • Exponential: a probability distribution characterized by a beta parameter. • Beta: a flexible probability distribution characterized by a lower bound and an upper bound and alpha and beta parameters. The uniform distribution is a special case. • Gamma: a flexible probability distribution characterized by alpha and beta parameters. The exponential distribution is a special case. • Gumbel: the Type I Largest Extreme Value probability distribution. Characterized by alpha and beta parameters. • Frechet: the Type II Largest Extreme Value probability distribution. Characterized by alpha and beta parameters. • Weibull: the Type III Smallest Extreme Value probability distribution. Characterized by alpha and beta parameters. • Histogram Bin: an empirically-based probability distribution characterized by a set of (x, y) pairs that map out histogram bins (a continuous interval with associated bin count). 9.3.1.2 Discrete Aleatory Uncertain Variables The following types of discrete aleatory uncertain variables are available: • Poisson: integer-valued distribution used to predict the number of discrete events that happen in a given time interval. • Binomial: integer-valued distribution used to predict the number of failures in a number of independent tests or trials. Dakota Version 6.7 User’s Manual generated on November 13, 2017 9.4. STATE VARIABLES 175 • Negative Binomial: integer-valued distribution used to predict the number of times to perform a test to have a target number of successes. • Geometric: integer-valued distribution used to model the number of successful trials that might occur before a failure is observed. • Hypergeometric: integer-valued distribution used to model the number of failures observed in a set of tests that has a known proportion of failures. • Histogram Point (integer, string, real): an empirically-based probability distribution characterized by a set of integer-valued (i, c), string-valued (s, c), and/or real-valued r, c pairs that map out histogram points (each a discrete point value i, s, or r, with associated count c). 9.3.2 Epistemic Uncertain Variables Epistemic uncertainties are reducible uncertainties resulting from a lack of knowledge. For epistemic uncertainties, data is generally sparse, making the use of probability theory questionable and leading to nonprobabilistic methods based on interval or fuzzy specifications. Dakota currently supports the following epistemic uncertain variable types. 9.3.2.1 Continuous Epistemic Uncertain Variables • Continuous Interval: a real-valued interval-based specification characterized by sets of lower and upper bounds and Basic Probability Assignments (BPAs) associated with each interval. The intervals may be overlapping, contiguous, or disjoint, and a single interval (with probability = 1) per variable is an important special case. The interval distribution is not a probability distribution, as the exact structure of the probabilities within each interval is not known. It is commonly used with epistemic uncertainty methods. 9.3.2.2 Discrete Epistemic Uncertain Variables • Discrete Interval: an integer-valued variant of the Continuous Interval described above ( 9.3.2.1). • Discrete Set (integer, string, and real): Similar to discrete design set variables 9.2.2, these epistemic variables admit a finite number of values (elements) for type integer, string, or real, each with an associated probability. 9.4 State Variables State variables consist of “other” variables which are to be mapped through the simulation interface, in that they are not to be used for design and they are not modeled as being uncertain. State variables provide a convenient mechanism for parameterizing additional model inputs which, in the case of a numerical simulator, might include solver convergence tolerances, time step controls, or mesh fidelity parameters. For additional model parameterizations involving strings (e.g., “mesh1.exo”), refer to the analysis components specification described in Section 9.6.1 and in the Interface Commands chapter of the Dakota Reference Manual [3]. Similar to the design variables discussed in Section 9.2, state variables can be specified with a continuous range (real-valued between bounds), a discrete range (integer-valued between bounds), a discrete integer-valued set, a discrete string-valued set, or a discrete real-valued set. Dakota Version 6.7 User’s Manual generated on November 13, 2017 176 CHAPTER 9. VARIABLES State variables, as with other types of variables, are viewed differently depending on the method in use. Since these variables are neither design nor uncertain variables, algorithms for optimization, least squares, and uncertainty quantification do not iterate on these variables; i.e., they are not active and are hidden from the algorithm. However, Dakota still maps these variables through the user’s interface where they affect the computational model in use. This allows optimization, least squares, and uncertainty quantification studies to be executed under different simulation conditions (which will result, in general, in different results). Parameter studies and design of experiments methods, on the other hand, are general-purpose iterative techniques which do not draw a distinction between variable types. They include state variables in the set of variables to be iterated, which allows these studies to explore the effect of state variable values on the response data of interest. In the future, state variables might be used in direct coordination with an optimization, least squares, or uncertainty quantification algorithm. For example, state variables could be used to enact model adaptivity through the use of a coarse mesh or loose solver tolerances in the initial stages of an optimization with continuous model refinement as the algorithm nears the optimal solution. 9.5 9.5.1 Management of Mixed Variables by Iterator View As alluded to in the previous section, the iterative method selected for use in Dakota partially determines what subset, or view, of the variables data is active in the iteration. (Section 9.5.3 contains a discussion of how user overrides, response function type, and method are used to determine active variable view.) The general case of having a mixture of various different types of variables is supported within all of the Dakota methods even though certain methods will only modify certain types of variables (e.g., optimizers and least squares methods only modify design variables, and uncertainty quantification methods typically only utilize uncertain variables). This implies that variables which are not under the direct control of a particular iterator will be mapped through the interface in an unmodified state. This allows for a variety of parameterizations within the model in addition to those which are being used by a particular iterator, which can provide the convenience of consolidating the control over various modeling parameters in a single file (the Dakota input file). An important related point is that the variable set that is active with a particular iterator is the same variable set for which derivatives are typically computed (see Section 11.3). There are certain situations where the user may want to explicitly control the subset of variables that is considered active for a certain Dakota method. This is done by specifying the keyword active in the variables specification block, followed by one of the following: all, design, uncertain, aleatory, epistemic, or state. Specifying one of these subsets of variables will allow the Dakota method to operate on the specified variable types and override the defaults. For example, the default behavior for a nondeterministic sampling method is to sample the uncertain variables. However, if the user specified active all in the variables specification block, the sampling would be performed over all variables (e.g. design and state variables as well as uncertain variables). This may be desired in situations such as surrogate based optimization under uncertainty, where a surrogate may be built over both design and uncertain variables. Another situation where one may want the fine-grained control available by specifying one of these variable types is when one has state variables but only wants to sample over the design variables when constructing a surrogate model. Finally, more sophisticated uncertainty studies may involve various combinations of epistemic vs. aleatory variables being active in nested models. Dakota Version 6.7 User’s Manual generated on November 13, 2017 9.6. DAKOTA PARAMETERS FILE DATA FORMAT 9.5.2 177 Domain Another control that the user can specify in the variables specification block controls the domain type. We have two domains currently: mixed and relaxed. Both domain types can have design, uncertain, and state variables. The domain specifies how the discrete variables are treated. If the user specifies mixed in the variable specification block, the continuous and discrete variables are treated separately. If the user specifies relaxed in the variable specification block, the discrete variables are relaxed and treated as continuous variables. This may be useful in optimization problems involving both continuous and discrete variables when a user would like to use an optimization method that is designed for continuous variable optimization. All Dakota methods have a default value of mixed for the domain type except for the branch-and-bound method which has a default domain type of relaxed. Note that the branch-and-bound method is experimental and still under development at this time. 9.5.3 Precedence If the user does not specify any explicit override of the active view of the variables, Dakota then considers the response function specification. If the user specifies objective functions or calibration terms in the response specification block, the active variables will be the design variables. If the user specifies the more generic response type, response functions, general response functions do not have a specific interpretation the way objective functions or calibration terms do. In the case of generic response functions, Dakota then tries to infer the active view from the method. If the method is a parameter study, or any of the methods available under dace, psuade, or fsu methods, the active view is set to all variables. For uncertainty quantification methods, if the method is sampling, then the view is set to aleatory if only aleatory variables are present, epistemic if only epistemic variables are present, or uncertain (covering both aleatory and epistemic) if both are present. If the uncertainty method involves interval estimation or evidence calculations, the view is set to epistemic. For other uncertainty quantification methods not mentioned in the previous sentences (e.g., reliability methods or stochastic expansion methods), the view is set to aleatory. Finally, for verification studies using the Richardson extrapolation method, the active view is set to state. Note that in surrogate-based optimization, where the surrogate is built on points defined by the method defined by the dace method pointer, the sampling used to generate the points is performed only over the design variables as a default unless otherwise specified (e.g. state variables will not be sampled for surrogate construction). With respect to domain type, if the user does not specify an explicit override of mixed or relaxed, Dakota infers the domain type from the method. As mentioned above, all methods currently use a mixed domain as a default, except the branch-and-bound method which is under development. 9.6 Dakota Parameters File Data Format Simulation interfaces which employ system calls and forks to create separate simulation processes must communicate with the simulation code through the file system. This is accomplished through the reading and writing of parameters and results files. Dakota uses a particular format for this data input/output. Depending on the user’s interface specification, Dakota will write the parameters file in either standard or APREPRO format (future XML formats are planned). The former option uses a simple “value tag” format, whereas the latter option uses a “{ tag = value }” format for compatibility with the APREPRO utility [122] (as well as DPrePro, BPREPRO, and JPrePost variants). Dakota Version 6.7 User’s Manual generated on November 13, 2017 178 CHAPTER 9. VARIABLES variables (i = 1 (i = 1 (i = 1 (i = 1 (i = 1 (i = 1 (i = 1 (i = 1 (i = 1 (i = 1 (i = 1 (i = 1 (i = 1 (i = 1 (i = 1 (i = 1 functions ASV_i:label_responsei derivative_variables DVV_i:label_cdvi analysis_components AC_i:analysis_driver_namei eval_id to to to to to to to to to to to to to to to to ncdv ) nddiv ) nddsv ) nddrv ) ncauv ) ndauiv ) ndausv ) ndaurv ) nceuv ) ndeuiv ) ndeusv ) ndeurv ) ncsv ) ndsiv ) ndssv ) ndsrv ) (i = 1 to m) (i = 1 to p) (i = 1 to q) Figure 9.1: Parameters file data format - standard option. 9.6.1 Parameters file format (standard) Prior to invoking a simulation, Dakota creates a parameters file which contains the current parameter values and a set of function requests. The standard format for this parameters file is shown in Figure 9.1. Integer values are denoted by “ ”, “ ” denotes a double precision value, and “ ” denotes a string value. Each of the colored blocks (black for variables, blue for active set vector, red for derivative variables vector, and green for analysis components) denotes an array which begins with an array length and a descriptive tag. These array lengths are useful for dynamic memory allocation within a simulator or filter program. The first array for variables begins with the total number of variables (n) with its identifier string “variables.” The next n lines specify the current values and descriptors of all of the variables within the parameter set in the following order: continuous design, discrete integer design (integer range, integer set), discrete string design (string set), discrete real design (real set), continuous aleatory uncertain (normal, lognormal, uniform, loguniform, triangular, exponential, beta, gamma, gumbel, frechet, weibull, histogram bin), discrete integer aleatory uncertain (poisson, binomial, negative binomial, geometric, hypergeometric, histogram point integer), discrete string aleatory uncertain (histogram point string), discrete real aleatory uncertain (histogram point real), continuous epistemic uncertain (real interval), discrete integer epistemic uncertain (interval, then set), discrete string epistemic uncertain (set), discrete real epistemic uncertain (set), continuous state, discrete integer state (integer range, integer set), discrete string state, and discrete real state (real set) variables. This ordering is consistent with the lists in Sections 9.2.2.1, 9.3.1.1 and 9.3.1.2 and the specification order in dakota.input.summary. The lengths Dakota Version 6.7 User’s Manual generated on November 13, 2017 9.6. DAKOTA PARAMETERS FILE DATA FORMAT 179 of these vectors add to a total of n (that is, n = ncdv + nddiv + nddsv + nddrv + ncauv + ndauiv + ndausv + ndaurv + nceuv + ndeuiv + ndeusv + ndeurv + ncsv + ndsiv + ndssv + ndsrv ). If any of the variable types are not present in the problem, then its block is omitted entirely from the parameters file. The tags are the variable descriptors specified in the user’s Dakota input file, or if no descriptors have been specified, default descriptors are used. The second array for the active set vector (ASV) begins with the total number of functions (m) and its identifier string “functions.” The next m lines specify the request vector for each of the m functions in the response data set followed by the tags “ASV i:label response”, where the label is either a user-provided response descriptor or a default-generated one. These integer codes indicate what data is required on the current function evaluation and are described further in Section 9.7. The third array for the derivative variables vector (DVV) begins with the number of derivative variables (p) and its identifier string “derivative variables.” The next p lines specify integer variable identifiers followed by the tags “DVV i:label cdv”. These integer identifiers are used to identify the subset of variables that are active for the calculation of derivatives (gradient vectors and Hessian matrices), and correspond to the list of variables in the first array (e.g., an identifier of 2 indicates that the second variable in the list is active for derivatives). The labels are again taken from user-provided or default variable descriptors. The final array for the analysis components (AC) begins with the number of analysis components (q) and its identifier string “analysis components.” The next q lines provide additional strings for use in specializing a simulation interface followed by the tags “AC i:analysis driver name”, where analysis driver name indicates the driver associated with this component. These strings are specified in a user’s input file for a set of analysis drivers using the analysis components specification. The subset of the analysis components used for a particular analysis driver is the set passed in a particular parameters file. The final entry eval id in the parameters file is the evaluation ID, by default an integer indicating interface evaluation ID number. When hierarchical tagging is enabled as described in 10.5.2, the identifier will be a colonseparated string, e.g., 4:9:2. Several standard-format parameters file examples are shown in Section 10.6. 9.6.2 Parameters file format (APREPRO) For the APREPRO format option, the same data is present and the same ordering is used as in the standard format. The only difference is that values are associated with their tags within “{ tag = value }” constructs as shown in Figure 9.2. An APREPRO-format parameters file example is shown in Section 10.6. The use of the APREPRO format option allows direct usage of these parameters files by the APREPRO utility, which is a file pre-processor that can significantly simplify model parameterization. Similar pre-processors include DPrePro, BPREPRO, and JPrePost. [Note: APREPRO is a Sandia-developed pre-processor that is not currently distributed with Dakota. DPrePro is a Perl script distributed with Dakota that performs many of the same functions as APREPRO, and is optimized for use with Dakota parameters files in either format. BPREPRO and JPrePost are additional Perl and JAVA tools, respectively, in use at other sites.] When a parameters file in APREPRO format is included within a template file (using an include directive), the APREPRO utility recognizes these constructs as variable definitions which can then be used to populate targets throughout the template file [122]. DPrePro, conversely, does not require the use of includes since it processes the Dakota parameters file and template simulation file separately to create a simulation input file populated with the variables data. Dakota Version 6.7 User’s Manual generated on November 13, 2017 180 CHAPTER 9. VARIABLES { { { { { { { { { { { { { { { { { { { { { { { { DAKOTA_VARS = } = } (i = 1 = } (i = 1 = } (i = 1 = } (i = 1 = } (i = 1 = } (i = 1 = } (i = 1 = } (i = 1 = } (i = 1 = } (i = 1 = } (i = 1 = } (i = 1 = } (i = 1 = } (i = 1 = } (i = 1 = } (i = 1 DAKOTA_FNS = } ASV_i:label_responsei = } DAKOTA_DER_VARS = } DVV_i:label_cdvi = } DAKOTA_AN_COMPS = } AC_i:analysis_driver_namei = } DAKOTA_EVAL_ID = } to to to to to to to to to to to to to to to to ncdv ) nddiv ) nddsv ) nddrv ) ncauv ) ndauiv ) ndausv ) ndaurv ) nceuv ) ndeuiv ) ndeusv ) ndeurv ) ncsv ) ndsiv ) ndssv ) ndsrv ) (i = 1 to m) (i = 1 to p) (i = 1 to q) Figure 9.2: Parameters file data format - APREPRO option. Dakota Version 6.7 User’s Manual generated on November 13, 2017 9.7. THE ACTIVE SET VECTOR 181 Table 9.1: Active set vector integer codes. Integer Code 7 6 5 4 3 2 1 0 9.7 Binary representation 111 110 101 100 011 010 001 000 Meaning Get Hessian, gradient, and value Get Hessian and gradient Get Hessian and value Get Hessian Get gradient and value Get gradient Get value No data required, function is inactive The Active Set Vector The active set vector contains a set of integer codes, one per response function, which describe the data needed on a particular execution of an interface. Integer values of 0 through 7 denote a 3-bit binary representation of all possible combinations of value, gradient, and Hessian requests for a particular function, with the most significant bit denoting the Hessian, the middle bit denoting the gradient, and the least significant bit denoting the value. The specific translations are shown in Table 9.1. The active set vector in Dakota gets its name from managing the active set, i.e., the set of functions that are active on a particular function evaluation. However, it also manages the type of data that is needed for functions that are active, and in that sense, has an extended meaning beyond that typically used in the optimization literature. 9.7.1 Active set vector control Active set vector control may be turned off to allow the user to simplify the supplied interface by removing the need to check the content of the active set vector on each evaluation. The Interface Commands chapter in the Dakota Reference Manual [3] provides additional information on this option (deactivate active set vector). Of course, this option trades some efficiency for simplicity and is most appropriate for those cases in which only a relatively small penalty occurs when computing and returning more data than may be needed on a particular function evaluation. Dakota Version 6.7 User’s Manual generated on November 13, 2017 182 CHAPTER 9. VARIABLES Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 10 Interfaces 10.1 Overview The interface specification in a Dakota input file controls details of function evaluations. The mechanisms currently in place for function evaluations involve interfacing with one or more computational simulation codes, computing algebraic mappings (refer to Section 16.1), or a combination of the two. This chapter will focus on mechanisms for simulation code invocation, starting with interface types in Section 10.2 and followed by a guide to constructing simulation-based interfaces in Section 10.3. This chapter also provides an overview of simulation interface components, covers issues relating to file management, and presents a number of example data mappings. For a detailed description of interface specification syntax, refer to the interface commands chapter in the Dakota Reference Manual [3]. 10.2 Simulation Interfaces The invocation of a simulation code is performed using either system calls or forks or via direct linkage. In the system call and fork cases, a separate process is created for the simulation and communication between Dakota and the simulation occurs through parameter and response files. For system call and fork interfaces, the interface section must specify the details of this data transfer. In the direct case, a separate process is not created and communication occurs in memory through a prescribed API. Sections 10.2.1 through 10.2.5 provide information on the simulation interfacing approaches. 10.2.1 Direct Simulation Interface The direct interface may be used to invoke simulations that are linked into the Dakota executable. This interface eliminates overhead from process creation and file I/O and can simplify operations on massively parallel computers. These advantages are balanced with the practicality of converting an existing simulation code into a library with a subroutine interface. Sandia codes for structural dynamics (Salinas), computational fluid dynamics (Sage), and circuit simulation (Xyce) and external codes such as Phoenix Integration’s ModelCenter framework and The Mathworks’ Matlab have been linked in this way, and a direct interface to Sandia’s SIERRA multiphysics frame- 184 CHAPTER 10. INTERFACES work is under development. In the latter case, the additional effort is particularly justified since SIERRA unifies an entire suite of physics codes. [Note: the “sandwich implementation” of combining a direct interface plug-in with Dakota’s library mode is discussed in the Dakota Developers Manual [2]]. In addition to direct linking with simulation codes, the direct interface also provides access to internal polynomial test functions that are used for algorithm performance and regression testing. The following test functions are available: cantilever, cyl head, log ratio, rosenbrock, short column, and text book (including text book1, text book2, text book3, and text book ouu). While these functions are also available as external programs in the Dakota/test directory, maintaining internally linked versions allows more rapid testing. See Chapter 20 for additional information on several of these test problems. An example input specification for a direct interface follows: interface, direct analysis_driver = ’rosenbrock’ Additional specification examples are provided in Section 2.3 and additional information on asynchronous usage of the direct function interface is provided in Section 17.2.1.1. Guidance for usage of some particular direct simulation interfaces is in Section 16.3 and the details of adding a simulation code to the direct interface are provided in Section 16.2. 10.2.2 System Call Simulation Interface Users are strongly encouraged to use the fork simulation interface if possible, though the system interface is still supported for portability and backward compatibility. The system call approach invokes a simulation code or simulation driver by using the system function from the standard C library [87]. In this approach, the system call creates a new process that communicates with Dakota through parameter and response files. The system call approach allows the simulation to be initiated via its standard invocation procedure (as a “black box”) and then coordinated with a variety of tools for pre- and post-processing. This approach has been widely used in previous studies [40, 42, 34]. The system call approach involves more process creation and file I/O overhead than the direct function approach, but this extra overhead is usually insignificant compared with the cost of a simulation. An example of a system call interface specification follows: interface, system analysis_driver = ’text_book’ parameters_file = ’text_book.in’ results_file = ’text_book.out’ file_tag file_save Information on asynchronous usage of the system interface is provided in Section 17.2.1.2. 10.2.3 Fork Simulation Interface The fork simulation interface uses the fork, exec, and wait families of functions to manage simulation codes or simulation drivers. (In a native MS Windows version of Dakota, similar Win32 functions, such as spawnvp(), are used instead.) Calls to fork or vfork create a copy of the Dakota process, execvp replaces this copy with the simulation code or driver process, and then Dakota uses the wait or waitpid functions to wait for completion of the new process. Transfer of variables and response data between Dakota and the simulator code or driver occurs through the file system in exactly the same manner as for the system call interface. An example of a fork interface specification follows: Dakota Version 6.7 User’s Manual generated on November 13, 2017 10.2. SIMULATION INTERFACES interface, fork input_filter output_filter analysis_driver parameters_file results_file file_tag 185 = = = = = ’test_3pc_if’ ’test_3pc_of’ ’test_3pc_ac’ ’tb.in’ ’tb.out’ More detailed examples of using the fork call interface are provided in Section 2.3.5.1 and in Section 10.3, and information on asynchronous usage of the fork call interface is provided in Section 17.2.1.3. 10.2.4 Syntax for Filter and Driver Strings With the fork interface, and on most systems, with the system interface as well, the string values supplied for input filter, output filter, and analysis driver can involve simple Bourne-shell syntax for specifying environment values that the filter or driver will see. For example, analysis_driver = ’opfile=myspec outlev=2 mydriver’ would cause mydriver to be invoked with environment variables opfile and outlev having the values “myspec” and “2”, respectively. If the driver is a shell script, it can access these values as $opfile and $outlev; a compiled driver can obtain these values from a function; drivers written in C or C++ can use the standard getenv function (e.g., invoking getenv("opfile")). Both the values assigned to environment variables and name of the file invoked as filter or driver can contain spaces, provided that the values in question are quoted. Within strings delimited by single quotes, you can use double quotes for quoting, and vice versa. For instance, analysis_driver = ’opfile="my spec" "my driver"’ and analysis_driver = "opfile=’my spec’ ’my driver’" both specify a driver named “my driver” and value “my spec” for $opfile. 10.2.5 Fork or System Call: Which to Use? The primary operational difference between the fork and system call simulation interfaces is that, in the fork interface, the fork/exec functions return a process identifier that the wait/waitpid functions can use to detect the completion of a simulation for either synchronous or asynchronous operations. The system call simulation interface, on the other hand, must use a response file detection scheme for this purpose in the asynchronous case. Thus, an important advantage of the fork interface over the system call interface is that it avoids the potential of a file race condition when employing asynchronous local parallelism (refer to Section 17.2.1). This condition can occur when the responses file has been created but the writing of the response data set to this file has not been completed (see Section 17.2.1.2). While significant care has been taken to manage this file race condition in the system call case, the fork interface still has the potential to be more robust when performing function evaluations asynchronously. Dakota Version 6.7 User’s Manual generated on November 13, 2017 186 CHAPTER 10. INTERFACES Another advantage of the fork interface is that it has additional asynchronous capabilities when a function evaluation involves multiple analyses. As shown in Table 17.1, the fork interface supports asynchronous local and hybrid parallelism modes for managing concurrent analyses within function evaluations, whereas the system call interface does not. These additional capabilities again stem from the ability to track child processes by their process identifiers. The only disadvantage to the fork interface compared with the system interface is that the fork/exec/wait functions are not part of the standard C library, whereas the system function is. As a result, support for implementations of the fork/exec/wait functions can vary from platform to platform. At one time, these commands were not available on some of Sandia’s massively parallel computers. However, in the more mainstream UNIX environments, availability of fork/exec/wait should not be an issue. In summary, the system call interface has been a workhorse for many years and is well tested and proven, but the fork interface supports additional capabilities and is recommended when managing asynchronous simulation code executions. Having both interfaces available has proven to be useful on a number of occasions and they will both continue to be supported for the foreseeable future. 10.3 Building a Black-Box Interface to a Simulation Code To interface a simulation code to Dakota using one of the black-box interfaces (system call or fork), pre- and post-processing functionality typically needs to be supplied (or developed) in order to transfer the parameters from Dakota to the simulator input file and to extract the response values of interest from the simulator’s output file for return to Dakota (see Figures 1.1 and 10.5). This is often managed through the use of scripting languages, such as C-shell [6], Bourne shell [13], Perl [144], or Python [99]. While these are common and convenient choices for simulation drivers/filters, it is important to recognize that any executable file can be used. If the user prefers, the desired pre- and post-processing functionality may also be compiled or interpreted from any number of programming languages (C, C++, F77, F95, JAVA, Basic, etc.). In the Dakota/examples/script interfaces/generic directory, a simple example uses the Rosenbrock test function as a mock engineering simulation code. Several scripts have been included to demonstrate ways to accomplish the pre- and post-processing needs. Actual simulation codes will, of course, have different pre- and post-processing requirements, and as such, this example serves only to demonstrate the issues associated with interfacing a simulator. Modifications will almost surely be required for new applications. 10.3.1 Generic Script Interface Files The Dakota/examples/script interfaces/generic directory contains four important files: dakota rosenbrock.in (the Dakota input file), simulator script (the simulation driver script), dprepro (a pre-processing utility), and templatedir/ros.template (a template simulation input file). The file dakota rosenbrock.in specifies the study that Dakota will perform and, in the interface section, describes the components to be used in performing function evaluations. In particular, it identifies simulator script as its analysis driver, as shown in Figure 10.1. The simulator script listed in Figure 10.2 is a short driver shell script that Dakota executes to perform each function evaluation. The names of the parameters and results files are passed to the script on its command line; they are referenced in the script by $1 and $2, respectively. The simulator script is divided into three parts: pre-processing, analysis, and post-processing. Dakota Version 6.7 User’s Manual generated on November 13, 2017 10.3. BUILDING A BLACK-BOX INTERFACE TO A SIMULATION CODE # # # DAKOTA INPUT FILE - dakota_rosenbrock.in This sample Dakota input file optimizes the Rosenbrock function. See p. 95 in Practical Optimization by Gill, Murray, and Wright. method, npsol_sqp # if NPSOL is not available, comment the above and try the following instead: ## conmin_frcg variables, continuous_design = 2 cdv_initial_point -1.0 cdv_lower_bounds -2.0 cdv_upper_bounds 2.0 cdv_descriptor ’x1’ 1.0 -2.0 2.0 ’x2’ interface, fork # asynchronous analysis_driver = ’simulator_script’ parameters_file = ’params.in’ results_file = ’results.out’ work_directory directory_tag copy_files = ’templatedir/*’ # uncomment to leave params.in and results.out files in work_dir subdirectories # named ’workdir’ file_save directory_save aprepro deactivate active_set_vector responses, num_objective_functions = 1 analytic_gradients no_hessians Figure 10.1: The dakota rosenbrock.in input file. Dakota Version 6.7 User’s Manual generated on November 13, 2017 187 188 CHAPTER 10. INTERFACES #!/bin/sh # Sample simulator to Dakota system call script # See Advanced Simulation Code Interfaces chapter in Users Manual # $1 is params.in FROM Dakota # $2 is results.out returned to Dakota # # # # # # -------------PRE-PROCESSING -------------Incorporate the parameters from DAKOTA into the template, writing ros.in Use the following line if SNL’s APREPRO utility is used instead of DPrePro. ../aprepro -c ’*’ -q --nowarning ros.template ros.in dprepro $1 ros.template ros.in # -------# ANALYSIS # -------rosenbrock_bb # --------------# POST-PROCESSING # --------------# extract function value from the simulation output grep ’Function value’ ros.out | cut -c 18- > results.tmp # extract gradients from the simulation output (in this case will be ignored # by DAKOTA if not needed) grep -i ’Function g’ ros.out | cut -c 21- >> results.tmp mv results.tmp $2 Figure 10.2: The simulator script sample driver script. Dakota Version 6.7 User’s Manual generated on November 13, 2017 10.3. BUILDING A BLACK-BOX INTERFACE TO A SIMULATION CODE 189 Usage: dprepro [options] parameters_file template_input_file new_input_file Options: --help print brief help message --man print full manual page and exit --left-delimiter=string, --right-delimiter=string override the default curly brace delimiters { and }; these may need to be quoted or escaped on the command line, e.g., require escaping with backslash: ‘ $ # require quotes or escaping: \ ’ ( ) | --output-format=string use the specified C-style format specification string as the default for numeric output in all substitutions; will be overridden by field-specific format strings; may need to be quoted or escaped Figure 10.3: Partial listing of the dprepro script. In the pre-processing portion, the simulator script uses dprepro, a parsing utility, to extract the current variable values from a parameters file ($1) and combine them with the simulator template input file (ros.template) to create a new input file (ros.in) for the simulator. Internal to Sandia, the APREPRO utility is often used for this purpose. For external sites where APREPRO is not available, the DPrePro utility mentioned above is an alternative with many of the capabilities of APREPRO that is specifically tailored for use with Dakota and is distributed with it (in Dakota/examples/script interfaces/generic/dprepro, or Dakota/bin in a binary distribution). Additionally, the BPREPRO utility is another alternative to APREPRO (see [146]), and at Lockheed Martin sites, the JPrePost utility is available as a JAVA pre- and post-processor [48]. The dprepro script (usage shown in Figure 10.3) will be used here for simplicity of discussion. It can use either Dakota’s aprepro parameters file format (see Section 9.6.2) or Dakota’s standard format (see Section 9.6.1), so either option may be selected in the interface section of the Dakota input file. The ros.template file listed in Figure 10.4 is a template simulation input file which contains targets for the incoming variable values, identified by the strings “{x1}” and “{x2}”. These identifiers match the variable descriptors specified in dakota rosenbrock.in. The template input file is contrived as Rosenbrock has nothing to do with finite element analysis; it only mimics a finite element code to demonstrate the simulator template process. The dprepro script will search the simulator template input file for fields marked with curly brackets and then create a new file (ros.in) by replacing these targets with the corresponding numerical values for the variables. As noted in the usage information for dprepro and shown in simulator script, the names for the Dakota parameters file ($1), template file (ros.template), and generated input file (ros.in) must be specified in the dprepro command line arguments. The second part of the script executes the rosenbrock bb simulator. The input and output file names, ros.in and ros.out, respectively, are hard-coded into the FORTRAN 77 program rosenbrock bb.f. When the rosenbrock bb simulator is executed, the values for x1 and x2 are read in from ros.in, the Rosenbrock function is evaluated, and the function value is written out to ros.out. The third part performs the post-processing and writes the response results to a file for Dakota to read. Using Dakota Version 6.7 User’s Manual generated on November 13, 2017 190 CHAPTER 10. INTERFACES Title of Model: Rosenbrock black box *************************************************************************** * Description: This is an input file to the Rosenbrock black box Fortran simulator. This simulator is structured so * as to resemble the input/output from an engineering * simulation code, even though Rosenbrock’s function * is a simple analytic function. The node, element, * and material blocks are dummy inputs. * * * Input: x1 and x2 * Output: objective function value *************************************************************************** node 1 location 0.0 0.0 node 2 location 0.0 1.0 node 3 location 1.0 0.0 node 4 location 1.0 1.0 node 5 location 2.0 0.0 node 6 location 2.0 1.0 node 7 location 3.0 0.0 node 8 location 3.0 1.0 element 1 nodes 1 3 4 2 element 2 nodes 3 5 6 4 element 3 nodes 5 7 8 6 element 4 nodes 7 9 10 8 material 1 elements 1 2 material 2 elements 3 4 variable 1 {x1} variable 2 {x2} end Figure 10.4: Listing of the ros.template file Dakota Version 6.7 User’s Manual generated on November 13, 2017 10.3. BUILDING A BLACK-BOX INTERFACE TO A SIMULATION CODE 191 the UNIX “grep” utility, the particular response values of interest are extracted from the raw simulator output and saved to a temporary file (results.tmp). When complete, this file is renamed $2, which in this example is always “results.out”. Note that moving or renaming the completed results file avoids any problems with read race conditions (see Section 17.2.1.2). Because the Dakota input file dakota rosenbrock.in (Figure 10.1) specifies work directory and directory tag in its interface section, each invocation of simulator script wakes up in its own temporary directory, which Dakota has populated with the contents of directory templatedir. Having a separate directory for each invocation of simulator script simplifies the script when the Dakota input file specifies asynchronous (so several instances of simulator script might run simultaneously), as fixed names such as ros.in, ros.out, and results.tmp can be used for intermediate files. If neither asynchronous nor file tag is specified, and if there is no need (e.g., for debugging) to retain intermediate files having fixed names, then directory tag offers no benefit and can be omitted. An alternative to directory tag is to proceed as earlier versions of this chapter — prior to Dakota 5.0’s introduction of work directory — recommended: add two more steps to the simulator script, an initial one to create a temporary directory explicitly and copy templatedir to it if needed, and a final step to remove the temporary directory and any files in it. When work directory is specified, Dakota adjusts the $PATH seen by simulator script so that simple program names (i.e., names not containing a slash) that are visible in Dakota’s directory will also be visible in the work directory. Relative path names — involving an intermediate slash but not an initial one, such as ./rosenbrock bb or a/bc/rosenbrock bb — will only be visible in the work directory if a link files or copy files specification (see §10.5.5) has made them visible there. As an example of the data flow on a particular function evaluation, consider evaluation 60. The parameters file for this evaluation consists of: { { { { { { { { { DAKOTA_VARS x1 x2 DAKOTA_FNS ASV_1 DAKOTA_DER_VARS DVV_1 DVV_2 DAKOTA_AN_COMPS = = = = = = = = = 2 1.638247697999295e-01 2.197298209103481e-02 1 1 2 1 2 0 } } } } } } } } } This file is called workdir/workdir.60/params.in if the line named ’workdir’ file_save directory_save in Figure 10.1 is uncommented. The first portion of the file indicates that there are two variables, followed by new values for variables x1 and x2, and one response function (an objective function), followed by an active set vector (ASV) value of 1. The ASV indicates the need to return the value of the objective function for these parameters (see Section 9.7). The dprepro script reads the variable values from this file, namely 1.638247697999295e-01 and 2.197298209103481e-02 for x1 and x2 respectively, and substitutes them in the {x1} and {x2} fields of the ros.template file. The final three lines of the resulting input file (ros.in) then appear as follows: variable 1 1.638247697999295e-01 variable 2 2.197298209103481e-02 end where all other lines are identical to the template file. The rosenbrock bb simulator accepts ros.in as its input file and generates the following output to the file ros.out: Dakota Version 6.7 User’s Manual generated on November 13, 2017 192 CHAPTER 10. INTERFACES Beginning execution of model: Rosenbrock black box Set up complete. Reading nodes. Reading elements. Reading materials. Checking connectivity...OK ***************************************************** Input value for x1 = Input value for x2 = 0.1638247697999295E+00 0.2197298209103481E-01 Computing solution...Done ***************************************************** Function value = 0.7015563957680092E+00 Function gradient = [ -0.1353509902591768E+01 -0.9731146217930163E+00 ] Next, the appropriate values are extracted from the raw simulator output and returned in the results file. This post-processing is relatively trivial in this case, and the simulator script uses the grep and cut utilities to extract the value from the “Function value” line of the ros.out output file and save it to $2, which is the results.out file for this evaluation. This single value provides the objective function value requested by the ASV. After 132 of these function evaluations, the following Dakota output shows the final solution using the rosenbrock bb simulator: Exit NPSOL - Optimal solution found. Final nonlinear objective value = 0.1165704E-06 NPSOL exits with INFORM code = 0 (see "Interpretation of output" section in NPSOL manual) NOTE: see Fortran device 9 file (fort.9 or ftn09) for complete NPSOL iteration history. <<<<< Iterator npsol_sqp completed. <<<<< Function evaluation summary: 132 total (132 new, 0 duplicate) <<<<< Best parameters = 9.9965861667e-01 x1 9.9931682203e-01 x2 <<<<< Best objective function = 1.1657044253e-07 <<<<< Best data captured at function evaluation 130 <<<<< Iterator npsol_sqp completed. <<<<< Single Method Strategy completed. Dakota execution time in seconds: Total CPU = 0.12 [parent = Total wall clock = 1.47497 10.3.2 0.116982, child = 0.003018] Adapting These Scripts to Another Simulation To adapt this approach for use with another simulator, several steps need to be performed: Dakota Version 6.7 User’s Manual generated on November 13, 2017 10.4. SIMULATION INTERFACE COMPONENTS 193 1. Create a template simulation input file by identifying the fields in an existing input file that correspond to the variables of interest and then replacing them with {} identifiers (e.g. {cdv 1}, {cdv 2}, etc.) which match the Dakota variable descriptors. Copy this template input file to a templatedir that will be used to create working directories for the simulation. 2. Modify the dprepro arguments in simulator script to reflect names of the Dakota parameters file (previously “$1”), template file name (previously “ros.template”) and generated input file (previously “ros.in”). Alternatively, use APREPRO, BPREPRO, or JPrePost to perform this step (and adapt the syntax accordingly). 3. Modify the analysis section of simulator script to replace the rosenbrock bb function call with the new simulator name and command line syntax (typically including the input and output file names). 4. Change the post-processing section in simulator script to reflect the revised extraction process. At a minimum, this would involve changing the grep command to reflect the name of the output file, the string to search for, and the characters to cut out of the captured output line. For more involved post-processing tasks, invocation of additional tools may have to be added to the script. 5. Modify the dakota rosenbrock.in input file to reflect, at a minimum, updated variables and responses specifications. These nonintrusive interfacing approaches can be used to rapidly interface with simulation codes. While generally custom for each new application, typical interface development time is on the order of an hour or two. Thus, this approach is scalable when dealing with many different application codes. Weaknesses of this approach include the potential for loss of data precision (if care is not taken to preserve precision in pre- and post-processing file I/O), a lack of robustness in post-processing (if the data capture is too simplistic), and scripting overhead (only noticeable if the simulation time is on the order of a second or less). If the application scope at a particular site is more focused and only a small number of simulation codes are of interest, then more sophisticated interfaces may be warranted. For example, the economy of scale afforded by a common simulation framework justifies additional effort in the development of a high quality Dakota interface. In these cases, more sophisticated interfacing approaches could involve a more thoroughly developed black box interface with robust support of a variety of inputs and outputs, or it might involve intrusive interfaces such as the direct simulation interface discussed below in Section 16.2 or the SAND interface described in Section 1.3. 10.3.3 Additional Examples A variety of additional examples of black-box interfaces to simulation codes are maintained in the Dakota/examples/script interfaces directory in the source code distribution. 10.4 Simulation Interface Components Figure 10.5 is an extension of Figure 1.1 that adds details of the components that make up each of the simulation interfaces (system call, fork, and direct). These components include an input filter (“IFilter”), one or more analysis drivers (“Analysis Code/Driver”), and an output filter (“OFilter”). The input and output filters provide optional facilities for managing simulation pre- and post-processing, respectively. More specifically, the input filter can be used to insert the Dakota parameters into the input files required by the simulator program, and the output filter can be used to recover the raw data from the simulation results and compute the Dakota Version 6.7 User’s Manual generated on November 13, 2017 194 CHAPTER 10. INTERFACES Figure 10.5: Components of the simulation interface desired response data set. If there is a single analysis code, it is often convenient to combine these pre- and postprocessing functions into a single simulation driver script, and the separate input and output filter facilities are rarely used in this case. If there are multiple analysis drivers, however, the input and output filter facilities provide a convenient means for managing non-repeated portions of the pre- and post-processing for multiple analyses. That is, pre- and post-processing tasks that must be performed for each analysis can be performed within the individual analysis drivers, and shared pre- and post-processing tasks that are only performed once for the set of analyses can be performed within the input and output filters. When spawning function evaluations using system calls or forks, Dakota must communicate parameter and response data with the analysis drivers and filters through use of the file system. This is accomplished by passing the names of the parameters and results files on the command line when executing an analysis driver or filter. The input filter or analysis driver read data from the parameters file and the output filter or analysis driver write the appropriate data to the responses file. While not essential when the file names are fixed, the file names must be retrieved from the command line when Dakota is changing the file names from one function evaluation to the next (i.e., using temporary files or root names tagged with numerical identifiers). In the case of a UNIX C-shell script, the two command line arguments are retrieved using $argv[1] and $argv[2] (see [6]). Similarly, Bourne shell scripts retrieve the two command line arguments using $1 and $2, and Perl scripts retrieve the two command line arguments using @ARGV[0] and @ARGV[1]. In the case of a C or C++ program, command line arguments are retrieved using argc (argument count) and argv (argument vector) [87], and for Fortran 77, the iargc function returns the argument count and the getarg subroutine returns command line arguments. 10.4.1 Single analysis driver without filters If a single analysis driver is selected in the interface specification and filters are not needed (as indicated by omission of the input filter and output filter specifications), then only one process will appear in the execution syntax of the simulation interface. An example of this syntax in the system call case is: driver params.in results.out Dakota Version 6.7 User’s Manual generated on November 13, 2017 10.4. SIMULATION INTERFACE COMPONENTS 195 where “driver” is the user-specified analysis driver and “params.in” and “results.out” are the names of the parameters and results files, respectively, passed on the command line. In this case, the user need not retrieve the command line arguments since the same file names will be used each time. For the same mapping, the fork simulation interface echoes the following syntax: blocking fork: driver params.in results.out for which only a single blocking fork is needed to perform the evaluation. Executing the same mapping with the direct simulation interface results in an echo of the following syntax: Direct function: invoking driver where this analysis driver must be linked as a function within Dakota’s direct interface (see Section 16.2). Note that no parameter or response files are involved, since such values are passed directly through the function argument lists. Both the system call and fork interfaces support asynchronous operations. The asynchronous system call execution syntax involves executing the system call in the background: driver params.in.1 results.out.1 & and the asynchronous fork execution syntax involves use of a nonblocking fork: nonblocking fork: driver params.in.1 results.out.1 where file tagging (see Section 10.5.2) has been user-specified in both cases to prevent conflicts between concurrent analysis drivers. In these cases, the user must retrieve the command line arguments since the file names change on each evaluation. Execution of the direct interface must currently be performed synchronously since multithreading is not yet supported (see Section 17.2.1.1). 10.4.2 Single analysis driver with filters When filters are used, the syntax of the system call that Dakota performs is: ifilter params.in results.out; driver params.in results.out; ofilter params.in results.out in which the input filter (“ifilter”), analysis driver (“driver”), and output filter (“ofilter”) processes are combined into a single system call through the use of semi-colons (see [6]). All three portions are passed the names of the parameters and results files on the command line. For the same mapping, the fork simulation interface echoes the following syntax: blocking fork: ifilter params.in results.out; driver params.in results.out; ofilter params.in results.out where a series of three blocking forks is used to perform the evaluation. Executing the same mapping with the direct simulation interface results in an echo of the following syntax: Dakota Version 6.7 User’s Manual generated on November 13, 2017 196 CHAPTER 10. INTERFACES Direct function: invoking { ifilter driver ofilter } where each of the three components must be linked as a function within Dakota’s direct interface. Since asynchronous operations are not yet supported, execution simply involves invocation of each of the three linked functions in succession. Again, no files are involved since parameter and response data are passed directly through the function argument lists. Asynchronous executions would appear as follows for the system call interface: (ifilter params.in.1 results.out.1; driver params.in.1 results.out.1; ofilter params.in.1 results.out.1) & and, for the fork interface, as: nonblocking fork: ifilter params.in.1 results.out.1; driver params.in.1 results.out.1; ofilter params.in.1 results.out.1 where file tagging of evaluations has again been user-specified in both cases. For the system call simulation interface, use of parentheses and semi-colons to bind the three processes into a single system call simplifies asynchronous process management compared to an approach using separate system calls. The fork simulation interface, on the other hand, does not rely on parentheses and accomplishes asynchronous operations by first forking an intermediate process. This intermediate process is then reforked for the execution of the input filter, analysis driver, and output filter. The intermediate process can be blocking or nonblocking (nonblocking in this case), and the second level of forks can be blocking or nonblocking (blocking in this case). The fact that forks can be reforked multiple times using either blocking or nonblocking approaches provides the enhanced flexibility to support a variety of local parallelism approaches (see Chapter 17). 10.4.3 Multiple analysis drivers without filters If a list of analysis drivers is specified and filters are not needed (i.e., neither input filter nor output filter appears), then the system call syntax would appear as: driver1 params.in results.out.1; driver2 params.in results.out.2; driver3 params.in results.out.3 where “driver1”, “driver2”, and “driver3” are the user-specified analysis drivers and “params.in” and “results.out” are the user-selected names of the parameters and results files. Note that the results files for the different analysis drivers have been automatically tagged to prevent overwriting. This automatic tagging of analyses (see Section 10.5.4) is a separate operation from user-selected tagging of evaluations (see Section 10.5.2). For the same mapping, the fork simulation interface echoes the following syntax: blocking fork: driver1 params.in results.out.1; driver2 params.in results.out.2; driver3 params.in results.out.3 for which a series of three blocking forks is needed (no reforking of an intermediate process is required). Executing the same mapping with the direct simulation interface results in an echo of the following syntax: Direct function: invoking { driver1 driver2 driver3 } Dakota Version 6.7 User’s Manual generated on November 13, 2017 10.4. SIMULATION INTERFACE COMPONENTS 197 where, again, each of these components must be linked within Dakota’s direct interface and no files are involved for parameter and response data transfer. Both the system call and fork interfaces support asynchronous function evaluations. The asynchronous system call execution syntax would be reported as (driver1 params.in.1 results.out.1.1; driver2 params.in.1 results.out.1.2; driver3 params.in.1 results.out.1.3) & and the nonblocking fork execution syntax would be reported as nonblocking fork: driver1 params.in.1 results.out.1.1; driver2 params.in.1 results.out.1.2; driver3 params.in.1 results.out.1.3 where, in both cases, file tagging of evaluations has been user-specified to prevent conflicts between concurrent analysis drivers and file tagging of the results files for multiple analyses is automatically used. In the fork interface case, an intermediate process is forked to allow a non-blocking function evaluation, and this intermediate process is then reforked for the execution of each of the analysis drivers. 10.4.4 Multiple analysis drivers with filters Finally, when combining filters with multiple analysis drivers, the syntax of the system call that Dakota performs is: ifilter params.in.1 results.out.1; driver1 params.in.1 results.out.1.1; driver2 params.in.1 results.out.1.2; driver3 params.in.1 results.out.1.3; ofilter params.in.1 results.out.1 in which all processes have again been combined into a single system call through the use of semi-colons and parentheses. Note that the secondary file tagging for the results files is only used for the analysis drivers and not for the filters. This is consistent with the filters’ defined purpose of managing the non-repeated portions of analysis pre- and post-processing (e.g., overlay of response results from individual analyses; see Section 10.5.4 for additional information). For the same mapping, the fork simulation interface echoes the following syntax: blocking fork: ifilter params.in.1 results.out.1; driver1 params.in.1 results.out.1.1; driver2 params.in.1 results.out.1.2; driver3 params.in.1 results.out.1.3; ofilter params.in.1 results.out.1 for which a series of five blocking forks is used (no reforking of an intermediate process is required). Executing the same mapping with the direct simulation interface results in an echo of the following syntax: Direct function: invoking { ifilter driver1 driver2 driver3 ofilter } where each of these components must be linked as a function within Dakota’s direct interface. Since asynchronous operations are not supported, execution simply involves invocation of each of the five linked functions in succession. Again, no files are involved for parameter and response data transfer since this data is passed directly through the function argument lists. Dakota Version 6.7 User’s Manual generated on November 13, 2017 198 CHAPTER 10. INTERFACES Asynchronous executions would appear as follows for the system call interface: (ifilter params.in.1 results.out.1; driver1 params.in.1 results.out.1.1; driver2 params.in.1 results.out.1.2; driver3 params.in.1 results.out.1.3; ofilter params.in.1 results.out.1) & and for the fork interface: nonblocking fork: ifilter params.in.1 results.out.1; driver1 params.in.1 results.out.1.1; driver2 params.in.1 results.out.1.2; driver3 params.in.1 results.out.1.3; ofilter params.in.1 results.out.1 where, again, user-selected file tagging of evaluations is combined with automatic file tagging of analyses. In the fork interface case, an intermediate process is forked to allow a non-blocking function evaluation, and this intermediate process is then reforked for the execution of the input filter, each of the analysis drivers, and the output filter. A complete example of these filters and multi-part drivers can be found in Dakota/test/dakota 3pc/dakota 3pc.in. 10.5 Simulation File Management This section describes some management features used for files that transfer data between Dakota and simulation codes (i.e., when the system call or fork interfaces are used). These features can generate unique filenames when Dakota executes programs in parallel and can help one debug the interface between Dakota and a simulation code. 10.5.1 File Saving Before driver execution: In Dakota 5.0 and newer, an existing results file will be removed immediately prior to executing the analysis driver. This new behavior addresses a common user problem resulting from starting Dakota with stale results files in the run directory. To override this default behavior and preserve any existing results files, specify allow existing results. After driver execution: The file save option in the interface specification allows the user to control whether parameters and results files are retained or removed from the working directory after the analysis completes. Dakota’s default behavior is to remove files once their use is complete to reduce clutter. If the method output setting is verbose, a file remove notification will follow the function evaluation echo, e.g., driver /usr/tmp/aaaa20305 /usr/tmp/baaa20305 Removing /usr/tmp/aaaa20305 and /usr/tmp/baaa20305 However, if file save appears in the interface specification, these files will not be removed. This latter behavior is often useful for debugging communication between Dakota and simulator programs. An example of a file save specification is shown in the file tagging example below. Dakota Version 6.7 User’s Manual generated on November 13, 2017 10.5. SIMULATION FILE MANAGEMENT 10.5.2 199 File Tagging for Evaluations When a user provides parameters file and results file specifications, the file tag option in the interface specification causes Dakota to make the names of these files unique by appending the function evaluation number to the root file names. Default behavior is to not tag these files, which has the advantage of allowing the user to ignore command line argument passing and always read to and write from the same file names. However, it has the disadvantage that files may be overwritten from one function evaluation to the next. When file tag appears in the interface specification, the file names are made unique by the appended evaluation number. This uniqueness requires the user’s interface to get the names of these files from the command line. The file tagging feature is most often used when concurrent simulations are running in a common disk space, since it can prevent conflicts between the simulations. An example specification of file tag and file save is shown below: interface, system analysis_driver = ’text_book’ parameters_file = ’text_book.in’ results_file = ’text_book.out’ file_tag file_save Special case: When a user specifies names for the parameters and results files and file save is used without file tag, untagged files are used in the function evaluation but are then moved to tagged files after the function evaluation is complete, to prevent overwriting files for which a file save request has been given. If the output control is set to verbose, then a notification similar to the following will follow the function evaluation echo: driver params.in results.out Files with non-unique names will be tagged to enable file_save: Moving params.in to params.in.1 Moving results.out to results.out.1 Hierarchical tagging: When a model’s specification includes the hierarchical tagging keyword, the tag applied to parameter and results file names of any subordinate interfaces will reflect any model hierarchy present. This option is useful for studies involving multiple models with a nested or hierarchical relationship. For example a nested model has a sub-method, which itself likely operates on a sub-model, or a hierarchical approximation involves coordination of low and high fidelity models. Specifying hierarchical tagging will yield function evaluation identifiers (“tags”) composed of the evaluation IDs of the models involved, e.g., outermodel.innermodel.interfaceid = 4.9.2. This communicates the outer contexts to the analysis driver when performing a function evaluation. For an example of using hierarhical tagging in a nested model context, see dakota/test/dakota uq timeseries * optinterf.in. 10.5.3 Temporary Files If parameters file and results file are not specified by the user, temporary files having generated names are used. For example, a system call to a single analysis driver might appear as: driver /tmp/dakota_params_aaaa2035 /tmp/dakota_results_baaa2030 and a system call to an analysis driver with filter programs might appear as: ifilter /tmp/dakota_params_aaaa2490 /tmp/dakota_results_baaa2490; driver /tmp/dakota_params_aaaa2490 tmp/dakota_results_baaa2490; ofilter /tmp/dakota_params_aaaa2490 /tmp/dakota_results_baa22490 Dakota Version 6.7 User’s Manual generated on November 13, 2017 200 CHAPTER 10. INTERFACES These files have unique names created by Boost filesystem utilities. This uniqueness requires the user’s interface to get the names of these files from the command line. File tagging with evaluation number is unnecessary with temporary files, but can be helpful for the user workflow to identify the evaluation number. Thus file tag requests will be honored. A file save request will be honored, but it should be used with care since the temporary file directory could easily become cluttered without the user noticing. 10.5.4 File Tagging for Analysis Drivers When multiple analysis drivers are involved in performing a function evaluation with either the system call or fork simulation interface, a secondary file tagging is automatically used to distinguish the results files used for the individual analyses. This applies to both the case of user-specified names for the parameters and results files and the default temporary file case. Examples for the former case were shown previously in Section 10.4.3 and Section 10.4.4. The following examples demonstrate the latter temporary file case. Even though Unix temporary files have unique names for a particular function evaluation, tagging is still needed to manage the individual contributions of the different analysis drivers to the response results, since the same root results filename is used for each component. For the system call interface, the syntax would be similar to the following: ifilter /var/tmp/aaawkaOKZ /var/tmp/baaxkaOKZ; driver1 /var/tmp/aaawkaOKZ /var/tmp/baaxkaOKZ.1; driver2 /var/tmp/aaawkaOKZ /var/tmp/baaxkaOKZ.2; driver3 /var/tmp/aaawkaOKZ /var/tmp/baaxkaOKZ.3; ofilter /var/tmp/aaawkaOKZ /var/tmp/baaxkaOKZ and, for the fork interface, similar to: blocking fork: ifilter /var/tmp/aaawkaOKZ driver1 /var/tmp/aaawkaOKZ driver2 /var/tmp/aaawkaOKZ driver3 /var/tmp/aaawkaOKZ ofilter /var/tmp/aaawkaOKZ /var/tmp/baaxkaOKZ; /var/tmp/baaxkaOKZ.1; /var/tmp/baaxkaOKZ.2; /var/tmp/baaxkaOKZ.3; /var/tmp/baaxkaOKZ Tagging of results files with an analysis identifier is needed since each analysis driver must contribute a userdefined subset of the total response results for the evaluation. If an output filter is not supplied, Dakota will combine these portions through a simple overlaying of the individual contributions (i.e., summing the results in /var/tmp/baaxkaOKZ.1, /var/tmp/baaxkaOKZ.2, and /var/tmp/baaxkaOKZ.3). If this simple approach is inadequate, then an output filter should be supplied to perform the combination. This is the reason why the results file for the output filter does not use analysis tagging; it is responsible for the results combination (i.e., combining /var/tmp/baaxkaOKZ.1, /var/tmp/baaxkaOKZ.2, and /var/tmp/baaxkaOKZ.3 into /var/tmp/baaxkaOKZ). In this case, Dakota will read only the results file from the output filter (i.e., /var/tmp/baaxkaOKZ) and interpret it as the total response set for the evaluation. Parameters files are not currently tagged with an analysis identifier. This reflects the fact that Dakota does not attempt to subdivide the requests in the active set vector for different analysis portions. Rather, the total active set vector is passed to each analysis driver and the appropriate subdivision of work must be defined by the user. This allows the division of labor to be very flexible. In some cases, this division might occur across response functions, with different analysis drivers managing the data requests for different response functions. And in other cases, the subdivision might occur within response functions, with different analysis drivers contributing portions to each of the response functions. The only restriction is that each of the analysis drivers must follow the response format dictated by the total active set vector. For response data for which an analysis driver has no contribution, 0’s must be used as placeholders. Dakota Version 6.7 User’s Manual generated on November 13, 2017 10.5. SIMULATION FILE MANAGEMENT 10.5.5 201 Work Directories Sometimes it is convenient for simulators and filters to run in a directory different from the one where Dakota is invoked. For instance, when performing concurrent evaluations and/or analyses, it is often necessary to cloister input and output files in separate directories to avoid conflicts. A simulator script used as an analysis driver can of course include commands to change to a different directory if desired (while still arranging to write a results file in the original directory), but Dakota has facilities that may simplify the creation of simulator scripts. When the work directory feature is enabled, Dakota will create a directory for each evaluation/analysis (with optional tagging and saving as with files). To enable the work directory feature an interface specification includes the keyword work_directory then Dakota will arrange for the simulator and any filters to wake up in the work directory, with $PATH adjusted (if necessary) so programs that could be invoked without a relative path to them (i.e., by a name not involving any slashes) from Dakota’s directory can also be invoked from the simulator’s (and filter’s) directory. On occasion, it is convenient for the simulator to have various files, e.g., data files, available in the directory where it runs. If, say, my/special/directory is such a directory (as seen from Dakota’s directory), the interface specification work_directory named ’my/special/directory’ would cause Dakota to start the simulator and any filters in that directory. If the directory did not already exist, Dakota would create it and would remove it after the simulator (or output filter, if specifed) finished, unless instructed not to do so by the appearance of directory save (or its deprecated synonym dir save) in the interface specification. If named ’...’ does not appear, then directory save cannot appear either, and Dakota creates a temporary directory (using the tmpnam function to determine its name) for use by the simulator and any filters. If you specify directory tag (or the deprecated dir tag), Dakota causes each invocation of the simulator and any filters to start in a a subdirectory of the work directory with a name composed of the work directory’s name followed by a period and the invocation number (1, 2, ...); this might be useful in debugging. Sometimes it can be helpful for the simulator and filters to start in a new directory populated with some files. Adding link_files ’templatedir/*’ to the work directory specification would cause the contents of directory templatedir to be linked into the work directory. Linking makes sense if files are large, but when practical, it is far more reliable to have copies of the files; adding copy files to the specification would cause the contents of the template directory to be copied to the work directory. The linking or copying does not overwrite existing files unless replace also appears in the specification. Here is a summary of possibilities for a work directory specification, with [...] denoting that ... is optional: work_directory [ named ’...’ ] [ directory_tag ] # (or dir_tag) [ directory_save ] # (or dir_save) [ link_files ’...’ ’...’ ] [ copy_files ’...’ ’...’ ] [ replace ] Dakota Version 6.7 User’s Manual generated on November 13, 2017 202 CHAPTER 10. INTERFACES 10.6 Parameter to Response Mapping Examples In this section, interface mapping examples are presented through the discussion of several parameters files and their corresponding results files. A typical input file for 2 variables (n = 2) and 3 functions (m = 3) using the standard parameters file format (see Section 9.6.1) is as follows: 2 1.500000000000000e+00 1.500000000000000e+00 3 1 1 1 2 1 2 0 variables cdv_1 cdv_2 functions ASV_1 ASV_2 ASV_3 derivative_variables DVV_1 DVV_2 analysis_components where numerical values are associated with their tags within “value tag” constructs. The number of design variables (n) and the string “variables” are followed by the values of the design variables and their tags, the number of functions (m) and the string “functions”, the active set vector (ASV) and its tags, the number of derivative variables and the string “derivative variables”, the derivative variables vector (DVV) and its tags, the number of analysis components and the string “analysis components”, and the analysis components array and its tags. The descriptive tags for the variables are always present and they are either the descriptors in the user’s variables specification, if given there, or are default descriptors. The length of the active set vector is equal to the number of functions (m). In the case of an optimization data set with an objective function and two nonlinear constraints (three response functions total), the first ASV value is associated with the objective function and the remaining two are associated with the constraints (in whatever consistent constraint order has been defined by the user). The DVV defines a subset of the variables used for computing derivatives. Its identifiers are 1-based and correspond to the full set of variables listed in the first array. Finally, the analysis components pass additional strings from the user’s analysis components specification in a Dakota input file through to the simulator. They allow the development of simulation drivers that are more flexible, by allowing them to be passed additional specifics at run time, e.g., the names of model files such as a particular mesh to use. For the APREPRO format option (see Section 9.6.2), the same set of data appears as follows: { { { { { { { { { { { DAKOTA_VARS cdv_1 cdv_2 DAKOTA_FNS ASV_1 ASV_2 ASV_3 DAKOTA_DER_VARS DVV_1 DVV_2 DAKOTA_AN_COMPS = = = = = = = = = = = 2 1.500000000000000e+00 1.500000000000000e+00 3 1 1 1 2 1 2 0 } } } } } } } } } } } where the numerical values are associated with their tags within “{ tag = value }” constructs. The user-supplied simulation interface, comprised of a simulator program or driver and (optionally) filter programs, is responsible for reading the parameters file and creating a results file that contains the response data Dakota Version 6.7 User’s Manual generated on November 13, 2017 10.6. PARAMETER TO RESPONSE MAPPING EXAMPLES 203 requested in the ASV. This response data is written in the format described in Section 11.2. Since the ASV contains all ones in this case, the response file corresponding to the above input file would contain values for the three functions: 1.250000000000000e-01 f 1.500000000000000e+00 c1 1.500000000000000e+00 c2 Since function tags are optional, the following would be equally acceptable: 1.250000000000000e-01 1.500000000000000e+00 1.500000000000000e+00 For the same parameters with different ASV components, 2 1.500000000000000e+00 1.500000000000000e+00 3 3 3 3 2 1 2 0 variables cdv_1 cdv_2 functions ASV_1 ASV_2 ASV_3 derivative_variables DVV_1 DVV_2 analysis_components the following response data is required: 1.250000000000000e-01 f 1.500000000000000e+00 c1 1.500000000000000e+00 c2 [ 5.000000000000000e-01 5.000000000000000e-01 ] [ 3.000000000000000e+00 -5.000000000000000e-01 ] [ -5.000000000000000e-01 3.000000000000000e+00 ] Here, we need not only the function values, but also each of their gradients. The derivatives are computed with respect to cdv 1 and cdv 2 as indicated by the DVV values. Another modification to the ASV components yields the following parameters file: 2 1.500000000000000e+00 1.500000000000000e+00 3 2 0 2 2 1 2 0 variables cdv_1 cdv_2 functions ASV_1 ASV_2 ASV_3 derivative_variables DVV_1 DVV_2 analysis_components Dakota Version 6.7 User’s Manual generated on November 13, 2017 204 CHAPTER 10. INTERFACES for which the following results file is needed: [ 5.000000000000000e-01 5.000000000000000e-01 ] [ -5.000000000000000e-01 3.000000000000000e+00 ] Here, we need gradients for functions f and c2, but not for c1, presumably since this constraint is inactive. A full Newton optimizer might make the following request: 2 1.500000000000000e+00 1.500000000000000e+00 1 7 2 1 2 0 variables cdv_1 cdv_2 functions ASV_1 derivative_variables DVV_1 DVV_2 analysis_components for which the following results file, 1.250000000000000e-01 f [ 5.000000000000000e-01 5.000000000000000e-01 ] [[ 3.000000000000000e+00 0.000000000000000e+00 0.000000000000000e+00 3.000000000000000e+00 ]] containing the objective function, its gradient vector, and its Hessian matrix, is needed. Again, the derivatives (gradient vector and Hessian matrix) are computed with respect to cdv 1 and cdv 2 as indicated by the DVV values. Lastly, a more advanced example could have multiple types of variables present; in this example, 2 continuous design and 3 discrete design range, 2 normal uncertain, and 3 continuous state and 2 discrete state range variables. When a mixture of variable types is present, the content of the DVV (and therefore the required length of gradient vectors and Hessian matrices) depends upon the type of study being performed (see Section 11.3). For a reliability analysis problem, the uncertain variables are the active continuous variables and the following parameters file would be typical: 12 1.500000000000000e+00 1.500000000000000e+00 2 2 2 5.000000000000000e+00 5.000000000000000e+00 3.500000000000000e+00 3.500000000000000e+00 3.500000000000000e+00 4 4 3 3 3 variables cdv_1 cdv_2 ddriv_1 ddriv_2 ddriv_3 nuv_1 nuv_2 csv_1 csv_2 csv_3 dsriv_1 dsriv_2 functions ASV_1 ASV_2 Dakota Version 6.7 User’s Manual generated on November 13, 2017 10.6. PARAMETER TO RESPONSE MAPPING EXAMPLES 3 2 6 7 2 mesh1.exo db1.xml 205 ASV_3 derivative_variables DVV_1 DVV_2 analysis_components AC_1 AC_2 Gradients are requested with respect to variable entries 6 and 7, which correspond to normal uncertain variables nuv 1 and nuv 2. The following response data would be appropriate: 7.943125000000000e+02 f 1.500000000000000e+00 c1 1.500000000000000e+00 c2 [ 2.560000000000000e+02 2.560000000000000e+02 ] [ 0.000000000000000e+00 0.000000000000000e+00 ] [ 0.000000000000000e+00 0.000000000000000e+00 ] In a parameter study, however, no distinction is drawn between different types of continuous variables, and derivatives would be needed with respect to all continuous variables (ndvv = 7 for the continuous design variables cdv 1 and cdv 2, the normal uncertain variables nuv 1 and nuv 2, and the continuous state variables csv 1, csv 2 and csv 3). The parameters file would appear as 12 1.500000000000000e+00 1.500000000000000e+00 2 2 2 5.000000000000000e+00 5.000000000000000e+00 3.500000000000000e+00 3.500000000000000e+00 3.500000000000000e+00 4 4 3 3 3 3 7 1 2 6 7 8 9 10 2 mesh1.exo db1.xml variables cdv_1 cdv_2 ddriv_1 ddriv_2 ddriv_3 nuv_1 nuv_2 csv_1 csv_2 csv_3 dsriv_1 dsriv_2 functions ASV_1 ASV_2 ASV_3 derivative_variables DVV_1 DVV_2 DVV_3 DVV_4 DVV_5 DVV_6 DVV_7 analysis_components AC_1 AC_2 and the corresponding results would appear as Dakota Version 6.7 User’s Manual generated on November 13, 2017 206 CHAPTER 10. INTERFACES 7.943125000000000e+02 f 1.500000000000000e+00 c1 1.500000000000000e+00 c2 [ 5.000000000000000e-01 2.560000000000000e+02 6.250000000000000e+01 [ 3.000000000000000e+00 0.000000000000000e+00 0.000000000000000e+00 [ -5.000000000000000e-01 0.000000000000000e+00 0.000000000000000e+00 10.7 5.000000000000000e-01 6.250000000000000e+01 ] -5.000000000000000e-01 0.000000000000000e+00 ] 3.000000000000000e+00 0.000000000000000e+00 ] 2.560000000000000e+02 6.250000000000000e+01 0.000000000000000e+00 0.000000000000000e+00 0.000000000000000e+00 0.000000000000000e+00 Parameters and Results Files with dakota.interfacing The Python module dakota.interfacing first was made available with Dakota 6.6. (It was released with Dakota 6.5 as the module dipy.) By providing a Python interface to read and write, respectively, Dakota parameters and results files, dakota.interfacing can greatly simplify development of black-box interfaces. The benefit may be greatest when one or more phases of the interface (pre-processing, execution, post-processing) is written in Python. The following sections describe the components of dakota.interfacing. These include a Parameters class for making variable information available, a Results class for collecting results and writing them to file, and a convenience function read parameters file, the preferred way to construct Parameters and Results objects from a Dakota parameters file. 10.7.1 Creating Parameters and Results objects dakota.interfacing has one free function, read parameters file, which creates Parameters and Results objects based on a Dakota parameters file. Its signature is: dakota.interfacing.read parameters file(parameters file=None, results file=None, ignore asv=False) parameters file and results file are the names of the parameters file that is to be read and the results file that ultimately is to be written. The names can be absolute or relative filepaths or just filenames. If a parameters file or results file is not provided, it will be obtained from the command line arguments. (The results filename is assumed to be the last command line argument, and the parameters file the second to last.) Note that if the working directory has changed since script invocation, filenames provided as command line arguments by Dakota’s fork or system interfaces may be incorrect. If results file is set to the constant dakota.interfacing.UNNAMED, the Results object is constructed without a results file name. In this case, an output stream must be provided when Results.write() is called. Unnamed results files are most helpful when no results file will be written, as with a script intended purely for pre-processing. By default, the returned Results object enforces the active set vector (see the Results class section). This behavior can be overriden, allowing any property (function, gradient, Hessian) of a response to be set, by setting ignore asv to True. Dakota Version 6.7 User’s Manual generated on November 13, 2017 10.7. PARAMETERS AND RESULTS FILES WITH DAKOTA.INTERFACING 10.7.2 207 Parameters objects Parameters objects make the variables, analysis components, evaluation ID, and evaluation number read from a Dakota parameters file available through a combination of key-value access and object attributes. Although Parameters objects may be constructed directly, it is advisable to use the read parameters file function instead. Variable values can be accessed by Dakota descriptor or by index using [] on the object itself. Variables types (integer, real, string) are inferred by first attempting to convert to int and then, if this fails, to float. Analysis components are accessible by index only using the an comps attribute. Parameters objects support iteration over the variables, yielding the index, variable descriptor, and variable value. Parameters objects have the attributes: • an comps List of the analysis components (strings). • eval id Evaluation id (string). • eval num Evaluation number (final token in eval id) (int). • aprepro format Boolean indicating whether the parameters file was in aprepro (True) or Dakota (False) format. • descriptors List of the variable descriptors • num variables Number of variables • num an comps Number of analysis components 10.7.3 Results objects Results objects: • communicate response requests from Dakota (active set vector and derivative variables) • collect response data (function values, gradients, and Hessians) • write Dakota results files Results objects are collections of Response objects, which are documented in the following section. Each Response can be accessed by name (Dakota descriptor) or by index using [] on the Results object itself. Results objects support iteration of the Responses, yielding the index, response descriptor, and Response. Although Results objects may be constructed directly, it is advisable to use the read parameters file function instead. Results objects have the attributes: • eval id Evaluation id (a string). • eval num Evaluation number (final token in eval id) (int). • aprepro format Boolean indicating whether the parameters file was in aprepro (True) or Dakota (False) format. Dakota Version 6.7 User’s Manual generated on November 13, 2017 208 CHAPTER 10. INTERFACES • descriptors List of the response descriptors (strings) • num responses Number of variables (read-only) • deriv vars List of the derivative variables (strings) • num deriv varsNumber of derivative variables (int) Results objects have the methods: • fail() Set the FAIL attribute. When the results file is written, it will contain only the word FAIL, triggering Dakota’s failure capturing behavior (See Chapter 19). • write(stream=None, ignore asv=None) Write the results to the Dakota results file. If stream is set, it overrides the results file name provided at construct time. It must be an open file-like object, rather than the name of a file. If ignore asv is True, the file will be written even if information requested via the active set vector is missing. 10.7.4 Response object Response objects store response information. They typically are instantiated and accessed through a Results object by index or response descriptor using []. Responses have the attributes: • asv a collections.namedtuple with three members, function, gradient, and hessian. Each is a boolean indicating whether Dakota requested the associated information for the response. namedtuples can be accessed by index or by member. • function Function value for the response. A ResponseError is raised if Dakota did not request the function value (and ignore asv is False). • gradient Gradient for the response. Gradients must be a 1D iterable of values that can be converted to floats, such as a list or 1D numpy array. A ResponseError is raised if Dakota did not request the gradient (and ignore asv is False), or if the number of elements does not equal the number of derivative variables. • hessian Hessian value for the response. Hessians must be an iterable of iterables (e.g. a 2D numpy array or list of lists). A ResponseError is raised if Dakota did not request the Hessian (and ignore asv is False), or if the dimension does not correspond correctly with the number of derivative variables. 10.7.5 dakota.interfacing Examples In addition to those in this section, the Dakota/examples/script interfaces/Python folder contains two runnable examples of Python analysis drivers. In one, rosenbrock bb.py, parameters file parsing is performed manually. The other, rosenbrock bb di.py, demonstrates the dakota.interfacing module. For most applications, using dakota.interfacing is straightforward. The first example, in Figure 10.6, is a mock analysis driver. Two variables with the descriptors x1 and x2 are read from the Dakota parameters file and used to evaluate the fictitious user function applic module.run(). The result, stored in f, is assigned to the Dakota Version 6.7 User’s Manual generated on November 13, 2017 10.7. PARAMETERS AND RESULTS FILES WITH DAKOTA.INTERFACING 209 import dakota.interfacing as di import applic_module params, results = di.read_parameters_file() # parameters can be accessed by descriptor, as shown here, or by index x1 = params["x1"] x2 = params["x2"] f = applic_module.run(x1,x2) # Responses also can be accessed by descriptor or index results["f"].function = f results.write() Figure 10.6: A simple analysis driver that uses dakota.interfacing function value of the appropriate response. (A common error is leaving off the function attribute, which is needed to distinguish the function value of the response from its gradient and Hessian.) The Results object exposes the active set vector read from the parameters file. When analytic gradients or Hessians are available for a response, the ASV should be queried to determine what Dakota has requested for an evaluation. If an attempt is made to addunrequested information to a response, a dakota.interface.ResponseError is raised. The same exception results if a requested piece of information is missing when Results.write() is called. The ignore asv option to read parameters file and Results.write() overrides ASV checks. In Figure 10.7, applic module.run() has been modified to return not only the function value of f, but also its gradient and Hessian. The asv attribute is examined to determine which of these to add to results["f"]. Dakota Version 6.7 User’s Manual generated on November 13, 2017 210 CHAPTER 10. INTERFACES import dakota.interfacing as di import applic_module # fictitious application params, results = di.read_parameters_file() x1 = params["x1"] x2 = params["x2"] f, df, df2 = applic_module.run(x1,x2) if Results.asv.function: results["f"].function = f if Results.asv.gradient: results["f"].gradient = df if Results.asv.hessian: results["f"] = df2 results.write() Figure 10.7: Examining the active set vector Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 11 Responses 11.1 Overview The responses specification in a Dakota input file controls the types of data that can be returned from an interface during Dakota’s execution. The specification includes the number and type of response functions (objective functions, nonlinear constraints, calibration terms, etc.) as well as availability of first and second derivatives (gradient vectors and Hessian matrices) for these response functions. This chapter will present a brief overview of the response data sets and their uses, as well as cover some user issues relating to file formats and derivative vector and matrix sizing. For a detailed description of responses section syntax and example specifications, refer to the Responses Commands chapter in the Dakota Reference Manual [3]. 11.1.1 Response function types The types of response functions listed in the responses specification should be consistent with the iterative technique called for in the method specification: • an optimization data set comprised of num objective functions, num nonlinear inequality constraints, and num nonlinear equality constraints. This data set is appropriate for use with optimization methods (e.g., the methods in Chapter 6). When using results files (11.2), the responses must be ordered: objectives, inequalities, then equalities. • a calibration data set comprised of calibration terms, num nonlinear inequality constraints, and num nonlinear equality constraints. This data set is appropriate for use with nonlinear least squares algorithms (e.g., the methods in Chapter 7). When using results files (11.2), the responses must be ordered: calibration terms, inequalities, then equalities. • a generic data set comprised of num response functions. This data set is appropriate for use with uncertainty quantification methods (e.g., the methods in Section 5). Certain general-purpose iterative techniques, such as parameter studies and design of experiments methods, can be used with any of these data sets. 212 CHAPTER 11. RESPONSES 11.1.2 Gradient availability Gradient availability for these response functions may be described by: • no gradients: gradients will not be used. • numerical gradients: gradients are needed and will be approximated by finite differences. • analytic gradients: gradients are needed and will be supplied by the simulation code (without any finite differencing by Dakota). • mixed gradients: the simulation will supply some gradient components and Dakota will approximate the others by finite differences. The gradient specification also links back to the iterative method used. Gradients commonly are needed when the iterative study involves gradient-based optimization, reliability analysis for uncertainty quantification, or local sensitivity analysis. 11.1.3 Hessian availability Hessian availability for the response functions is similar to the gradient availability specifications, with the addition of support for “quasi-Hessians”: • no hessians: Hessians will not be used. • numerical gradients: Hessians are needed and will be approximated by finite differences. These finite differences may involve first-order differences of gradients (if analytic gradients are available for the response function of interest) or second-order differences of function values (in all other cases). • quasi hessians: Hessians are needed and will be approximated by secant updates (BFGS or SR1) from a series of gradient evaluations. • analytic hessians: Hessians are needed and are available directly from the simulation code. • mixed hessians: Hessians are needed and will be obtained from a mix of numerical, analytic, and “quasi” sources. The Hessian specification also links back to the iterative method in use; Hessians commonly would be used in gradient-based optimization by full Newton methods or in reliability analysis with second-order limit state approximations or second-order probability integrations. 11.1.4 Field Data Prior to Dakota 6.1, Dakota responses have been treated as scalar responses. That is, if the user specifies response functions=5, Dakota treats the five responses as five separate scalar quantities. There are some cases where responses are a “field” quantity, meaning that the responses are a function of one or more independent variables such as time and/or spatial location. In these cases, the responses should be treated as field data. For example, it can become extremely cumbersome to represent 5000 values from a time-temperature trace or a Dakota Version 6.7 User’s Manual generated on November 13, 2017 11.2. DAKOTA RESULTS FILE DATA FORMAT 213 current-voltage curve in Dakota. With scalar response quantities, we ignore the independent variable(s). For example, if we have a response R as a function of time t, the user currently gives Dakota a set of discrete responses at particular times and Dakota doesn’t know the times. With the field data capability, the user can specify that they have one field response of size 5000 x 1 (for example). Dakota will have a large set of data R = f (t), with both the response R and independent coordinates t specified to Dakota. The independent variable(s) can be useful in interpolation between simulation responses and experimental observations. It also can be useful in surrogate construction. We plan to handle correlation or structure between field responses, which is currently not handled when we treat the responses as individual, separate scalar quantities. For all three major response types (objective functions, calibration terms, and response functions), one can specify field responses (e.g. with field objectives, field calibration terms, and field responses). For each type of field response, one can specify the length of the field (e.g. lengths=5000) and the number of independent coordinates (num coordinates per field). The user can specify the independent coordinates by specifying read field coordinates and providing the coordinates in files named response descriptor.coords. In the case of field data from physical experiments used to calibrate field data from simulation experiments, the specification is more involved: the user should refer to the Dakota Reference manual to get the syntax. Note that at this time, field responses may be specified by the user as outlined above. All methods can handle field data, but currently the calibration methods are the only methods specialized for field data, specifically they interpolate the simulation field data to the experiment field data points to calculate the residual terms. This is applicable to nl2sol, nlssol, optpp g newton, the MCMC Bayesian methods, as well as general optimization methods that recast the residuals into a sum-of-squares error. The other methods simply handle the field responses as a number of scalar responses currently. In future versions, we are planning some additional features with methods that can handle field data, including reduction of the field data. 11.2 Dakota Results File Data Format Simulation interfaces using system calls and forks to create separate simulation processes must communicate with the simulation through the file system. This is done by reading and writing files of parameters and results. Dakota uses its own format for this data input/output. For the results file, only one format is supported (versus the two parameter-file formats described in Section 9.6). Ordering of response functions is as listed in Section 11.1.1 (e.g., objective functions or calibration terms are first, followed by nonlinear inequality constraints, followed by nonlinear equality constraints). After a simulation, Dakota expects to read a file containing responses reflecting the current parameters and corresponding to the function requests in the active set vector. The response data must be in the format shown in Figure 11.1. The first block of data (shown in black) conveys the requested function values and is followed by a block of requested gradients (shown in blue), followed by a block of requested Hessians (shown in red). If the amount of data in the file does not match the function request vector, Dakota will abort execution with an error message. Function values have no bracket delimiters, but each may be followed by its own non-numeric label. Labels must be separated from numeric function values by white space (one or more blanks, tabs, or newline characters) and they must not contain any white space themselves (e.g., use “response1” or “response 1,” but not “response 1”). Labels also must not resemble numerical values. By default, function value labels are optional and are ignored by Dakota; they are permitted only as a convenience to the user. However, if strict checking is activated by including the labeled keyword in the interface section of the Dakota input file, then labels are required for every function value. Further, labels must exactly match the Dakota Version 6.7 User’s Manual generated on November 13, 2017 214 CHAPTER 11. RESPONSES ... [ .. ] [ .. ] ... [ .. ] [[ .. ]] [[ .. ]] ... [[ .. ]] Figure 11.1: Results file data format. response descriptors of their corresponding function values. These stricter labeling requirements enable Dakota to detect and report when function values are returned out-of-order, or when specific function values are repeated or missing. Gradient vectors are surrounded by single brackets [. . . ndvv -vector of doubles. . . ]. Labels are not used and must not be present. White space separating the brackets from the data is optional. Hessian matrices are surrounded by double brackets [[. . . ndvv ×ndvv matrix of doubles. . . ]]. Hessian components (numeric values for second partial derivatives) are listed by rows and separated by white space; in particular, they can be spread across multiple lines for readability. Labels are not used and must not be present. White space after the initial double bracket and before the final one is optional, but none can appear within the double brackets. The format of the numeric fields may be floating point or scientific notation. In the latter case, acceptable exponent characters are “E” or “e.” A common problem when dealing with Fortran programs is that a C++ read of a numeric field using “D” or “d” as the exponent (i.e., a double precision value from Fortran) may fail or be truncated. In this case, the “D” exponent characters must be replaced either through modifications to the Fortran source or compiler flags or through a separate post-processing step (e.g., using the UNIX sed utility). 11.3 Active Variables for Derivatives An important question for proper management of both gradient and Hessian data is: if several different types of variables are used, for which variables are response function derivatives needed? That is, how is ndvv determined? The short answer is that the derivative variables vector (DVV) specifies the set of variables to be used for computing derivatives, and ndvv is the length of this vector. In most cases, the DVV is defined directly from the set of active continuous variables for the iterative method in use. Since methods operate over a subset, or view, of the variables that is active in the iteration, it is this same set of variables for which derivatives are most commonly computed. Derivatives are never needed with respect to any discrete variables (since these derivatives do not in general exist) and the active continuous variables depend on view override specifications, inference by response type, and inference by method type, in that order, as described in Section 9.5. In a few cases, derivatives are needed with respect to the inactive continuous variables. This occurs for nested Dakota Version 6.7 User’s Manual generated on November 13, 2017 11.3. ACTIVE VARIABLES FOR DERIVATIVES 215 iteration where a top-level iterator sets derivative requirements (with respect to its active continuous variables) on the final solution of the lower-level iterator (for which the top-level active variables are inactive). For example, in an uncertainty analysis within a nested design under uncertainty algorithm, derivatives of the lower-level response functions may be needed with respect to the design variables, which are active continuous variables at the top level but are inactive within the uncertainty quantification. These instances are the reason for the creation and inclusion of the DVV vector — to clearly indicate the variables whose partial derivatives are needed. In all cases, if the DVV is honored, then the correct derivative components are returned. In simple cases, such as optimization and calibration studies that only specify design variables and for nondeterministic analyses that only specify uncertain variables, derivative component subsets are not an issue and the exact content of the DVV may be safely ignored. Dakota Version 6.7 User’s Manual generated on November 13, 2017 216 CHAPTER 11. RESPONSES Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 12 Inputs to Dakota 12.1 Overview of Inputs Dakota supports a number of command-line arguments, as described in Section 2.4. Among these are specifications for the Dakota input file and, optionally, a restart file. The syntax of the Dakota input file is described in detail in the Dakota Reference Manual [3], and the restart file is described in Chapter 18. A Dakota input file may be prepared with a text editor such as Emacs, Vi, or WordPad, or with the Dakota graphical user interface. The Dakota GUI is built on the Java-based Eclipse Framework [33] and presents the Dakota input specification options in synchronized text-editing and graphical views. The Dakota GUI includes templates and wizards for helping create Dakota studies and can invoke Dakota to run an analysis. The Dakota GUI for Linux, Windows, and Mac, is available for download from the Dakota website http://dakota.sandia.gov/, along with licensing information, separate GUI documentation, and installation tips. 12.1.1 Tabular Data Formats The Dakota input file and/or command line may identify additional text files for tabular data import in contexts described in Section 12.2. Examples include data from which to build a surrogate, points at which to run a list parameter study, post-run input data, and least squares and Bayesian calibration data. Dakota writes and reads tabular data with C++ stream operators/conversions, so most integer and floating point formats are acceptable for imported numeric data. Dakota supports the following tabular formats: • Annotated: In most contexts, Dakota tabular data defaults to “annotated” tabular format. An annotated tabular file is a whitespace-separated text file with one leading header row of comments/column labels. In most imports/exports, each subsequent row contains an evaluation ID and interface ID, followed by data for variables, or variables followed by responses, depending on context. This example shows 5 variables, followed by the 1 text book response: %eval_id interface 1 I1 2 I1 3 I1 TF1ln 0.97399 0.94468 1.0279 TF1ln 1.0476 1.0636 1.0035 hpu_r1 12 4.133 12 hpu_r2 ModelForm 4.133 3 12 3 4.133 3 text_book 14753 14753 14753 Another example is shown in Figure 13.2. Note: Dakota 6.1 and newer include a column for the interface ID. See the discussion of custom-annotated format for importing/exporting Dakota 6.0 format files. 218 CHAPTER 12. INPUTS TO DAKOTA For scalar experiment data files, each subsequent row contains an experiment ID, followed by data for configuration variables, observations, and/or observation errors, depending on context. This example shows 3 data points for each of two experiments. %experiment d1 d2 d3 1 82 15.5 2.02 2 82.2 15.45 2 • Free-form: When optionally specifying freeform for a given tabular import, the data file must be provided in a free-form format, omitting the leading header row and ID column(s). The raw num rows x num cols numeric data entries may appear separated with any whitespace including arbitrary spaces, tabs, and newlines. In this format, vectors may therefore appear as a single row or single column (or mixture; entries will populate the vector in order). This example shows the free-form version of the annotated data above: 0.97399 0.94468 1.0279 1.0476 1.0636 1.0035 12 4.133 12 4.133 12 4.133 3 3 3 14753 14753 14753 • Custom-annotated: In Dakota 6.2 and newer, a custom-annotated format is supported, to allow backwardcompatibility with Dakota 6.0 tabular formats, which had a header and evaluation ID, but no interface ID. This can be specified, for example, with method list_parameter_study import_points_file = ’dakota_pstudy.3.dat’ custom_annotated header eval_id The custom annotated keyword has options to control header row, eval id column, and interface id column. In tabular files, variables appear in input specification order as documented in the reference manual. As of Dakota 6.1, tabular I/O has columns for all of the variables (active and inactive), not only the active variables as in previous versions. To import data corresponding only to the active variables, use the keyword active only when specifying the import file. Attention: Prior to October 2011, samples, calibration, and surrogate data files were free-form format. They now default to annotated format, though there are freeform and custom annotated options. For both formats, a warning will be generated if a specific number of data are expected, but extra is found and an error generated when there is insufficient data. Some TPLs like SCOLIB and JEGA manage their own file I/O and only support the free-form option. 12.2 Data Imports The Dakota input file and/or command line may identify additional files used to import data into Dakota. 12.2.1 AMPL algebraic mappings: stub.nl, stub.row, and stub.col As described in Section 16.1, an AMPL specification of algebraic input-to-output relationships may be imported into Dakota and used to define or augment the mappings of a particular interface. Dakota Version 6.7 User’s Manual generated on November 13, 2017 12.2. DATA IMPORTS 12.2.2 219 Genetic algorithm population import Genetic algorithms (GAs) from the JEGA and SCOLIB packages support a population import feature using the keywords initialization type flat file = STRING. This is useful for warm starting GAs from available data or previous runs. Refer to the Method Specification chapter in the Dakota Reference Manual [3] for additional information on this specification. The flat file must be in free-form format as described in Section 12.1.1. 12.2.3 Calibration data import Calibration methods (deterministic least squares and Bayesian) require residuals, or differences between model predictions q(θ) and data d: r(θ) = q(θ) − d, (12.1) By default, if a Dakota input file specifies responses, calibration terms, the simulation interface is required to return a vector of residuals to Dakota. If in addition the input file includes calibration data or calibration data file, Dakota assumes the interface will return the model predictions q(θ) themselves and Dakota will compute residuals based on the provided data. There are two calibration data import mechanisms: 1. Scalar responses only with calibration data file: This uses a single tabular text file to import data values and (optionally) experiment numbers, configurations, and observation variances. Each row of the data file expresses this information for a single experiment. 2. Field and/or scalar responses with calibration data: In order to accommodate the richer structure of field-valued responses, this specification requires separate data files per response group (descriptor) DESC, per experiment NUM. The files are named DESC.NUM.* and must each be in a tabular text format. The tabular data files may be specified to be annotated (default), custom annotated, or freeform format. Calibration data imports include the following information: • Configuration variables (optional): state variable values indicating the configuration at which this experiment was conducted; length must agree with the number of state variables active in the study. • Experimental observations (required): experimental data values to difference with model responses; length equal to the total response length (number of scalars + sum(field lengths)). • Experimental variances (optional): measurement errors (variances/covariances) associated with the experimental observations For specifics on specifying calibration data imports, see 7.5 and the responses >calibration terms keyword in the Dakota Reference Manual [3]. Note on variance: Field responses may optionally have scalar, diagonal, or matrix-valued error covariance information. As an example, Figure 12.1 shows an observation vector with 5 responses; 2 scalar + 3 field (each field of length >1). The corresponding covariance matrix has scalar variances σ12 , σ22 for each of the scalars s1, s2, diagonal covariance D3 for field f 3, scalar covariance σ42 for field f 4, and full matrix covariance C5 for field f 5. In total, Dakota supports block diagonal covariance Σ across the responses, with blocks Σi , which could be fully dense within a given field response group. Covariance across the highest-level responses (off-diagonal blocks) is not supported, nor is covariance between experiments. Dakota Version 6.7 User’s Manual generated on November 13, 2017 220 CHAPTER 12. INPUTS TO DAKOTA Figure 12.1: An example of scalar and field response data, with associated block-diagonal observation error covariance. 12.2.4 PCE coefficient import Polynomial chaos expansion (PCE) methods compute coefficients for response expansions which employ a basis of multivariate orthogonal polynomials. Normally, the polynomial chaos method calculates these coefficients based either on a spectral projection or a linear regression (see Section 5.4). However, Dakota also supports the option of importing a set of response PCE coefficients from a file specified with import expansion file = STRING. Each row of the free-form formatted file must be comprised of a coefficient followed by its associated multi-index (the same format used for output described in Section 13.6). This file import can be used to evaluate moments analytically or compute probabilities numerically from a known response expansion. Refer to the Method Specification chapter in the Dakota Reference Manual [3] for additional information on this specification. 12.2.5 Surrogate construction and evaluation data Global data fit surrogates, including some stochastic expansions, may be constructed from a variety of data sources. One of these sources is an auxiliary data file, as specified by the keyword import points file = STRING. The file may be in annotated (default), custom annotated, or free-form format with columns corresponding to variables and responses. Surfpack global surrogate models may also be evaluated at a user-provided file containing challenge points. Refer to the Model Specification chapter in the Dakota Reference Manual [3] for additional information on this specification. 12.2.6 Variables/responses import to post-run The post-run mode (supported only for sampling, parameter study, and DACE methods) requires specification of a file containing parameter and response data. Annotated is the default format (see Section 12.1.1), where leading columns for evaluation and interface IDs are followed by columns for variables (active and inactive by default), then those for responses, with an ignored header row of labels and then one row per evaluation. Typically this file would be generated by executing dakota -i dakota.in -pre run ::variables.dat and then adding columns of response data to variables.dat to make varsresponses.dat. The file is specified at the command line with: Dakota Version 6.7 User’s Manual generated on November 13, 2017 12.2. DATA IMPORTS 221 dakota -i dakota.in -post_run varsresponses.dat:: To import post-run data in other formats, specify post run in the input file instead of at the command-line, and provide a format option. Dakota Version 6.7 User’s Manual generated on November 13, 2017 222 CHAPTER 12. INPUTS TO DAKOTA Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 13 Output from Dakota 13.1 Overview of Output Formats While Dakota primarily targets complex numerical simulation codes run on massively parallel supercomputers, Dakota’s output focuses on succinct, text-based reporting of the iterations and function evaluations performed by an algorithm. In a number of contexts, Dakota generates tabular output useful for data analysis and visualization with external tools. Unix versions of Dakota have a basic graphical output capability that is useful as a monitoring tool. The Dakota GUI is an emerging capability that will provide more advanced visualization facilities in time. 13.2 Standard Output Dakota prints information on algorithm progress, function evaluations, and summary results to standard out and error (screen, terminal, or console) as it runs. Detailed information for a function evaluation may include an evaluation number, parameter values, execution syntax, the active set vector, and the response data set. This output may be redirected to a file using the command-line options described in Section 2.4. Here, optimization of the “container” problem (see Chapter 20) is used as an example to describe Dakota output. The input file shown in Figure 13.1 specifies one equality constraint, and that Dakota’s finite difference algorithm will provide central difference numerical gradients to the NPSOL optimizer. 224 CHAPTER 13. OUTPUT FROM DAKOTA # Dakota Input File: container_opt_npsol.in environment graphics tabular_data tabular_data_file = ’container_opt_npsol.dat’ method npsol_sqp variables continuous_design = descriptors ’H’ initial_point 4.5 lower_bounds 0.0 2 ’D’ 4.5 0.0 interface analysis_drivers = ’container’ fork parameters_file = ’container.in’ results_file = ’container.out’ file_tag responses objective_functions = 1 nonlinear_equality_constraints = 1 numerical_gradients method_source dakota interval_type central fd_step_size = 0.001 no_hessians Figure 13.1: Dakota input file for the Dakota/examples/users/container opt npsol.in “container” test problem – see Dakota Version 6.7 User’s Manual generated on November 13, 2017 13.2. STANDARD OUTPUT 225 A partial listing of the Dakota output for the container optimization example follows: Dakota version 5.4+ (stable) released May 2 2014. Subversion revision 2508 built May 2 2014 15:26:14. Running MPI Dakota executable in serial mode. Start time: Fri May 2 15:38:11 2014 ---------------------------------------------------------------Begin DAKOTA input file /home/briadam/dakota/build/examples/users/container_opt_npsol.in ---------------------------------------------------------------# Dakota Input File: container_opt_npsol.in environment graphics --------------------End DAKOTA input file --------------------Using Dakota input file ’/home/briadam/dakota/build/examples/users/container_opt_npsol.in’ Writing new restart file dakota.rst >>>>> Executing environment. >>>>> Running npsol_sqp iterator. -----------------------------------------Begin Dakota derivative estimation routine ----------------------------------------->>>>> Initial map for analytic portion of response: --------------------Begin Evaluation 1 --------------------Parameters for evaluation 1: 4.5000000000e+00 H 4.5000000000e+00 D blocking fork: container container.in.1 container.out.1 Active response data for evaluation 1: Active set vector = { 1 1 } 1.0713145108e+02 obj_fn 8.0444076396e+00 nln_eq_con_1 >>>>> Dakota finite difference gradient evaluation for x[1] + h: --------------------Begin Evaluation 2 --------------------Dakota Version 6.7 User’s Manual generated on November 13, 2017 226 CHAPTER 13. OUTPUT FROM DAKOTA Parameters for evaluation 2: 4.5045000000e+00 H 4.5000000000e+00 D blocking fork: container container.in.2 container.out.2 Active response data for evaluation 2: Active set vector = { 1 1 } 1.0719761302e+02 obj_fn 8.1159770472e+00 nln_eq_con_1 >>>>> Dakota finite difference gradient evaluation for x[1] - h: --------------------Begin Evaluation 3 --------------------Parameters for evaluation 3: 4.4955000000e+00 H 4.5000000000e+00 D blocking fork: container container.in.3 container.out.3 Active response data for evaluation 3: Active set vector = { 1 1 } 1.0706528914e+02 obj_fn 7.9728382320e+00 nln_eq_con_1 >>>>> Dakota finite difference gradient evaluation for x[2] + h: --------------------Begin Evaluation 4 --------------------Parameters for evaluation 4: 4.5000000000e+00 H 4.5045000000e+00 D blocking fork: container container.in.4 container.out.4 Active response data for evaluation 4: Active set vector = { 1 1 } 1.0727959301e+02 obj_fn 8.1876180243e+00 nln_eq_con_1 >>>>> Dakota finite difference gradient evaluation for x[2] - h: --------------------Begin Evaluation 5 --------------------Parameters for evaluation 5: 4.5000000000e+00 H 4.4955000000e+00 D Dakota Version 6.7 User’s Manual generated on November 13, 2017 13.2. STANDARD OUTPUT 227 blocking fork: container container.in.5 container.out.5 Active response data for evaluation 5: Active set vector = { 1 1 } 1.0698339109e+02 obj_fn 7.9013403937e+00 nln_eq_con_1 >>>>> Total response returned to iterator: Active set vector = { 3 3 } Deriv vars 1.0713145108e+02 8.0444076396e+00 [ 1.4702653619e+01 3.2911324639e+01 [ 1.5904312809e+01 3.1808625618e+01 vector = { 1 2 } obj_fn nln_eq_con_1 ] obj_fn gradient ] nln_eq_con_1 gradient >>>>> Dakota finite difference gradient evaluation for x[2] - h: --------------------Begin Evaluation 40 --------------------Parameters for evaluation 40: 4.9873894231e+00 H 4.0230575428e+00 D blocking fork: container container.in.40 container.out.40 Active response data for evaluation 40: Active set vector = { 1 1 } 9.8301287596e+01 obj_fn -1.2698647501e-01 nln_eq_con_1 >>>>> Total response returned to iterator: Active set vector = { 3 3 } Deriv vars 9.8432498116e+01 -9.6918029158e-12 [ 1.3157517860e+01 3.2590159623e+01 [ 1.2737124497e+01 3.1548877601e+01 vector = { 1 2 } obj_fn nln_eq_con_1 ] obj_fn gradient ] nln_eq_con_1 gradient NPSOL exits with INFORM code = 0 (see "Interpretation of output" section in NPSOL manual) NOTE: see Fortran device 9 file (fort.9 or ftn09) for complete NPSOL iteration history. <<<<< Function evaluation summary: 40 total (40 new, 0 duplicate) <<<<< Best parameters = Dakota Version 6.7 User’s Manual generated on November 13, 2017 228 CHAPTER 13. OUTPUT FROM DAKOTA 4.9873894231e+00 H 4.0270846274e+00 D <<<<< Best objective function = 9.8432498116e+01 <<<<< Best constraint values = -9.6918029158e-12 <<<<< Best data captured at function evaluation 36 <<<<< Iterator npsol_sqp completed. <<<<< Environment execution completed. DAKOTA execution time in seconds: Total CPU = 0.03 [parent = Total wall clock = 0.090703 Exit graphics window to terminate DAKOTA. 0.023997, child = 0.006003] The output begins with information on the Dakota version, compilation date, and run mode. It then echos the user input file before proceeding to execution phase. The lines following “>>>>> Running npsol sqp iterator.” show Dakota performing function evaluations 1–5 that have been requested by NPSOL. Evaluations 6 through 39 have been omitted from the listing for brevity. Immediately following the line “Begin Function Evaluation 1”, the initial values of the design variables, the syntax of the blocking fork function evaluation, and the resulting objective and constraint function values returned by the simulation are listed. The values of the design variables are labeled with the tags H and D, respectively, according to the descriptors given in the input file, Figure 13.1. The values of the objective function and volume constraint are labeled with the tags obj fn and nln eq con 1, respectively. Note that the initial design parameters are infeasible since the equality constraint is violated (6= 0). However, by the end of the run, the optimizer finds a design that is both feasible and optimal for this example. Between the design variables and response values, the content of the system call to the simulator is displayed as “(container container.in.1 container.out.1)”, with container being the name of the simulator and container.in.1 and container.out.1 being the names of the parameters and results files, respectively. Just preceding the output of the objective and constraint function values is the line “Active set vector = {1 1}”. The active set vector indicates the types of data that are required from the simulator for the objective and constraint functions, and values of “1” indicate that the simulator must return values for these functions (gradient and Hessian data are not required). For more information on the active set vector, see Section 9.7. Since finite difference gradients have been specified, Dakota computes their values by making additional function evaluation requests to the simulator at perturbed parameter values. Examples of the gradient-related function evaluations have been included in the sample output, beginning with the line that reads “>>>>> Dakota finite difference evaluation for x[1] + h:”. The resulting finite difference gradients are listed after function evaluation 5 beginning with the line “>>>>> Total response returned to iterator:”. Here, another active set vector is displayed in the Dakota output file. The line “Active set vector = { 3 3 }” indicates that the total response resulting from the finite differencing contains function values and gradients. The final lines of the Dakota output, beginning with the line “<<<<< Function evaluation summary:”, summarize the results of the optimization study. The best values of the optimization parameters, objective function, and volume constraint are presented along with the function evaluation number where they occurred, total function evaluation counts, and a timing summary. In the end, the objective function has been minimized and the equality constraint has been satisfied (driven to zero within the constraint tolerance). The Dakota results may be intermixed with iteration information from the NPSOL library. For example lines with Dakota Version 6.7 User’s Manual generated on November 13, 2017 13.3. TABULAR OUTPUT DATA %eval_id interface 1 NO_ID 2 NO_ID 3 NO_ID 4 NO_ID 5 NO_ID 6 NO_ID 7 NO_ID 8 NO_ID H 4.5 5.801246882 5.197920019 4.932877133 4.989328733 4.987494493 4.987391669 4.987389423 229 D 4.5 3.596476363 3.923577479 4.044776216 4.026133158 4.027041977 4.02708372 4.027084627 obj_fn 107.1314511 94.33737399 97.7797214 98.28930566 98.4270019 98.43249058 98.43249809 98.43249812 nln_eq_con_1 8.04440764 -4.59103645 -0.6780884711 -0.1410680284 -0.005324671422 -7.307058453e-06 -2.032538049e-08 -9.691802916e-12 Figure 13.2: Dakota’s tabular output file showing the iteration history of the “container” optimization problem. the heading “Majr Minr Step Fun Merit function Norm gZ Violtn nZ Penalty Conv” come from Fortran write statements within NPSOL. The output is mixed since both Dakota and NPSOL are writing to the same standard output stream. The relative locations of these output contributions can vary depending on the specifics of output buffering and flushing on a particular platform and depending on whether or not the standard output is being redirected to a file. In some cases, output from the optimization library may appear on each iteration (as in this example), and in other cases, it may appear at the end of the Dakota output. Finally, a more detailed summary of the NPSOL iterations is written to the Fortran device 9 file (e.g., fort.9 or ftn09). 13.3 Tabular Output Data In a number of contexts, Dakota can output information in a whitespace-separated columnar data file, a tabular data file. The most common usage, to capture the iteration history in a tabular file, is enabled by including the keyword tabular data in the environment specification (see Figure 13.1). This output format facilitates the transfer of Dakota’s iteration history data to external mathematical analysis and/or graphics plotting packages (e.g., MATLAB, TECplot, Excel, S-plus, Minitab). The default file name for the top-level tabular output data is “dakota tabular.dat,” though tabular data file allows specification of an alternate name. Example tabular output from the “container” optimization problem is shown in Figure 13.2. This annotated tabular format (see Section 12.1.1) file contains the complete history of data requests from NPSOL (8 requests map into a total of 40 function evaluations when including the central finite differencing). The first column is the data request number, the second column is the interface ID (which is NO ID if the user does not specify a name for the interface), the third and fourth columns are the design parameter values (labeled in the example as “H” and “D”), the fifth column is the objective function (labeled “obj fn”), and the sixth column is the nonlinear equality constraint (labeled “nln eq con 1”). Attention: The second column labeled “interface” is new as of Dakota 6.1. It identifies which interface was used to map the variables to responses on each line of the tabular file (recall that the interface defines which simulation is being run though the analysis driver specification). Disambiguating the interface is important when using hybrid methods, multi-fidelity methods, or nested models. In more common, simpler analyses, users typically ignore the first two columns and only focus on the columns of inputs (variables) and outputs (responses). To generate tabular output in Dakota 6.0 format, use the custom-annotated format described in Section 12.1.1. Attention: As of Dakota 6.1, the tabular file will include columns for all of the variables (both active and inactive) present in a given interface. Previously, Dakota only wrote the “active” variables. Recall that some variables may be inactive if they are not operated on by a particular method (e.g. uncertain variables might not be active in an optimization, design variables may not be active in a sampling study). The order of the variables printed out will Dakota Version 6.7 User’s Manual generated on November 13, 2017 230 CHAPTER 13. OUTPUT FROM DAKOTA Figure 13.3: Dakota 2D graphics for “container” problem showing history of an objective function, an equality constraint, and two variables. be in Dakota’s standard variable ordering, which is indicated by the input specification order, and summarized in the Dakota Reference Manual. Any evaluations from Dakota’s internal finite differencing are suppressed, to facilitate rapid plotting of the most critical data. This suppression of lower level data is consistent with the data that is sent to the graphics windows, as described in Section 13.4. If this data suppression is undesirable, Section 18.2.3 describes an approach where every function evaluation, even the ones from finite differencing, can be saved to a file in tabular format by using the Dakota restart utility. 13.4 Graphics Output Graphics capabilities are available for monitoring the progress of an iterative study. The graphics option is invoked by adding the graphics flag in the environment specification of the Dakota input file (see Figure 13.1). The graphics display the values of each response function (e.g., objective and constraint functions) and each parameter for the function evaluations in the study. As for the tabular output described in Section 13.3, internal finite difference evaluations are suppressed in order to omit this clutter from the graphics. Figure 13.3 shows the optimization iteration history for the container example. If Dakota is executed on a remote machine, the DISPLAY variable in the user’s UNIX environment [58] may need to be set to the local machine in order to display the graphics window. The scroll bars which are located on each graph below and to the right of each plot may be operated by dragging on the bars or pressing the arrows, both of which result in expansion/contraction of the axis scale. Clicking on the “Options” button results in the window shown in Figure 13.4, which allows the user to include min/max markers on the vertical axis, vertical and horizontal axis labels, and a plot legend within the corresponding graphics plot. In addition, the values of either or both axes may be plotted using a logarithmic scale (so long as all plot values are greater than zero) and an encapsulated postscript (EPS) file, named dakota graphic i.eps where i is the plot window number, can be created using the “Print” button. 13.5 Error Messages Output A variety of error messages are printed by Dakota in the event that an error is detected in the input specification. Some of the more common input errors, and the associated error messages, are described below. See also the Common Specification Mistakes section in the Dakota Reference Manual [3]. Dakota Version 6.7 User’s Manual generated on November 13, 2017 13.5. ERROR MESSAGES OUTPUT 231 Figure 13.4: Options for Dakota 2D graphics. Incorrectly spelled specifications, such as ‘‘numericl gradients’’, will result in error messages of the form: Input line 31: unrecognized identifier ’numericl_gradients’. Input line 31: unrecognized identifier ’method_source’. Input line 31: unrecognized identifier ’dakota’. Input line 31: unrecognized identifier ’interval_type’. Input line 31: unrecognized identifier ’central’. Input line 31: unrecognized identifier ’fd_gradient_step_size’. Input line 31: One of the following 4 entities must be specified for responses... analytic_gradients mixed_gradients no_gradients numerical_gradients In this example the line numbers given are approximate, as all input following an errant keywords is considered a single line through the end of the block. The input parser catches syntax errors, but not logic errors. The fact that certain input combinations are erroneous must be detected after parsing, at object construction time. For example, if a no gradients specification for a response data set is combined with selection of a gradient-based optimization method, then this error must be detected during set-up of the optimizer (see last line of listing): Error: gradient-based minimizers require a gradient specification. Many such errors can be detected earlier by running dakota -check. Another common mistake involves a mismatch between the amount of data expected on a function evaluation and the data returned by the user’s simulation code or driver. The available response data is specified in the responses keyword block, and the subset of this data needed for a particular evaluation is managed by the active set vector. For example, if Dakota expects function values and gradients to be returned (as indicated by an active set vector containing 3’s), but the user’s simulation code only returns function values, then the following error message is generated: At EOF: insufficient data for functionGradient 1 Dakota Version 6.7 User’s Manual generated on November 13, 2017 232 CHAPTER 13. OUTPUT FROM DAKOTA Unfortunately, descriptive error messages are not available for all possible failure modes of Dakota. If you encounter core dumps, segmentation faults, or other failures, please request help using the support mechanisms described on the Dakota website. 13.6 Stochastic expansion exports Polynomial chaos expansion (PCE) methods compute coefficients for response expansions which employ a basis of multivariate orthogonal polynomials. The polynomial chaos method calculates these coefficients based on a number of approaches described in Section 5.4). One may output the PCE coefficients to a file using the keyword export expansion file = STRING. Each row of the exported file will contain a coefficient, followed by the multi-index indicating which basis terms correspond to it. Only free-form format (Section 12.1.1) is supported for this file. When using numerical integration schemes with structured rules, Dakota can also output the integration points and corresponding weights to a tabular file. This output is generated when method output is verbose or higher. Weights and points are printed to a file dakota quadrature tabular.dat (tensor product quadrature), dakota sparse tabular.dat (sparse grids), or dakota cubature tabular.dat (cubature methods), with one line per integration point. 13.7 Surrogate Model Exports Most Dakota surrogate models, including all those implemented in Surfpack and some stochastic expansion approaches support the keyword export approx points file = STRING. When specified, any approximate evaluations performed on the surrogate model will be output to the specified data file. The data file can be exported in any of the tabular formats described in Section 12.1.1 (default annotated). This facilitates plotting or external diagnostics of the surrogate model. In addition, the Surfpack family of global surrogate models supports export to a human-readable and selfdocumenting algebraic form, suitable for reuse in user-developed tools, and to non-human readable model files, which are intended for use with the surfpack executable or library API. The export model keyword group is used to specify model export filenames and formats. It is fully described in the Dakota Reference Manual [3]. 13.8 Variables Output from Pre-run The pre-run mode (supported only for select methods) permits specification of an output file to which Dakota will write parameter (variables) data in any supported tabular format (default annotated; see Section 12.1.1) with data columns corresponding to each variable. This file can be generated with sampling, parameter study, and DACE methods by invoking dakota -i dakota.in -pre_run ::variables.dat for example, to output the variables (samples) in an LHS study. If a user adds the corresponding response values to this file, it may then be imported using Dakota’s post-run mode. Command line pre-run will always export in annotated format. To export pre-run data in other formats, specify pre run in the input file instead of at the command-line, and provide a format option. Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 14 Advanced Methods 14.1 Overview A variety of “meta-algorithm” capabilities have been developed in order to provide a mechanism for employing individual iterators and models as reusable components within higher-level solution approaches. This capability allows the use of existing iterative algorithm and computational model software components as building blocks to accomplish more sophisticated studies, such as hybrid, multistart, Pareto, or surrogate-based minimization. Further multi-component capabilities are enabled by the model recursion capabilities described in Chapter 8 with specific examples in Chapter 15. 14.2 Hybrid Minimization In the hybrid minimization method (keyword: hybrid), a sequence of minimization methods are applied to find an optimal design point. The goal of this method is to exploit the strengths of different minimization algorithms through different stages of the minimization process. Global/local optimization hybrids (e.g., genetic algorithms combined with nonlinear programming) are a common example in which the desire for a global optimum is balanced with the need for efficient navigation to a local optimum. An important related feature is that the sequence of minimization algorithms can employ models of varying fidelity. In the global/local case, for example, it would often be advantageous to use a low-fidelity model in the global search phase, followed by use of a more refined model in the local search phase. The specification for hybrid minimization involves a list of method identifier strings, and each of the corresponding method specifications has the responsibility for identifying the model specification (which may in turn identify variables, interface, and responses specifications) that each method will use (see the Dakota Reference Manual [3] and the example discussed below). Currently, only the sequential hybrid approach is available. The embedded and collaborative approaches are not fully functional at this time. In the sequential hybrid minimization approach, a sequence of minimization methods is invoked in the order specified in the Dakota input file. After each method completes execution, the best solution or solutions from that method are used as the starting point(s) for the following method. The number of solutions transferred is defined by how many that method can generate and how many the user specifies with the individual method keyword final solutions. For example, currently only a few of the global optimization methods such as genetic algorithms (e.g. moga and coliny ea) and sampling methods return multiple solutions. In this case, 234 CHAPTER 14. ADVANCED METHODS the specified number of solutions from the previous method will be used to initialize the subsequent method. If the subsequent method cannot accept multiple input points (currently only a few methods such as the genetic algorithms in JEGA allow multiple input points), then multiple instances of the subsequent method are generated, each one initialized by one of the optimal solutions from the previous method. For example, if LHS sampling were run as the first method and the number of final solutions was 10 and the DOT conjugate gradient was the second method, there would be 10 instances of dot frcg started, each with a separate LHS sample solution as its initial point. Method switching is governed by the separate convergence controls of each method; that is, each method is allowed to run to its own internal definition of completion without interference. Individual method completion may be determined by convergence criteria (e.g., convergence tolerance) or iteration limits (e.g., max iterations). Figure 14.1 shows a Dakota input file that specifies a sequential hybrid optimization method to solve the “textbook” optimization test problem. The textbook hybrid strat.in file provided in Dakota/examples/users starts with a coliny ea solution which feeds its best point into a coliny pattern search optimization which feeds its best point into optpp newton. While this approach is overkill for such a simple problem, it is useful for demonstrating the coordination between multiple sub-methods in the hybrid minimization algorithm. The three optimization methods are identified using the method list specification in the hybrid method section of the input file. The identifier strings listed in the specification are ‘GA’ for genetic algorithm, ‘PS’ for pattern search, and ‘NLP’ for nonlinear programming. Following the hybrid method keyword block are the three corresponding method keyword blocks. Note that each method has a tag following the id method keyword that corresponds to one of the method names listed in the hybrid method keyword block. By following the identifier tags from method to model and from model to variables, interface, and responses, it is easy to see the specification linkages for this problem. The GA optimizer runs first and uses model ‘M1’ which includes variables ‘V1’, interface ‘I1’, and responses ‘R1’. Note that in the specification, final solutions=1, so only one (the best) solution is returned from the GA. However, it is possible to change this to final solutions=5 and get five solutions passed from the GA to the Pattern Search (for example). Once the GA is complete, the PS optimizer starts from the best GA result and again uses model ‘M1’. Since both GA and PS are nongradient-based optimization methods, there is no need for gradient or Hessian information in the ‘R1’ response keyword block. The NLP optimizer runs last, using the best result from the PS method as its starting point. It uses model ‘M2’ which includes the same ‘V1’ and ‘I1’ keyword blocks, but uses the responses keyword block ‘R2’ since the full Newton optimizer used in this example (optpp newton) needs analytic gradient and Hessian data to perform its search. 14.3 Multistart Local Minimization A simple, heuristic, global minimization technique is to use many local minimization runs, each of which is started from a different initial point in the parameter space. This is known as multistart local minimization. This is an attractive method in situations where multiple local optima are known or expected to exist in the parameter space. However, there is no theoretical guarantee that the global optimum will be found. This approach combines the efficiency of local minimization methods with a user-specified global stratification (using a specified starting points list, a number of specified random starts, or both; see the Dakota Reference Manual [3] for additional specification details). Since solutions for different starting points are independent, parallel computing may be used to concurrently run the local minimizations. An example input file for multistart local optimization on the “quasi sine” test function (see quasi sine fcn.C in Dakota Source/test) is shown in Figure 14.2. The method keyword block in the input file contains the keyword multi start, along with the set of starting points (3 random and 5 listed) that will be used for the optimization runs. The other keyword blocks in the input file are similar to what would be used in a single Dakota Version 6.7 User’s Manual generated on November 13, 2017 14.3. MULTISTART LOCAL MINIMIZATION 235 # Dakota Input File: textbook_hybrid_strat.in environment top_method_pointer = ’HS’ graphics method id_method = ’HS’ hybrid sequential method_pointer_list = ’GA’ ’PS’ ’NLP’ method id_method = ’GA’ coliny_ea seed = 1234 population_size = 5 model_pointer = ’M1’ final_solutions = 1 output verbose method id_method = ’PS’ coliny_pattern_search stochastic seed = 1234 initial_delta = 0.1 threshold_delta = 1.e-4 solution_target = 1.e-10 exploratory_moves basic_pattern model_pointer = ’M1’ output verbose method id_method = ’NLP’ optpp_newton gradient_tolerance = 1.e-12 convergence_tolerance = 1.e-15 model_pointer = ’M2’ output verbose model id_model = ’M1’ single interface_pointer = ’I1’ variables_pointer = ’V1’ responses_pointer = ’R1’ model id_model = ’M2’ single interface_pointer = ’I1’ variables_pointer = ’V1’ responses_pointer = ’R2’ variables id_variables = ’V1’ continuous_design = 2 initial_point 0.6 upper_bounds 5.8 lower_bounds 0.5 descriptors ’x1’ 0.7 2.9 -2.9 ’x2’ interface id_interface = ’I1’ analysis_drivers = ’text_book’ direct responses id_responses = ’R1’ objective_functions = 1 no_gradients no_hessians responses id_responses = ’R2’ objective_functions = 1 analytic_gradients analytic_hessians Figure 14.1: Dakota input file for a sequential hybrid Dakota/examples/users/textbook hybrid strat.in Dakota Version 6.7 User’s Manual generated on November 13, 2017 optimization method – see 236 CHAPTER 14. ADVANCED METHODS optimization run. The quasi sine test function has multiple local minima, but there is an overall trend in the function that tends toward the global minimum at (x1, x2) = (0.177, 0.177). See [61] for more information on this test function. Figure 14.3 shows the results summary for the eight local optimizations performed. From the five specified starting points and the 3 random starting points (as identified by the x1, x2 headers), the eight local optima (as identified by the x1*, x2* headers) are all different and only one of the local optimizations finds the global minimum. 14.4 Pareto Optimization The Pareto optimization method (keyword: pareto set) is one of three multiobjective optimization capabilities discussed in Section 6.3.1. In the Pareto optimization method, multiple sets of multiobjective weightings are evaluated. The user can specify these weighting sets in the method keyword block using a multi objective weight sets list, a number of random weight sets, or both (see the Dakota Reference Manual [3] for additional specification details). Dakota performs one multiobjective optimization problem for each set of multiobjective weights. The collection of computed optimal solutions form a Pareto set, which can be useful in making trade-off decisions in engineering design. Since solutions for different multiobjective weights are independent, parallel computing may be used to concurrently execute the multiobjective optimization problems. Figure 14.5 shows the results summary for the Pareto-set optimization method. For the four multiobjective weighting sets (as identified by the w1, w2, w3 headers), the local optima (as identified by the x1, x2 headers) are all different and correspond to individual objective function values of (f1 , f2 , f3 ) = (0.0,0.5,0.5), (13.1,1.2,8.16), (532.,33.6,-2.9), and (0.125,0.0,0.0) (note: the composite objective function is tabulated under the obj fn header). The first three solutions reflect exclusive optimization of each of the individual objective functions in turn, whereas the final solution reflects a balanced weighting and the lowest sum of the three objectives. Plotting these (f1 , f2 , f3 ) triplets on a 3-dimensional plot results in a Pareto surface (not shown), which is useful for visualizing the trade-offs in the competing objectives. 14.5 Mixed Integer Nonlinear Programming (MINLP) Many nonlinear optimization problems involve a combination of discrete and continuous variables. These are known as mixed integer nonlinear programming (MINLP) problems. A typical MINLP optimization problem is formulated as follows: minimize: f (x, d) subject to: gL ≤ g(x, d) ≤ gU h(x, d) = ht (14.1) xL ≤ x ≤ xU d ∈ {−2, −1, 0, 1, 2} where d is a vector whose elements are integer values. In situations where the discrete variables can be temporarily relaxed (i.e., noncategorical discrete variables, see Section 9.2.2), the branch-and-bound algorithm can be applied. Categorical variables (e.g., true/false variables, feature counts, etc.) that are not relaxable cannot be used with the branch and bound method. During the branch and bound process, the discrete variables are treated Dakota Version 6.7 User’s Manual generated on November 13, 2017 14.5. MIXED INTEGER NONLINEAR PROGRAMMING (MINLP) 237 # Dakota Input File: qsf_multistart_strat.in environment top_method_pointer = ’MS’ graphics method id_method = ’MS’ multi_start method_pointer = ’NLP’ random_starts = 3 seed = 123 starting_points = -0.8 -0.8 -0.8 0.8 0.8 -0.8 0.8 0.8 0.0 0.0 method id_method = ’NLP’ dot_bfgs variables continuous_design = 2 lower_bounds -1.0 upper_bounds 1.0 descriptors ’x1’ -1.0 1.0 ’x2’ interface analysis_drivers = ’quasi_sine_fcn’ fork #asynchronous responses objective_functions = 1 analytic_gradients no_hessians Figure 14.2: Dakota input file for a multistart Dakota/examples/users/qsf multistart strat.in <<<<< Results summary: set_id x1 1 -0.8 2 -0.8 3 0.8 4 0.8 5 0 6 -0.7767971993 7 -0.3291571008 8 0.8704730469 x2 -0.8 0.8 -0.8 0.8 0 0.01810943539 -0.7697378755 0.7720679005 x1* -0.8543728666 -0.9998398719 0.177092822 0.1770928217 0.03572926375 -0.7024118387 0.3167607374 0.177092899 local optimization x2* -0.8543728666 0.177092822 -0.9998398719 0.1770928217 0.03572926375 0.03572951143 -0.4009188363 0.3167611757 method obj_fn 0.5584096919 0.291406596 0.291406596 0.0602471946 0.08730499239 0.3165522387 0.2471403213 0.08256082751 Figure 14.3: Dakota results summary for a multistart local optimization method. Dakota Version 6.7 User’s Manual generated on November 13, 2017 – see 238 CHAPTER 14. ADVANCED METHODS # Dakota Input File: textbook_pareto_strat.in environment top_method_pointer = ’PS’ graphics method id_method = ’PS’ pareto_set method_pointer = ’NLP’ weight_sets = 1.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 1.0 0.333 0.333 0.333 method id_method = ’NLP’ dot_bfgs model single variables continuous_design = 2 initial_point 0.9 upper_bounds 5.8 lower_bounds 0.5 descriptors ’x1’ 1.1 2.9 -2.9 ’x2’ interface analysis_drivers = ’text_book’ direct responses objective_functions = 3 analytic_gradients no_hessians Figure 14.4: Dakota input file for the Pareto Dakota/examples/users/textbook pareto strat.in <<<<< Results summary: set_id w1 1 1 2 0 3 0 4 0.333 w2 0 1 0 0.333 w3 0 0 1 0.333 optimization method – see x1 x2 obj_fn 0.9996554048 0.997046351 7.612301561e-11 0.5 2.9 -1.2 5.8 1.12747589e-11 -2.9 0.5 0.5000000041 0.041625 Figure 14.5: Dakota results summary for the Pareto-set optimization method. Dakota Version 6.7 User’s Manual generated on November 13, 2017 14.5. MIXED INTEGER NONLINEAR PROGRAMMING (MINLP) 239 as continuous variables and the integrality conditions on these variables are incrementally enforced through a sequence of optimization subproblems. By the end of this process, an optimal solution that is feasible with respect to the integrality conditions is computed. Dakota’s branch and bound method (keyword: branch and bound) can solve optimization problems having either discrete or mixed continuous/discrete variables. This method uses the parallel branch-and-bound algorithm from the PEBBL software package [32] to generate a series of optimization subproblems (“branches”). These subproblems are solved as continuous variable problems using any of Dakota’s nonlinear optimization algorithms (e.g., DOT, NPSOL). When a solution to a branch is feasible with respect to the integrality constraints, it provides an upper bound on the optimal objective function, which can be used to prune branches with higher objective functions that are not yet feasible. Since solutions for different branches are independent, parallel computing may be used to concurrently execute the optimization subproblems. PEBBL, by itself, targets the solution of mixed integer linear programming (MILP) problems, and through coupling with Dakota’s nonlinear optimizers, is extended to solution of MINLP problems. In the case of MILP problems, the upper bound obtained with a feasible solution is an exact bound and the branch and bound process is provably convergent to the global minimum. For nonlinear problems which may exhibit nonconvexity or multimodality, the process is heuristic in general, since there may be good solutions that are missed during the solution of a particular branch. However, the process still computes a series of locally optimal solutions, and is therefore a natural extension of the results from local optimization techniques for continuous domains. Only with rigorous global optimization of each branch can a global minimum be guaranteed when performing branch and bound on nonlinear problems of unknown structure. In cases where there are only a few discrete variables and when the discrete values are drawn from a small set, then it may be reasonable to perform a separate optimization problem for all of the possible combinations of the discrete variables. However, this brute force approach becomes computationally intractable if these conditions are not met. The branch-and-bound algorithm will generally require solution of fewer subproblems than the brute force method, although it will still be significantly more expensive than solving a purely continuous design problem. 14.5.1 Example MINLP Problem As an example, consider the following MINLP problem [43]: minimize: f (x) = 6 X (xi − 1.4)4 i=1 x2 ≤0 2 x 1 g2 = x22 − ≤0 2 −10 ≤ x1 , x2 , x3 , x4 ≤ 10 g1 = x21 − (14.2) x5 , x6 ∈ {0, 1, 2, 3, 4} This problem is a variant of the textbook test problem described in Section 20.1. In addition to the introduction of two integer variables, a modified value of 1.4 is used inside the quartic sum to render the continuous solution a non-integral solution. Figure 14.6 shows the sequence of branches generated for this problem. The first optimization subproblem relaxes the integrality constraint on parameters x5 and x6 , so that 0 ≤ x5 ≤ 4 and 0 ≤ x6 ≤ 4. The values for x5 and x6 Dakota Version 6.7 User’s Manual generated on November 13, 2017 240 CHAPTER 14. ADVANCED METHODS Figure 14.6: Branching history for example MINLP optimization problem. at the solution to this first subproblem are x5 = x6 = 1.4. Since x5 and x6 must be integers, the next step in the solution process “branches” on parameter x5 to create two new optimization subproblems; one with 0 ≤ x5 ≤ 1 and the other with 2 ≤ x5 ≤ 4. Note that, at this first branching, the bounds on x6 are still 0 ≤ x6 ≤ 4. Next, the two new optimization subproblems are solved. Since they are independent, they can be performed in parallel. The branch-and-bound process continues, operating on both x5 and x6 , until a optimization subproblem is solved where x5 and x6 are integer-valued. At the solution to this problem, the optimal values for x5 and x6 are x5 = x6 = 1. In this example problem, the branch-and-bound algorithm executes as few as five and no more than seven optimization subproblems to reach the solution. For comparison, the brute force approach would require 25 optimization problems to be solved (i.e., five possible values for each of x5 and x6 ). In the example given above, the discrete variables are integer-valued. In some cases, the discrete variables may be real-valued, such as x ∈ {0.0, 0.5, 1.0, 1.5, 2.0}. The branch-and-bound algorithm is restricted to work with integer values. Therefore, it is up to the user to perform a transformation between the discrete integer values from Dakota and the discrete real values that are passed to the simulation code (see Section 9.2.2). When integrality is not being relaxed, a common mapping is to use the integer value from Dakota as the index into a vector of discrete real values. However, when integrality is relaxed, additional logic for interpolating between the discrete real values is needed. 14.6 Surrogate-Based Minimization Surrogate models approximate an original, high fidelity “truth” model, typically at reduced computational cost. In Dakota, several surrogate model selections are possible, which are categorized as data fits, multifidelity models, Dakota Version 6.7 User’s Manual generated on November 13, 2017 14.6. SURROGATE-BASED MINIMIZATION 241 and reduced-order models, as described in Section 8.4. In the context of minimization (optimization or calibration), surrogate models can speed convergence by reducing function evaluation cost or smoothing noisy response functions. Three categories of surrogate-based minimization are discussed in this chapter: • Trust region-managed surrogate-based local minimization, with data fit surrogate, multifidelity models, or reduced-order models. • Surrogate-based global minimization, where a single surrogate is built (and optionally iteratively updated) over the whole design space. • Efficient global minimization: nongradient-based constrained and unconstrained optimization and nonlinear least squares based on Gaussian process models, guided by an expected improvement function. 14.6.1 Surrogate-Based Local Minimization In the surrogate-based local minimization method (keyword: surrogate based local) the minimization algorithm operates on a surrogate model instead of directly operating on the computationally expensive simulation model. The surrogate model can be based on data fits, multifidelity models, or reduced-order models, as described in Section 8.4. Since the surrogate will generally have a limited range of accuracy, the surrogate-based local algorithm periodically checks the accuracy of the surrogate model against the original simulation model and adaptively manages the extent of the approximate optimization cycles using a trust region approach. Refer to the Dakota Theory Manual [4] for algorithmic details on iterate acceptance, merit function formulations, convergence assessment, and constraint relaxation. 14.6.1.1 SBO with Data Fits When performing SBO with local, multipoint, and global data fit surrogates, it is necessary to regenerate or update the data fit for each new trust region. In the global data fit case, this can mean performing a new design of experiments on the original high-fidelity model for each trust region, which can effectively limit the approach to use on problems with, at most, tens of variables. Figure 14.7 displays this case. However, an important benefit of the global sampling is that the global data fits can tame poorly-behaved, nonsmooth, discontinuous response variations within the original model into smooth, differentiable, easily navigated surrogates. This allows SBO with global data fits to extract the relevant global design trends from noisy simulation data. When enforcing local consistency between a global data fit surrogate and a high-fidelity model at a point, care must be taken to balance this local consistency requirement with the global accuracy of the surrogate. In particular, performing a correction on an existing global data fit in order to enforce local consistency can skew the data fit and destroy its global accuracy. One approach for achieving this balance is to include the consistency requirement within the data fit process by constraining the global data fit calculation (e.g., using constrained linear least squares). This allows the data fit to satisfy the consistency requirement while still addressing global accuracy with its remaining degrees of freedom. Embedding the consistency within the data fit also reduces the sampling requirements. For example, a quadratic polynomial normally requires at least (n + 1)(n + 2)/2 samples for n variables to perform the fit. However, with an embedded first-order consistency constraint at a single point, the minimum number of samples is reduced by n + 1 Figure 14.7: SBO iteration proto (n2 + n)/2. gression for global data fits. Dakota Version 6.7 User’s Manual generated on November 13, 2017 242 CHAPTER 14. ADVANCED METHODS In the local and multipoint data fit cases, the iteration progression will appear as in Fig. 14.9. Both cases involve a single new evaluation of the original high-fidelity model per trust region, with the distinction that multipoint approximations reuse information from previous SBO iterates. Like model hierarchy surrogates, these techniques scale to larger numbers of design variables. Unlike model hierarchy surrogates, they generally do not require surrogate corrections, since the matching conditions are embedded in the surrogate form (as discussed for the global Taylor series approach above). The primary disadvantage to these surrogates is that the region of accuracy tends to be smaller than for global data fits and multifidelity surrogates, requiring more SBO cycles with smaller trust regions. More information on the design of experiments methods is available in Chapter 4, and the data fit surrogates are described in Section 8.4.3. Figure 14.8 shows a Dakota input file that implements surrogate-based optimization on Rosenbrock’s function. The first method keyword block contains the SBO keyword surrogate based local, plus the commands for specifying the trust region size and scaling factors. The optimization portion of SBO, using the CONMIN Fletcher-Reeves conjugate gradient method, is specified in the following keyword blocks for method, model, variables, and responses. The model used by the optimization method specifies that a global surrogate will be used to map variables into responses (no interface specification is used by the surrogate model). The global surrogate is constructed using a DACE method which is identified with the ‘SAMPLING’ identifier. This data sampling portion of SBO is specified in the final set of keyword blocks for method, model, interface, and responses (the earlier variables specification is reused). This example problem uses the Latin hypercube sampling method in the LHS software to select 10 design points in each trust region. A single surrogate model is constructed for the objective function using a quadratic polynomial. The initial trust region is centered at the design point (x1 , x2 ) = (−1.2, 1.0), and extends ±0.4 (10% of the global bounds) from this point in the x1 and x2 coordinate directions. If this input file is executed in Dakota, it will converge to the optimal design point at (x1 , x2 ) = (1, 1) in approximately 800 function evaluations. While this solution is correct, it is obtained at a much higher cost than a traditional gradient-based optimizer (e.g., see the results obtained in Section 2.3.3). This demonstrates that the SBO method with global data fits is not really intended for use with smooth continuous optimization problems; direct gradient-based optimization can be more efficient for such applications. Rather, SBO with global data fits is best-suited for the types of problems that occur in engineering design where the response quantities may be discontinuous, nonsmooth, or may have multiple local optima [60]. In these types of engineering design problems, traditional gradient-based optimizers often are ineffective, whereas global data fits can extract the global trends of interest despite the presence of local nonsmoothness (for an example problem with multiple local optima, look in Dakota/test for the file dakota sbo sine fcn.in [61]). The surrogate-based local minimizer is only mathematically guaranteed to find a local minimum. However, in practice, SBO can often find the global minimum. Due to the random sampling method used within the SBO algorithm, the SBO method will solve a given problem a little differently each time it is run (unless the user specifies a particular random number seed in the dakota input file as is shown in Figure 14.8). Our experience on the quasi-sine function mentioned above is that if you run this problem 10 times with the same starting conditions but different seeds, then you will find the global minimum in about 70-80% of the trials. This is good performance for what is mathematically only a local optimization method. 14.6.1.2 SBO with Multifidelity Models When performing SBO with model hierarchies, the low-fidelity model is normally fixed, requiring only a single high-fidelity evaluation to compute a new correction for each new trust region. Figure 14.9 displays this case. This renders the multifidelity SBO technique more scalable to larger numbers of design variables since the number of high-fidelity evaluations per iteration (assuming no finite differencing for derivatives) is independent of the scale of the design problem. However, the ability to smooth poorly-behaved response variations in the high-fidelity Dakota Version 6.7 User’s Manual generated on November 13, 2017 14.6. SURROGATE-BASED MINIMIZATION 243 # Dakota Input File: rosen_opt_sbo.in environment tabular_data tabular_data_file = ’rosen_opt_sbo.dat’ top_method_pointer = ’SBLO’ method id_method = ’SBLO’ surrogate_based_local model_pointer = ’SURROGATE’ method_pointer = ’NLP’ max_iterations = 500 trust_region initial_size = 0.10 minimum_size = 1.0e-6 contract_threshold = 0.25 expand_threshold = 0.75 contraction_factor = 0.50 expansion_factor = 1.50 method id_method = ’NLP’ conmin_frcg max_iterations = 50 convergence_tolerance = 1e-8 model id_model = ’SURROGATE’ surrogate global correction additive zeroth_order polynomial quadratic dace_method_pointer = ’SAMPLING’ responses_pointer = ’SURROGATE_RESP’ variables continuous_design = 2 initial_point -1.2 1.0 lower_bounds -2.0 -2.0 upper_bounds 2.0 2.0 descriptors ’x1’ ’x2’ responses id_responses = ’SURROGATE_RESP’ objective_functions = 1 numerical_gradients method_source dakota interval_type central fd_step_size = 1.e-6 no_hessians method id_method = ’SAMPLING’ sampling samples = 10 seed = 531 sample_type lhs model_pointer = ’TRUTH’ model id_model = ’TRUTH’ single interface_pointer = ’TRUE_FN’ responses_pointer = ’TRUE_RESP’ interface id_interface = ’TRUE_FN’ analysis_drivers = ’rosenbrock’ direct deactivate evaluation_cache restart_file responses id_responses = ’TRUE_RESP’ objective_functions = 1 no_gradients no_hessians Figure 14.8: Dakota input file for the surrogate-based Dakota/examples/users/rosen opt sbo.in Dakota Version 6.7 User’s Manual generated on November 13, 2017 local optimization example – see 244 CHAPTER 14. ADVANCED METHODS model is lost, and the technique becomes dependent on having a well-behaved low-fidelity model1 . In addition, the parameterizations for the low and high-fidelity models may differ, requiring the use of a mapping between these parameterizations. Space mapping, corrected space mapping, POD mapping, and hybrid POD space mapping are being explored for this purpose [112, 113]. When applying corrections to the low-fidelity model, there is no concern for balancing global accuracy with the local consistency requirements. However, with only a single high-fidelity model evaluation at the center of each trust region, it is critical to use the best correction possible on the low-fidelity model in order to achieve rapid convergence rates to the optimum of the highfidelity model [37]. A multifidelity test problem named dakota sbo hierarchical.in is available in Dakota/test to demonstrate this SBO approach. This test problem uses the Rosenbrock function as the high fidelity model and a function named “lf rosenbrock” as the low fidelity model. Here, lf rosenbrock is a variant of the Rosenbrock function (see Dakota Source/test/lf rosenbrock.C for formulation) with the minimum point at (x1 , x2 ) = (0.80, 0.44), whereas the minimum of the original Rosenbrock function is (x1 , x2 ) = (1, 1). Multifidelity SBO locates the Figure 14.9: SBO iteration prohigh-fidelity minimum in 11 high fidelity evaluations for additive second- gression for model hierarchies. order corrections and in 208 high fidelity evaluations for additive first-order corrections, but fails for zeroth-order additive corrections by converging to the low-fidelity minimum. 14.6.1.3 SBO with Reduced Order Models When performing SBO with reduced-order models (ROMs), the ROM is mathematically generated from the highfidelity model. A critical issue in this ROM generation is the ability to capture the effect of parametric changes within the ROM. Two approaches to parametric ROM are extended ROM (E-ROM) and spanning ROM (S-ROM) techniques [147]. Closely related techniques include tensor singular value decomposition (SVD) methods [92]. In the single-point and multipoint E-ROM cases, the SBO iteration can appear as in Fig. 14.9, whereas in the S-ROM, global E-ROM, and tensor SVD cases, the SBO iteration will appear as in Fig. 14.7. In addition to the high-fidelity model analysis requirements, procedures for updating the system matrices and basis vectors are also required. Relative to data fits and multifidelity models, ROMs have some attractive advantages. Compared to data fits such as regression-based polynomial models, they are more physics-based and would be expected to be more predictive (e.g., in extrapolating away from the immediate data). Compared to multifidelity models, ROMS may be more practical in that they do not require multiple computational models or meshes which are not always available. The primary disadvantage is potential invasiveness to the simulation code for projecting the system using the reduced basis. 14.6.2 Surrogate-Based Global Minimization Surrogate-based global minimization differs from the surrogate-based local minimization approach discussed in the previous section in several ways: it is not a trust-region approach; initially there is one global surrogate 1 It is also possible to use a hybrid data fit/multifidelity approach in which a smooth data fit of a noisy low fidelity model is used in combination with a high fidelity model Dakota Version 6.7 User’s Manual generated on November 13, 2017 14.6. SURROGATE-BASED MINIMIZATION 245 constructed over a set of sample points and the optimizer operates on that surrogate (as opposed to adaptively selecting points and re-building a surrogate in each trust region); and there is no guarantee of convergence. The surrogate based global method was developed to address two needs. The first is the case where a user wishes to use existing function evaluations or a fixed sample size (perhaps based on computational cost and allocation of resources) to build a surrogate once and optimize on it. In this case (a single global optimization on a surrogate model), the set of surrogate building points is determined in advance as opposed to the trust-region local surrogate optimization in which the number of “true” function evaluations depends on the location and size of the trust region, the goodness of the surrogate within the trust-region, and problem characteristics. In the second surrogate based global use case, we want to update the surrogate, but globally. That is, we add points to the sample set used to create the surrogate, rebuild the surrogate, and then perform another global optimization on the new surrogate. Thus, surrogate-based global optimization can be used in an iterative scheme. In one iteration, minimizers of the surrogate model are found, and a selected subset of these are passed to the next iteration. In the next iteration, these surrogate points are evaluated with the “truth” model, and then added to the set of points upon which the next surrogate is constructed. This presents a more accurate surrogate to the minimizer at each subsequent iteration, presumably driving to optimality quickly. Note that a global surrogate is constructed using the same bounds in each iteration. This approach has no guarantee of convergence. The surrogate-based global method was originally designed for MOGA (a multi-objective genetic algorithm). Since genetic algorithms often need thousands or tens of thousands of points to produce optimal or near-optimal solutions, surrogates can help by reducing the necessary truth model evaluations. Instead of creating one set of surrogates for the individual objectives and running the optimization algorithm on the surrogate once, the idea is to select points along the (surrogate) Pareto frontier, which can be used to supplement the existing points. In this way, one does not need to use many points initially to get a very accurate surrogate. The surrogate becomes more accurate as the iterations progress. Most single objective optimization methods will return only a single optimal point. In that case, only one point from the surrogate model will be evaluated with the “true” function and added to the pointset upon which the surrogate is based. In this case, it will take many iterations of the surrogate-based global optimization for the approach to converge, and its utility may not be as great as for the multi-objective case when multiple optimal solutions are passed from one iteration to the next to supplement the surrogate. Note that the user has the option of appending the optimal points from the surrogate model to the current set of truth points or using the optimal points from the surrogate model to replace the optimal set of points from the previous iteration. Although appending to the set is the default behavior, at this time we strongly recommend using the option replace points because it appears to be more accurate and robust. When using the surrogate-based global method, we first recommend running one optimization on a single surrogate model. That is, set max iterations to 1. This will allow one to get a sense of where the optima are located and also what surrogate types are the most accurate to use for the problem. Note that by fixing the seed of the sample on which the surrogate is built, one can take a Dakota input file, change the surrogate type, and re-run the problem without any additional function evaluations by specifying the use of the dakota restart file which will pick up the existing function evaluations, create the new surrogate type, and run the optimization on that new surrogate. Also note that one can specify that surrogates be built for all primary functions and constraints or for only a subset of these functions and constraints. This allows one to use a ”truth” model directly for some of the response functions, perhaps due to them being much less expensive than other functions. Finally, a diagnostic threshold can be used to stop the method if the surrogate is so poor that it is unlikely to provide useful points. If the goodness-of-fit has an R-squared value less than 0.5, meaning that less than half the variance of the output can be explained or accounted for by the surrogate model, the surrogate-based global optimization stops and outputs an error message. This is an arbitrary threshold, but generally one would want to have an R-squared value as close to 1.0 as possible, and an R-squared value below 0.5 indicates a very poor fit. For the surrogate-based global method, we initially recommend a small number of maximum iterations, such as Dakota Version 6.7 User’s Manual generated on November 13, 2017 246 CHAPTER 14. ADVANCED METHODS 3–5, to get a sense of how the optimization is evolving as the surrogate gets updated globally. If it appears to be changing significantly, then a larger number (used in combination with restart) may be needed. Figure 14.10 shows a Dakota input file that implements surrogate-based global optimization on a multi-objective test function. The first method keyword block contains the keyword surrogate based global, plus the commands for specifying five as the maximum iterations and the option to replace points in the global surrogate construction. The method block identified as MOGA specifies a multi-objective genetic algorithm optimizer and its controls. The model keyword block specifies a surrogate model. In this case, a gaussian process model is used as a surrogate. The dace method pointer specifies that the surrogate will be build on 100 Latin Hypercube samples with a seed = 531. The remainder of the input specification deals with the interface to the actual analysis driver and the 2 responses being returned as objective functions from that driver. Dakota Version 6.7 User’s Manual generated on November 13, 2017 14.6. SURROGATE-BASED MINIMIZATION # Dakota Input File: mogatest1_opt_sbo.in environment tabular_data tabular_data_file = ’mogatest1_opt_sbo.dat’ top_method_pointer = ’SBGO’ method id_method = ’SBGO’ surrogate_based_global model_pointer = ’SURROGATE’ method_pointer = ’MOGA’ max_iterations = 5 replace_points output verbose method id_method = ’MOGA’ moga seed = 10983 population_size = 300 max_function_evaluations = 5000 initialization_type unique_random crossover_type shuffle_random num_offspring = 2 num_parents = 2 crossover_rate = 0.8 mutation_type replace_uniform mutation_rate = 0.1 fitness_type domination_count replacement_type below_limit = 6 shrinkage_percentage = 0.9 niching_type distance 0.05 0.05 postprocessor_type orthogonal_distance 0.05 0.05 convergence_type metric_tracker percent_change = 0.05 num_generations = 10 output silent model id_model = ’SURROGATE’ surrogate global dace_method_pointer = ’SAMPLING’ correction additive zeroth_order gaussian_process dakota method id_method = ’SAMPLING’ sampling samples = 100 seed = 531 sample_type lhs model_pointer = ’TRUTH’ model id_model = ’TRUTH’ single interface_pointer = ’TRUE_FN’ variables continuous_design = 3 initial_point 0 upper_bounds 4 lower_bounds -4 descriptors ’x1’ 0 4 -4 ’x2’ 0 4 -4 ’x3’ interface id_interface = ’TRUE_FN’ analysis_drivers = ’mogatest1’ direct responses objective_functions = 2 no_gradients no_hessians Dakota Version 6.7 User’s Manual generated on November 13, 2017 Figure 14.10: MOGA example – see Dakota/examples/users/mogatest1 opt sbo.in 247 248 CHAPTER 14. ADVANCED METHODS Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 15 Advanced Model Recursions The surrogate and nested model constructs admit a wide variety of multi-iterator, multi-model solution approaches. For example, optimization within optimization (for hierarchical multidisciplinary optimization), uncertainty quantification within uncertainty quantification (for interval-valued probability, second-order probability, or Dempster-Shafer approaches to mixed aleatory-epistemic UQ), uncertainty quantification within optimization (for optimization under uncertainty), and optimization within uncertainty quantification (for uncertainty of optima) are all supported, with and without surrogate model indirection. Three important examples are highlighted: mixed aleatory-epistemic UQ, optimization under uncertainty, and surrogate-based UQ. Starting with Dakota version 6.1, concurrency can now be exploited across sub-iteration instances. For example, multiple inner loop UQ assessments can be performed simultaneously within optimization under uncertainty or mixed aleatory-epistemic UQ studies, provided the outer loop algorithm supports concurrency in its evaluations. Both meta-iterators and nested models support iterator servers, processors per iterator, and iterator scheduling specifications which can be used to define a parallel configuration that partitions servers for supporting sub-iteration concurrency. Refer to Chapter 17 for additional information on parallel configurations, and to the Methods and Models chapters of the Reference Manual [3] for additional information on these specifications. 15.1 Mixed Aleatory-Epistemic UQ Mixed UQ approaches employ nested models to embed one uncertainty quantification (UQ) within another. The outer level UQ is commonly linked to epistemic uncertainties (also known as reducible uncertainties) resulting from a lack of knowledge, and the inner UQ is commonly linked to aleatory uncertainties (also known as irreducible uncertainties) that are inherent in nature. The outer level generates sets of realizations of the epistemic parameters, and each set of these epistemic parameters in used within a separate inner loop probabilistic analysis over the aleatory random variables. In this manner, ensembles of aleatory statistics are generated, one set for each realization of the epistemic parameters. In Dakota, we support interval-valued probability (IVP), second-order probability (SOP), and Dempster-Shafer theory of evidence (DSTE) approaches to mixed uncertainty. These three approaches differ in how they treat the epistemic variables in the outer loop: they are treated as intervals in IVP, as belief structures in DSTE, and as subjective probability distributions in SOP. This set of techniques provides a spectrum of assumed epistemic structure, from strongest assumptions in SOP to weakest in IVP. 250 15.1.1 CHAPTER 15. ADVANCED MODEL RECURSIONS Interval-valued probability (IVP) In IVP (also known as probability bounds analysis [47, 85, 9]), we employ an outer loop of interval estimation in combination with an aleatory inner loop. In interval analysis, it is assumed that nothing is known about the uncertain input variables except that they lie within certain intervals. The problem of uncertainty propagation then becomes an interval analysis problem: given inputs that are defined within intervals, what are the corresponding intervals on the outputs? Starting from a specification of intervals and probability distributions on the inputs, the intervals may augment the probability distributions, insert into the probability distributions, or some combination (refer to Section 8.5 and to the Models chapter of the Reference Manual [3]). We generate an ensemble of cumulative distribution functions (CDF) or Complementary Cumulative Distribution Functions (CCDF), one CDF/CCDF result for each aleatory analysis. Plotting an entire ensemble of CDFs or CCDFs in a “horsetail” plot allows one to visualize the upper and lower bounds on the family of distributions (see Figure 15.1). Given that the ensemble stems Figure 15.1: Example CDF ensemble. Commonly referred to as a “horsetail” plot. from multiple realizations of the epistemic uncertainties, the interpretation is that each CDF/CCDF instance has no relative probability of occurrence, only that each instance is possible. For prescribed response levels on the CDF/CCDF, an interval on the probability is computed based on the bounds of the ensemble at that level, and vice versa for prescribed probability levels. This interval on a statistic is interpreted simply as a possible range, where the statistic could take any of the possible values in the range. A sample input file is shown in Figure 15.2, in which the outer epistemic level variables are defined as intervals. Samples will be generated from these intervals to select means for X and Y that are employed in an inner level reliability analysis of the cantilever problem (see Section 20.7). Figure 15.3 shows excerpts from the resulting output. In this particular example, the outer loop generates 50 possible realizations of epistemic variables, which are then sent to the inner loop to calculate statistics such as the mean weight, and cumulative distribution function for the stress and displacement reliability indices. Thus, the outer loop has 50 possible values for the mean weight, but since there is no distribution structure on these observations, only the minimum and maximum value are reported. Similarly, the minimum and maximum values of the CCDF for the stress and displacement reliability indices are reported. When performing a mixed aleatory-epistemic analysis, response levels and probability levels should only be Dakota Version 6.7 User’s Manual generated on November 13, 2017 15.1. MIXED ALEATORY-EPISTEMIC UQ 251 defined in the (inner) aleatory loop. For example, if one wants to generate an interval around possible CDFs or CCDFS, we suggest defining a number of probability levels in the inner loop (0.1, 0.2, 0.3, etc). For each epistemic instance, these will be calculated during the inner loop and reported back to the outer loop. In this way, there will be an ensemble of CDF percentiles (for example) and one will have interval bounds for each of these percentile levels defined. Finally, although the epistemic variables are often values defining distribution parameters for the inner loop, they are not required to be: they can just be separate uncertain variables in the problem. As compared to aleatory quantities of interest (e.g., mean, variance, probability) that must be integrated over a full probability domain, we observe that the desired minima and maxima of the output ranges are local point solutions in the epistemic parameter space, such that we may employ directed optimization techniques to compute these extrema and potentially avoid the cost of sampling the full epistemic space. In dakota/test, test input files such as dakota uq cantilever ivp exp.in and dakota uq short column ivp exp.in replace the outer loop sampling with the local and global interval optimization methods described in Section 5.7.1. In these cases, we no longer generate horse tails and infer intervals, but rather compute the desired intervals directly. 15.1.2 Second-order probability (SOP) SOP is similar to IVP in its segregation of aleatory and epistemic uncertainties and its use of nested iteration. However, rather than modeling epistemic uncertainty with a single interval per variable and computing intervalvalued statistics, we instead employ subjective probability distributions and compute epistemic statistics on the aleatory statistics (for example, probabilities on probabilities – the source of the “second-order” terminology [66]). Now the different hairs of the horsetail shown in Figure 15.1 have a relative probability of occurrence and stronger inferences may be drawn. In particular, mean, 5th percentile, and 95th percentile probability values are a common example. Second-order probability is sometimes referred to as probability of frequency (PoF) analysis, referring to a probabilistic interpretation of the epistemic variables and a frequency interpretation of the aleatory variables. The PoF terminology is used in a recent National Academy of Sciences report on the Quantification of Margins and Uncertainties (QMU) [105]. Rather than employing interval estimation techniques at the outer loop in SOP, we instead apply probabilistic methods, potentially the same ones as used for the aleatory propagation on the inner loop. The previous example in Figure 15.2 can be modified to define the epistemic outer loop using uniform variables instead of interval variables (annotated test #1 in dakota/test/dakota uq cantilever sop rel.in). The process of generating the epistemic values is essentially the same in both cases; however, the interpretation of results is quite different. In IVP, each “hair” or individual CDF in the horsetail plot in Figure 15.1 would be interpreted as a possible realization of aleatory uncertainty conditional on a particular epistemic sample realization. The ensemble then indicates the influence of the epistemic variables (e.g. by how widespread the ensemble is). However, if the outer loop variables are defined to be uniformly distributed in SOP, then the outer loop results will be reported as statistics (such as mean and standard deviation) and not merely intervals. It is important to emphasize that these outer level output statistics are only meaningful to the extent that the outer level input probability specifications are meaningful (i.e., to the extent that uniform distributions are believed to be representative of the epistemic variables). In dakota/test, additional test input files such as dakota uq cantilever sop exp.in and dakota uq short column sop exp.in explore other outer/inner loop probabilistic analysis combinations, particulary using stochastic expansion methods. Dakota Version 6.7 User’s Manual generated on November 13, 2017 252 CHAPTER 15. ADVANCED MODEL RECURSIONS # Dakota Input File: cantilever_uq_sop_rel.in environment top_method_pointer = ’EPISTEMIC’ method id_method = ’EPISTEMIC’ sampling samples = 50 seed = 12347 model_pointer = ’EPIST_M’ model id_model = ’EPIST_M’ nested sub_method_pointer = ’ALEATORY’ primary_variable_mapping = ’X’ ’Y’ secondary_variable_mapping = ’mean’ ’mean’ primary_response_mapping = 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1. variables_pointer = ’EPIST_V’ responses_pointer = ’EPIST_R’ variables id_variables = ’EPIST_V’ continuous_interval_uncertain = 2 num_intervals = 1 1 interval_probabilities = 1.0 lower_bounds = 400.0 800.0 upper_bounds = 600.0 1200.0 descriptors ’X_mean’ ’Y_mean’ 1.0 responses id_responses = ’EPIST_R’ response_functions = 3 descriptors = ’mean_wt’ ’ccdf_beta_s’ ’ccdf_beta_d’ no_gradients no_hessians method id_method = ’ALEATORY’ local_reliability mpp_search no_approx response_levels = 0.0 0.0 num_response_levels = 0 1 1 compute reliabilities distribution complementary model_pointer = ’ALEAT_M’ model id_model = ’ALEAT_M’ single interface_pointer = ’ALEAT_I’ variables_pointer = ’ALEAT_V’ responses_pointer = ’ALEAT_R’ variables id_variables = ’ALEAT_V’ continuous_design = 2 initial_point 2.4522 3.8826 descriptors ’w’ ’t’ normal_uncertain = 4 means = 40000. 29.E+6 500. 1000. std_deviations = 2000. 1.45E+6 100. 100. descriptors = ’R’ ’E’ ’X’ ’Y’ interface id_interface = ’ALEAT_I’ analysis_drivers = ’cantilever’ direct deactivate evaluation_cache restart_file responses id_responses = ’ALEAT_R’ response_functions = 3 descriptors = ’weight’ ’stress’ ’displ’ analytic_gradients no_hessians Figure 15.2: Dakota input file for the interval-valued Dakota/examples/users/cantilever uq sop rel.in probability example – see Dakota Version 6.7 User’s Manual generated on November 13, 2017 15.2. OPTIMIZATION UNDER UNCERTAINTY (OUU) 253 Statistics based on 50 samples: Min and Max values for each response function: mean_wt: Min = 9.5209117200e+00 Max = 9.5209117200e+00 ccdf_beta_s: Min = 1.7627715524e+00 Max = 4.2949468386e+00 ccdf_beta_d: Min = 2.0125192955e+00 Max = 3.9385559339e+00 Figure 15.3: Interval-valued statistics for cantilever beam reliability indices. 15.1.3 Dempster-Shafer Theory of Evidence In IVP, we estimate a single epistemic output interval for each aleatory statistic. This same nested analysis procedure may be employed within the cell computations of a DSTE approach. Instead of a single interval, we now compute multiple output intervals, one for each combination of the input basic probability assignments, in order to define epistemic belief and plausibility functions on the aleatory statistics computed in the inner loop. While this can significantly increase the computational requirements, belief and plausibility functions provide a more finely resolved epistemic characterization than a basic output interval. The single-level DSTE approach for propagating epistemic uncertainties is described in Section 5.7.2 and in the Dakota Theory Manual [4]. An example of nested DSTE for propagating mixed uncertainties can be seen in dakota/test in the input file dakota uq ishigami dste exp.in. 15.2 Optimization Under Uncertainty (OUU) Optimization under uncertainty (OUU) approaches incorporate an uncertainty quantification method within the optimization process. This is often needed in engineering design problems when one must include the effect of input parameter uncertainties on the response functions of interest. A typical engineering example of OUU would minimize the probability of failure of a structure for a set of applied loads, where there is uncertainty in the loads and/or material properties of the structural components. In OUU, a nondeterministic method is used to evaluate the effect of uncertain variable distributions on response functions of interest (refer to Chapter 5 for additional information on nondeterministic analysis). Statistics on these response functions are then included in the objective and constraint functions of an optimization process. Different UQ methods can have very different features from an optimization perspective, leading to the tailoring of optimization under uncertainty approaches to particular underlying UQ methodologies. If the UQ method is sampling based, then three approaches are currently supported: nested OUU, surrogate-based OUU, and trust-region surrogate-based OUU. Additional details and computational results are provided in [38]. Another class of OUU algorithms is called reliability-based design optimization (RBDO). RBDO methods are used to perform design optimization accounting for reliability metrics. The reliability analysis capabilities described in Section 5.3 provide a rich foundation for exploring a variety of RBDO formulations. [35] investigated bi-level, fully-analytic bi-level, and first-order sequential RBDO approaches employing underlying first-order reliability assessments. [36] investigated fully-analytic bi-level and second-order sequential RBDO approaches employing underlying second-order reliability assessments. When using stochastic expansions for UQ, analytic moments and analytic design sensitivities can be exploited as described in [45]. Several approaches for obtaining design sensitivities of statistical metrics are discussed in Section 15.2.5. Dakota Version 6.7 User’s Manual generated on November 13, 2017 254 CHAPTER 15. ADVANCED MODEL RECURSIONS Figure 15.4: Formulation 1: Nested OUU. Finally, when employing epistemic methods for UQ, the set of statistics available for use within optimization are interval-based. Robustness metrics typically involve the width of the intervals, and reliability metrics typically involve the worst case upper or lower bound of the interval. Each of these OUU methods is overviewed in the following sections. 15.2.1 Nested OUU In the case of a nested approach, the optimization loop is the outer loop which seeks to optimize a nondeterministic quantity (e.g., minimize probability of failure). The uncertainty quantification (UQ) inner loop evaluates this nondeterministic quantity (e.g., computes the probability of failure) for each optimization function evaluation. Figure 15.4 depicts the nested OUU iteration where d are the design variables, u are the uncertain variables characterized by probability distributions, ru (d, u) are the response functions from the simulation, and su (d) are the statistics generated from the uncertainty quantification on these response functions. Figure 15.5 shows a Dakota input file for a nested OUU example problem that is based on the textbook test problem. In this example, the objective function contains two probability of failure estimates, and an inequality constraint contains another probability of failure estimate. For this example, failure is defined to occur when one of the textbook response functions exceeds its threshold value. The environment keyword block at the top of the input file identifies this as an OUU problem. The environment keyword block is followed by the optimization specification, consisting of the optimization method, the continuous design variables, and the response quantities that will be used by the optimizer. The mapping matrices used for incorporating UQ statistics into the optimization response data are described in the Dakota Reference Manual [3]. The uncertainty quantification specification includes the UQ method, the uncertain variable probability distributions, the interface to the simulation code, and the UQ response attributes. As with other complex Dakota input files, the identification tags given in each keyword block can be used to follow the relationships among the different keyword blocks. Latin hypercube sampling is used as the UQ method in this example problem. Thus, each evaluation of the response functions by the optimizer entails 50 Latin hypercube samples. In general, nested OUU studies can easily generate several thousand function evaluations and gradient-based optimizers may not perform well due to noisy or insensitive statistics resulting from under-resolved sampling. These observations motivate the use of surrogate-based approaches to OUU. Other nested OUU examples in the dakota/test directory include dakota ouu1 tbch.in, which adds an additional interface for including deterministic data in the textbook OUU problem, and dakota ouu1 cantilever.in, which solves the cantilever OUU problem (see Section 20.7) with a nested approach. For each of these files, the “1” identifies formulation 1, which is short-hand for the nested approach. Dakota Version 6.7 User’s Manual generated on November 13, 2017 15.2. OPTIMIZATION UNDER UNCERTAINTY (OUU) 255 # Dakota Input File: textbook_opt_ouu1.in environment top_method_pointer = ’OPTIM’ method id_method = ’OPTIM’ npsol_sqp convergence_tolerance = 1.e-10 model_pointer = ’OPTIM_M’ model id_model = ’OPTIM_M’ nested sub_method_pointer = ’UQ’ primary_response_mapping = 0. 0. 1. 0. 0. 1. 0. 0. 0. secondary_response_mapping = 0. 0. 0. 0. 0. 0. 0. 0. 1. variables_pointer = ’OPTIM_V’ responses_pointer = ’OPTIM_R’ variables id_variables = ’OPTIM_V’ continuous_design = 2 initial_point 1.8 upper_bounds 2.164 lower_bounds 1.5 descriptors ’d1’ 1.0 4.0 0.0 ’d2’ responses id_responses = ’OPTIM_R’ objective_functions = 1 nonlinear_inequality_constraints = 1 upper_bounds = .1 numerical_gradients method_source dakota interval_type central fd_step_size = 1.e-1 no_hessians method id_method = ’UQ’ sampling model_pointer = ’UQ_M’ samples = 50 sample_type lhs seed = 1 response_levels = 3.6e+11 1.2e+05 3.5e+05 distribution complementary model id_model = ’UQ_M’ single interface_pointer = ’UQ_I’ variables_pointer = ’UQ_V’ responses_pointer = ’UQ_R’ variables id_variables = ’UQ_V’ continuous_design = 2 normal_uncertain = 2 means = 248.89 593.33 std_deviations = 12.4 29.7 descriptors = ’nuv1’ ’nuv2’ uniform_uncertain = 2 lower_bounds = 199.3 474.63 upper_bounds = 298.5 712. descriptors = ’uuv1’ ’uuv2’ weibull_uncertain = 2 alphas = 12. 30. betas = 250. 590. descriptors = ’wuv1’ ’wuv2’ interface id_interface = ’UQ_I’ analysis_drivers = ’text_book_ouu’ direct # fork asynch evaluation_concurrency = 5 responses id_responses = ’UQ_R’ response_functions = 3 no_gradients no_hessians Figure 15.5: Dakota input file for the Dakota/examples/users/textbook opt ouu1.in Dakota Version 6.7 User’s Manual generated on November 13, 2017 nested OUU example – see 256 CHAPTER 15. ADVANCED MODEL RECURSIONS Figure 15.6: Formulations 2, 3, and 4 for Surrogate-based OUU. 15.2.2 Surrogate-Based OUU (SBOUU) Surrogate-based optimization under uncertainty strategies can be effective in reducing the expense of OUU studies. Possible formulations include use of a surrogate model at the optimization level, at the uncertainty quantification level, or at both levels. These surrogate models encompass both data fit surrogates (at the optimization or UQ level) and model hierarchy surrogates (at the UQ level only). Figure 15.6 depicts the different surrogate-based formulations where r̂u and ŝu are approximate response functions and approximate response statistics, respectively, generated from the surrogate models. SBOUU examples in the dakota/test directory include dakota sbouu2 tbch.in, dakota sbouu3 tbch.in, and dakota sbouu4 tbch.in, which solve the textbook OUU problem, and dakota sbouu2 cantilever.in, dakota sbouu3 cantilever.in, and dakota sbouu4 cantilever.in, which solve the cantilever OUU problem (see Section 20.7). For each of these files, the “2,” “3,” and “4” identify formulations 2, 3, and 4, which are short-hand for the “layered containing nested,” “nested containing layered,” and “layered containing nested containing layered” surrogatebased formulations, respectively. In general, the use of surrogates greatly reduces the computational expense of these OUU study. However, without restricting and verifying the steps in the approximate optimization cycles, weaknesses in the data fits can be exploited and poor solutions may be obtained. The need to maintain accuracy of results leads to the use of trust-region surrogate-based approaches. 15.2.3 Trust-Region Surrogate-Based OUU (TR-SBOUU) The TR-SBOUU approach applies the trust region logic of deterministic SBO (see Section 14.6.1) to SBOUU. Trust-region verifications are applicable when surrogates are used at the optimization level, i.e., formulations 2 and 4. As a result of periodic verifications and surrogate rebuilds, these techniques are more expensive than SBOUU; however they are more reliable in that they maintain the accuracy of results. Relative to nested OUU (formulation 1), TR-SBOUU tends to be less expensive and less sensitive to initial seed and starting point. TR-SBOUU examples in the dakota/test directory include dakota trsbouu2 tbch.in and dakota trsbouu4 tbch.in, which solve the textbook OUU problem, and dakota trsbouu2 cantilever.in and dakota trsbouu4 cantilever.in, which solve the cantilever OUU problem (see Section 20.7). Dakota Version 6.7 User’s Manual generated on November 13, 2017 15.2. OPTIMIZATION UNDER UNCERTAINTY (OUU) 257 Computational results for several example problems are available in [38]. 15.2.4 RBDO Bi-level and sequential approaches to reliability-based design optimization (RBDO) and their associated sensitivity analysis requirements are described in the Optimization Under Uncertainty chapter of the Dakota Theory Manual [4]. A number of bi-level RBDO examples are provided in dakota/test. The dakota rbdo cantilever.in, dakota rbdo short column.in, and dakota rbdo steel column.in input files solve the cantilever (see Section 20.7), short column (see Section 20.10.4), and steel column (see Section 20.10.5) OUU problems using a bi-level RBDO approach employing numerical design gradients. The dakota rbdo cantilever analytic.in and dakota rbdo short column analytic.in input files solve the cantilever and short column OUU problems using a bi-level RBDO approach with analytic design gradients and first-order limit state approximations. The dakota rbdo cantilever analytic2.in, dakota rbdo short column analytic2.in, and dakota rbdo steel column analytic2.in input files also employ analytic design gradients, but are extended to employ second-order limit state approximations and integrations. Sequential RBDO examples are also provided in dakota/test. The dakota rbdo cantilever trsb.in and dakota rbdo short column trsb.in input files solve the cantilever and short column OUU problems using a first-order sequential RBDO approach with analytic design gradients and first-order limit state approximations. The dakota rbdo cantilever trsb2.in, dakota rbdo short column trsb2.in, and dakota rbdo steel column trsb2.in input files utilize second-order sequential RBDO approaches that employ second-order limit state approximations and integrations (from analytic limit state Hessians with respect to the uncertain variables) and quasi-Newton approximations to the reliability metric Hessians with respect to design variables. 15.2.5 Stochastic Expansion-Based Design Optimization For stochastic expansion-based approaches to optimization under uncertainty, bi-level, sequential, and multifidelity approaches and their associated sensitivity analysis requirements are described in the Optimization Under Uncertainty chapter of the Dakota Theory Manual [4]. In dakota/test, the dakota pcbdo cantilever.in, dakota pcbdo rosenbrock.in, dakota pcbdo short column.in, and dakota pcbdo steel column.in input files solve cantilever (see Section 20.7), Rosenbrock, short column (see Section 20.10.4), and steel column (see Section 20.10.5) OUU problems using a bi-level polynomial chaos-based approach, where the statistical design metrics are reliability indices based on moment projection (see Mean Value section in Reliability Methods Chapter of Dakota Theory Manual [4]). The test matrix in the former three input files evaluate design gradients of these reliability indices using several different approaches: analytic design gradients based on a PCE formed over only over the random variables, analytic design gradients based on a PCE formed over all variables, numerical design gradients based on a PCE formed only over the random variables, and numerical design gradients based on a PCE formed over all variables. In the cases where the expansion is formed over all variables, only a single PCE construction is required for the complete PCBDO process, whereas the expansions only over the random variables must be recomputed for each change in design variables. Sensitivities for “augmented” design variables (which are separate from and augment the random variables) may be handled using either analytic approach; however, sensitivities for “inserted” design variables (which define distribution parameters for the random variables) must be computed dx using dR dx ds (refer to Stochastic Sensitivity Analysis section in Optimization Under Uncertainty chapter of Dakota Theory Manual [4]). Additional test input files include: Dakota Version 6.7 User’s Manual generated on November 13, 2017 258 CHAPTER 15. ADVANCED MODEL RECURSIONS • dakota scbdo cantilever.in, dakota scbdo rosenbrock.in, dakota scbdo short column.in, and dakota scbdo steel column.in input files solve cantilever, Rosenbrock, short column, and steel column OUU problems using a bi-level stochastic collocationbased approach. • dakota pcbdo cantilever trsb.in, dakota pcbdo rosenbrock trsb.in, dakota pcbdo short column trsb.in, dakota pcbdo steel column trsb.in, dakota scbdo cantilever trsb.in, dakota scbdo rosenbrock trsb.in, dakota scbdo short column trsb.in, and dakota scbdo steel column trsb.in input files solve cantilever, Rosenbrock, short column, and steel column OUU problems using sequential polynomial chaos-based and stochastic collocation-based approaches. • dakota pcbdo cantilever mf.in, dakota pcbdo rosenbrock mf.in, dakota pcbdo short column mf.in, dakota scbdo cantilever mf.in, dakota scbdo rosenbrock mf.in, and dakota scbdo short column mf.in input files solve cantilever, Rosenbrock, and short column OUU problems using multifidelity polynomial chaos-based and stochastic collocation-based approaches. 15.2.6 Epistemic OUU An emerging capability is optimization under epistemic uncertainty. As described in the Nested Model section of the Reference Manual [3], epistemic and mixed aleatory/epistemic uncertainty quantification methods generate lower and upper interval bounds for all requested response, probability, reliability, and generalized reliability level mappings. Design for robustness in the presence of epistemic uncertainty could simply involve minimizing the range of these intervals (subtracting lower from upper using the nested model response mappings), and design for reliability in the presence of epistemic uncertainty could involve controlling the worst case upper or lower bound of the interval. We now have the capability to perform epistemic analysis by using interval optimization on the “outer loop” to calculate bounding statistics of the aleatory uncertainty on the “inner loop.” Preliminary studies [44] have shown this approach is more efficient and accurate than nested sampling (which was described in Section 15.1.2). This approach uses an efficient global optimization method for the outer loop and stochastic expansion methods (e.g. polynomial chaos or stochastic collocation on the inner loop). The interval optimization is described in Section 5.7.1. Example input files demonstrating the use of interval estimation for epistemic analysis, specifically in epistemic-aleatory nesting, are: dakota uq cantilever sop exp.in, and dakota short column sop exp.in. Both files are in Dakota/test. 15.3 Surrogate-Based Uncertainty Quantification Many uncertainty quantification (UQ) methods are computationally costly. For example, sampling often requires many function evaluations to obtain accurate estimates of moments or percentile values of an output distribution. One approach to overcome the computational cost of sampling is to evaluate the true function (e.g. run the analysis driver) on a fixed, small set of samples, use these sample evaluations to create a response surface approximation (e.g. a surrogate model or meta-model) of the underlying “true” function, then perform random sampling (using thousands or millions of samples) on the approximation to obtain estimates of the mean, variance, and percentiles of the response. This approach, called “surrogate-based uncertainty quantification” is easy to do in Dakota, and one can set up input files to compare the results using no approximation (e.g. determine the mean, variance, and percentiles Dakota Version 6.7 User’s Manual generated on November 13, 2017 15.3. SURROGATE-BASED UNCERTAINTY QUANTIFICATION 259 of the output directly based on the initial sample values) with the results obtained by sampling a variety of surrogate approximations. Example input files of a standard UQ analysis based on sampling alone vs. sampling a surrogate are shown in the textbook uq sampling.in and textbook uq surrogate.in in the Dakota/examples/users directory. Note that one must exercise some caution when using surrogate-based methods for uncertainty quantification. In general, there is not a single, straightforward approach to incorporate the error of the surrogate fit into the uncertainty estimates of the output produced by sampling the surrogate. Two references which discuss some of the related issues are [62] and [130]. The first reference shows that statistics of a response based on a surrogate model were less accurate, and sometimes biased, for surrogates constructed on very small sample sizes. In many cases, however, [62] shows that surrogate-based UQ performs well and sometimes generates more accurate estimates of statistical quantities on the output. The second reference goes into more detail about the interaction between sample type and response surface type (e.g., are some response surfaces more accurate when constructed on a particular sample type such as LHS vs. an orthogonal array?) In general, there is not a strong dependence of the surrogate performance with respect to sample type, but some sample types perform better with respect to some metrics and not others (for example, a Hammersley sample may do well at lowering root mean square error of the surrogate fit but perform poorly at lowering the maximum absolute deviation of the error). Much of this work is empirical and application dependent. If you choose to use surrogates in uncertainty quantification, we strongly recommend trying a variety of surrogates and examining diagnostic goodness-of-fit metrics. Known Issue: When using discrete variables, there have been sometimes significant differences in data fit surrogate behavior observed across computing platforms in some cases. The cause has not yet been fully diagnosed and is currently under investigation. In addition, guidance on appropriate construction and use of surrogates with discrete variables is under development. In the meantime, users should therefore be aware that there is a risk of inaccurate results when using surrogates with discrete variables. Dakota Version 6.7 User’s Manual generated on November 13, 2017 260 CHAPTER 15. ADVANCED MODEL RECURSIONS Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 16 Advanced Simulation Code Interfaces This chapter extends the interface discussion in Chapter 10 and its discussion of generic black-box interfaces to simulations (Section 10.3). It describes specialized, tightly integrated, and advanced interfaces through which Dakota can perform function evaluation mappings. It describes AMPL-based algebraic mappings (Section 16.1), tight integration of a simulation code into Dakota (Section 16.2), and specialized interfaces to Matlab, Python, and Scilab (Sections 16.3 and 16.4). 16.1 Algebraic Mappings If desired, one can define algebraic input-output mappings using the AMPL code [49] and save these mappings in 3 files: stub.nl, stub.col, and stub.row, where stub is a particular root name describing a particular problem. These files names can be communicated to Dakota using the algebraic mappings input. Dakota will use stub.col and stub.row to obtain input and output identifier strings, respectively, and will use the AMPL solver library [52] to evaluate expressions conveyed in stub.nl, and, if needed, their first and second derivatives. As a simple example (from Dakota/test/dakota ampl*), consider algebraic mappings based on Newton’s law F = ma. The following is an AMPL input file of variable and expression declarations and output commands: var mass; var a; var v; minimize force: mass*a; minimize energy: 0.5 * mass * vˆ2; option auxfiles rc; # request stub.row and stub.col write gfma; # write stub.nl, stub.row, stub.col When processed by an AMPL processor, three files are created (as requested by the “option auxfiles” command). The first is the dakota ampl fma.nl file containing problem statistics, expression graphs, bounds, etc.: 262 g3 0 3 0 0 2 0 0 0 3 0 0 0 0 0 4 6 4 0 0 O0 0 o2 v0 v1 O1 0 o2 o2 n0.5 v0 o5 v2 n2 b 3 3 3 k2 0 0 G0 2 0 0 1 0 G1 2 0 0 2 0 CHAPTER 16. ADVANCED SIMULATION CODE INTERFACES 1 0 # problem fma 2 0 0 # vars, constraints, objectives, ranges, eqns # nonlinear constraints, objectives # network constraints: nonlinear, linear 0 # nonlinear vars in constraints, objectives, both 0 1 # linear network variables; functions; arith, flags 0 0 0 # discrete variables: binary, integer, nonlinear (b,c,o) # nonzeros in Jacobian, gradients # max name lengths: constraints, variables 0 0 0 # common exprs: b,c,o,c1,o1 Next, the dakota ampl fma.col file contains the set of variable descriptor strings: mass a v and the dakota ampl fma.row file contains the set of response descriptor strings: force energy The variable and objective function names declared within AMPL should be a subset of the variable descriptors and response descriptors used by Dakota (see the Dakota Reference Manual [3] for information on Dakota variable and response descriptors). Ordering of the inputs and outputs within the AMPL declaration is not important, as Dakota Version 6.7 User’s Manual generated on November 13, 2017 16.1. ALGEBRAIC MAPPINGS 263 Dakota will reorder data as needed. The following listing shows an excerpt from Dakota/test/dakota ampl fma.in, which demonstrates a combined algebraic/simulation-based mapping in which algebraic mappings from the fma definition are overlaid with simulation-based mappings from text book: variables, continuous_design = 5 descriptor ’x1’ ’mass’ ’a’ ’x4’ ’v’ initial_point 0.0 2.0 1.0 0.0 3.0 lower_bounds -3.0 0.0 -5.0 -3.0 -5.0 upper_bounds 3.0 10.0 5.0 3.0 5.0 interface, algebraic_mappings = ’dakota_ampl_fma.nl’ system analysis_driver = ’text_book’ parameters_file = ’tb.in’ results_file = ’tb.out’ file_tag responses, response_descriptors = ’force’ ’ineq1’ ’energy’ num_objective_functions = 1 num_nonlinear_inequality_constraints = 1 num_nonlinear_equality_constraints = 1 nonlinear_equality_targets = 20.0 analytic_gradients no_hessians Note that the algebraic inputs and outputs are a subset of the total inputs and outputs and that Dakota will track the algebraic contributions to the total response set using the order of the descriptor strings. In the case where both the algebraic and simulation-based components contribute to the same function, they are added together. To solve text book algebraically (refer to Section 20.1 for definition), the following AMPL model file could be used Dakota Version 6.7 User’s Manual generated on November 13, 2017 264 # # # # CHAPTER 16. ADVANCED SIMULATION CODE INTERFACES Problem : Textbook problem used in DAKOTA testing Constrained quartic, 2 continuous variables Solution: x=(0.5, 0.5), obj = .125, c1 = 0, c2 = 0 # continuous variables var x1 >= 0.5 <= 5.8 := 0.9; var x2 >= -2.9 <= 2.9 := 1.1; # objective function minimize obj: (x1 - 1)ˆ4 + (x2 - 1)ˆ4; # constraints (current required syntax for DAKOTA/AMPL interface) minimize c1: x1ˆ2 - 0.5*x2; minimize c2: x2ˆ2 - 0.5*x1; # required for output of *.row and *.col files option auxfiles rc; Note that the nonlinear constraints should not currently be declared as constraints within AMPL. Since the Dakota variable bounds and constraint bounds/targets currently take precedence over any AMPL specification, the current approach is to declare all AMPL outputs as objective functions and then map them into the appropriate response function type (objectives, least squares terms, nonlinear inequality/equality constraints, or generic response functions) within the Dakota input specification. 16.2 Developing a Direct Simulation Interface If a more efficient interface to a simulation is desired (e.g., to eliminate process creation and file I/O overhead) or if a targeted computer architecture cannot accommodate separate optimization and simulation processes (e.g., due to lightweight operating systems on compute nodes of large parallel computers), then linking a simulation code directly with Dakota may be desirable. This is an advanced capability of Dakota, and it requires a user to have access to (and knowledge of) the Dakota source code, as well as the source code of the simulation code. Three approaches are outlined below for developing direct linking between Dakota and a simulation: extension, derivation, and sandwich. For additional information, refer to “Interfacing with Dakota as a Library” in the Dakota Developers Manual [2]. Once performed, Dakota can bind with the new direct simulation interface using the direct interface specification in combination with an analysis driver, input filter or output filter specification that corresponds to the name of the new subroutine. 16.2.1 Extension The first approach to using the direct function capability with a new simulation (or new internal test function) involves extension of the existing DirectFnApplicInterface class to include new simulation member functions. In this case, the following steps are performed: 1. The functions to be invoked (analysis programs, input and output filters, internal testers) must have their main programs changed into callable functions/subroutines. Dakota Version 6.7 User’s Manual generated on November 13, 2017 16.2. DEVELOPING A DIRECT SIMULATION INTERFACE 265 2. The resulting callable function can then be added directly to the private member functions in DirectFnApplicInterface if this function will directly access the Dakota data structures (variables, active set, and response attributes of the class). It is more common to add a wrapper function to DirectFnApplicInterface which manages the Dakota data structures, but allows the simulator subroutine to retain a level of independence from Dakota (see Salinas, ModelCenter, and Matlab wrappers as examples). 3. The if-else blocks in the derived map if(), derived map ac(), and derived map of() member functions of the DirectFnApplicInterface class must be extended to include the new function names as options. If the new functions are class member functions, then Dakota data access may be performed through the existing class member attributes and data objects do not need to be passed through the function parameter list. In this case, the following function prototype is appropriate: int function_name(); If, however, the new function names are not members of the DirectFnApplicInterface class, then an extern declaration may additionally be needed and the function prototype should include passing of the Variables, ActiveSet, and Response data members: int function_name(const Dakota::Variables& vars, const Dakota::ActiveSet& set, Dakota::Response& response); 4. The Dakota system must be recompiled and linked with the new function object files or libraries. Various header files may have to be included, particularly within the DirectFnApplicInterface class, in order to recognize new external functions and compile successfully. Refer to the Dakota Developers Manual [2] for additional information on the DirectFnApplicInterface class and the Dakota data types. 16.2.2 Derivation As described in “Interfacing with Dakota as a Library” in the Dakota Developers Manual [2], a derivation approach can be employed to further increase the level of independence between Dakota and the host application. In this case, rather than adding a new function to the existing DirectFnApplicInterface class, a new interface class is derived from DirectFnApplicInterface which redefines the derived map if(), derived map ac(), and derived map of() virtual functions. In the approach of Section 16.2.3 below, the class derivation approach avoids the need to recompile the Dakota library when the simulation or its direct interface class is modified. 16.2.3 Sandwich In a “sandwich” implementation, a simulator provides both the “front end” and “back end” with Dakota sandwiched in the middle. To accomplish this approach, the simulation code is responsible for interacting with the user (the front end), links Dakota in as a library (refer to “Interfacing with Dakota as a Library” in the Dakota Developers Manual [2]), and plugs in a derived direct interface class to provide a closely-coupled mechanism for performing function evaluations (the back end). This approach makes Dakota services available to other codes and frameworks and is currently used by Sandia codes such as Xyce (electrical simulation), Sage (CFD), and SIERRA (multiphysics). Dakota Version 6.7 User’s Manual generated on November 13, 2017 266 16.3 CHAPTER 16. ADVANCED SIMULATION CODE INTERFACES Existing Direct Interfaces to External Simulators In addition to built-in polynomial test functions described in Section 10.2.1, Dakota includes direct interfaces to Sandia’s Salinas code for structural dynamics, Phoenix Integration’s ModelCenter framework, The Mathworks’ Matlab scientific computing environment, Scilab (as described in Section 16.4), and Python. While these can be interfaced to with a script-based approach, some usability and efficiency gains may be realized by re-compiling Dakota with these direct interfaces enabled. Some details on Matlab and Python interfaces are provided here. Note that these capabilities permit using Matlab or Python to evaluate a parameter to response mapping; they do not make Dakota algorithms available as a service, i.e., as a Matlab toolbox or Python module. 16.3.1 Matlab Dakota’s direct function interface includes the capability to invoke Matlab for function evaluations, using the Matlab engine API. When using this close-coupling, the Matlab engine is started once when Dakota initializes, and then during analysis function evaluations are performed exchanging parameters and results through the Matlab C API. This eliminates the need to use the file system and the expense of initializing the Matlab engine for each function evaluation. The Dakota/Matlab interface has been built and tested on 32-bit Linux with Matlab 7.0 (R14) and on 64-bit Linux with Matlab 7.1 (R14SP3). Configuration support for other platforms is included, but is untested. Builds on other platforms or with other versions of Matlab may require modifications to Dakota including its build system To use the Dakota/Matlab interface, Dakota must be configured and compiled with the Matlab feature enabled. The Mathworks only provides shared object libraries for its engine API, so Dakota must be dynamically linked to at least the Matlab libraries. To compile Dakota with the Matlab interface enabled, set the CMake variable DAKOTA MATLAB:BOOL=ON, possibly with MATLAB DIR:FILEPATH=/path/to/matlab, where MATLAB DIR is the root of your Matlab installation (it should be a directory containing directories bin/YOURPLATFORM and extern/include). Since the Matlab libraries are linked dynamically, they must be accessible at compile time and at run time. Make sure the path to the appropriate Matlab shared object libraries is on your LD LIBRARY PATH. For example to accomplish this in BASH on 32-bit Linux, one might type export LD_LIBRARY_PATH=/usr/local/matlab/bin/glnx86:$LD_LIBRARY_PATH or add such a command to the .bashrc file. Then proceed with compiling as usual. Example files corresponding to the following tutorial are available in Dakota/examples/linked interfaces/Matlab. 16.3.1.1 Dakota/Matlab input file specification The Matlab direct interface is specified with direct, matlab keywords in an interface specification. The Matlab m-file which performs the analysis is specified through the analysis drivers keyword. Here is a sample Dakota interface specification: interface, matlab analysis_drivers = ’myanalysis.m’ Dakota Version 6.7 User’s Manual generated on November 13, 2017 16.3. EXISTING DIRECT INTERFACES TO EXTERNAL SIMULATORS Dakota. numFns numVars numACV numADIV numADRV numDerivVars xC xDI xDR xCLabels xDILabels xDRLabels directFnASV directFnDVV fnFlag gradFlag hessFlag currEvalId 267 number of functions (responses, constraints) total number of variables number active continuous variables number active discrete integer variables number active discrete real variables number of derivative variables specified in directFnDVV continuous variable values ([1 x numACV]) discrete integer variable values ([1 x numADIV]) discrete real variable values ([1 x numADRV]) continuous var labels (cell array of numACV strings) discrete integer var labels (cell array of numADIV strings) discrete real var labels (cell array of numADIV strings) active set vector ([1 x numFns]) derivative variables vector ([1 x numDerivVars]) nonzero if function values requested nonzero if gradients requested nonzero if hessians requested current evaluation ID Figure 16.1: Dakota/Matlab parameter data structure. Multiple Matlab analysis drivers are supported. Multiple analysis components are supported as for other interfaces as described in Section 10.4. The .m extension in the analysis drivers specification is optional and will be stripped by the interface before invoking the function. So myanalysis and myanalysis.m will both cause the interface to attempt to execute a Matlab function myanalysis for the evaluation. 16.3.1.2 Matlab .m file specification The Matlab analysis file myanalysis.m must define a Matlab function that accepts a Matlab structure as its sole argument and returns the same structure in a variable called Dakota. A manual execution of the call to the analysis in Matlab should therefore look like: >> Dakota = myanalysis(Dakota) Note that the structure named Dakota will be pushed into the Matlab workspace before the analysis function is called. The structure passed from Dakota to the analysis m-function contains essentially the same information that would be passed to a Dakota direct function included in DirectApplicInterface.C, with fields shown in Figure 16.1. The structure Dakota returned from the analysis must contain a subset of the fields shown in Figure 16.2. It may contain additional fields and in fact is permitted to be the structure passed in, augmented with any required outputs. An example Matlab analysis driver rosenbrock.m for the Rosenbrock function is shown in Figure 16.3. Dakota Version 6.7 User’s Manual generated on November 13, 2017 268 CHAPTER 16. ADVANCED SIMULATION CODE INTERFACES Dakota. fnVals fnGrads fnHessians fnLabels failure ([1 x numFns], required if function values requested) ([numFns x numDerivVars], required if gradients requested) ([numFns x numDerivVars x numDerivVars], required if hessians requested) (cell array of numFns strings, optional) (optional: zero indicates success, nonzero failure Figure 16.2: Dakota/Matlab response data structure. 16.3.2 Python Dakota’s Python direct interface has been tested on Linux with Python 2.x. When enabled, it allows Dakota to make function evaluation calls directly to an analysis function in a user-provided Python module. Data may flow between Dakota and Python either in multiply-subscripted lists or NumPy arrays. The Python direct interface must be enabled when compiling Dakota. Set the CMake variable DAKOTA PYTHON:BOOL=ON, and optionally DAKOTA PYTHON NUMPY:BOOL=ON (default is ON) to use Dakota’s NumPy array interface (requires NumPy installation providing arrayobject.h). If NumPy is not enabled, Dakota will use multiply-subscripted lists for data flow. An example of using the Python direct interface with both lists and arrays is included in Dakota/examples/linked interfaces/Python. The Python direct driver is selected with, for example, interface, python # numpy analysis_drivers = ’python_module:analysis_function’ where python module denotes the module (file python module.py) Dakota will attempt to import into the Python environment and analysis function denotes the function to call when evaluating a parameter set. If the Python module is not in the directory from which Dakota is started, setting the PYTHONPATH environment variable to include its location can help the Python engine find it. The optional numpy keyword indicates Dakota will communicate with the Python analysis function using numarray data structures instead of the default lists. Whether using the list or array interface, data from Dakota is passed (via kwargs) into the user function in a dictionary containing the entries shown in Table 16.1. The analysis function must return a dictionary containing the data specified by the active set vector with fields “fns”, “fnGrads”, and “fnHessians”, corresponding to function values, gradients, and Hessians, respectively. The function may optionally include a failure code in “failure” (zero indicates success, nonzero failure) and function labels in “fnLabels”. See the linked interfaces example referenced above for more details. 16.4 Scilab Script and Direct Interfaces Scilab is open source computation software which can be used to perform function evaluations during Dakota studies, for example to calculate the objective function in optimization. Dakota includes three Scilab interface variants: scripted, linked, and compiled. In each mode, Dakota calls Scilab to perform a function evaluation and Dakota Version 6.7 User’s Manual generated on November 13, 2017 16.4. SCILAB SCRIPT AND DIRECT INTERFACES 269 function Dakota = rosenbrock(Dakota) Dakota.failure = 0; if ( Dakota.numVars ˜= 2 | Dakota.numADV | ... ( ˜isempty( find(Dakota.directFnASM(2,:)) | ... find(Dakota.directFnASM(3,:)) ) & Dakota.numDerivVars ˜= 2 ) ) sprintf(’Error: Bad number of variables in rosenbrock.m fn.\n’); Dakota.failure = 1; elseif (Dakota.numFns > 2) % 1 fn -> opt, 2 fns -> least sq sprintf(’Error: Bad number of functions in rosenbrock.m fn.\n’); Dakota.failure = 1; else if Dakota.numFns > 1 least_sq_flag = true; else least_sq_flag = false; end f0 = Dakota.xC(2)-Dakota.xC(1)*Dakota.xC(1); f1 = 1.-Dakota.xC(1); % **** f: if (least_sq_flag) if Dakota.directFnASM(1,1) Dakota.fnVals(1) = 10*f0; end if Dakota.directFnASM(1,2) Dakota.fnVals(2) = f1; end else if Dakota.directFnASM(1,1) Dakota.fnVals(1) = 100.*f0*f0+f1*f1; end end % **** df/dx: if (least_sq_flag) if Dakota.directFnASM(2,1) Dakota.fnGrads(1,1) = -20.*Dakota.xC(1); Dakota.fnGrads(1,2) = 10.; end if Dakota.directFnASM(2,2) Dakota.fnGrads(2,1) = -1.; Dakota.fnGrads(2,2) = 0.; end else if Dakota.directFnASM(2,1) Dakota.fnGrads(1,1) = -400.*f0*Dakota.xC(1) - 2.*f1; Dakota.fnGrads(1,2) = 200.*f0; end end % **** dˆ2f/dxˆ2: if (least_sq_flag) if Dakota.directFnASM(3,1) Dakota.fnHessians(1,1,1) = -20.; Dakota.fnHessians(1,1,2) = 0.; Dakota.fnHessians(1,2,1) = 0.; Dakota.fnHessians(1,2,2) = 0.; end if Dakota.directFnASM(3,2) Dakota.fnHessians(2,1:2,1:2) = 0.; end else if Dakota.directFnASM(3,1) fx = Dakota.xC(2) - 3.*Dakota.xC(1)*Dakota.xC(1); Dakota.fnHessians(1,1,1) = -400.*fx + 2.0; Dakota.fnHessians(1,1,2) = -400.*Dakota.xC(1); Dakota.fnHessians(1,2,1) = -400.*Dakota.xC(1); Dakota.fnHessians(1,2,2) = 200.; end end Dakota.fnLabels = {’f1’}; end Figure 16.3: Sample Matlab implementation of the Rosenbrock test function for the Dakota/Matlab interface. Dakota Version 6.7 User’s Manual generated on November 13, 2017 270 CHAPTER 16. ADVANCED SIMULATION CODE INTERFACES Table 16.1: Data dictionary passed to Python direct interface. Entry Name functions variables cv div drv av cv labels div labels drv labels av labels asv dvv analysis components currEvalId Description number of functions (responses, constraints) total number of variables list/array of continuous variable values list/array of discrete integer variable values list/array of discrete real variable values single list/array of all variable values continuous variable labels discrete integer variable labels discrete real variable labels all variable labels active set vector derivative variables vector list of analysis components as strings current evaluation ID number then retrieves the Scilab results. Dakota’s Scilab interface was contributed in 2011 by Yann Collette and Yann Chapalain. The Dakota/Scilab interface variants are described next. 16.4.1 Scilab Script Interface Dakota distributions include a directory Dakota/examples/script interfaces/Scilab which demonstrates script-based interfacing to Scilab. The Rosenbrock subdirectory contains four notable files: • dakota scilab rosenbrock.in (the Dakota input file), • rosenbrock.sci (the Scilab computation code), • scilab rosen bb simulator.sh (the analysis driver), and • scilab rosen wrapper.sci (Scilab script). The dakota scilab rosenbrock.in file specifies the Dakota study to perform. The interface type is external (fork) and the shell script scilab rosen bb simulator.sh is the analysis driver used to perform function evaluations. The Scilab file rosenbrock.sci accepts variable values and computes the objective, gradient, and Hessian values of the Rosenbrock function as requested by Dakota. The scilab rosen bb simulator.sh is a short shell driver script, like that described in Section 10.3, that Dakota executes to perform each function evaluation. Dakota passes the names of the parameters and results files to this script as $argv[1] and $argv[2], respectively. The scilab rosen bb simulator.sh is divided into three parts: pre-processing, analysis, and post-processing. In the analysis portion, the scilab rosen bb simulator.sh uses scilab rosen wrapper.sci to extract the current variable values from the input parameters file ($argv[1]) and communicate them to the computation code in rosenbrock.sci. The resulting objective function is transmitted to Dakota via the output result file ($argv[1]), and the driver script cleans up any temporary files. The directory also includes PID and FemTRUSS examples, which are run in a similar way. Dakota Version 6.7 User’s Manual generated on November 13, 2017 16.4. SCILAB SCRIPT AND DIRECT INTERFACES 16.4.2 271 Scilab Linked Interface The Dakota/Scilab linked interface allows Dakota to communicate directly with Scilab through in-memory data structures, typically resulting in faster communication, as it does not rely on files or pipes. In this mode, Dakota publishes a data structure into the Scilab workspace, and then invokes the specified Scilab analysis driver directly. In Scilab, this structure is an mlist (http://help.scilab.org/docs/5.3.2/en_US/mlist.html), with the same fields as in the Matlab interface 16.1, with the addition of a field dakota type, which is used to validate the names of fields in the data structure. The linked interface is implemented in source files src/ScilabInterface.[CH] directory, and must be enabled at compile time when building Dakota from source by setting DAKOTA SCILAB:BOOL=ON, and setting appropriate environment variables at compile and run time as described in README.Scilab in Dakota/examples/linked interfaces/Scilab/. This directory also contains examples for the Rosenbrock and PID problems. These examples are similar to those in Dakota/examples/script interfaces, with a few notable exceptions: 1. There is no shell driver script 2. The Dakota input file specifies the interface as ’scilab’, indicating a direct, internal interface to Scilab using the Dakota data structure described above: interface, scilab analysis_driver = ’rosenbrock.sci’ 16.4.3 Scilab Compiled Interface In “compiled interface” mode, the Dakota analysis driver is a lightweight shim, which communicates with the running application code such as Scilab via named pipes. It is similar to that for Matlab in Dakota/examples/compiled interfa whose README is likely instructive. An example of a Scilab compiled interface is included in Dakota/examples/compiled interfaces/Scilab/Rosenbrock. As with the other Scilab examples, there are computation code and Dakota input files. Note the difference in the Dakota input file rosenbrock.in, where the analysis driver starts the dakscilab shim program and always evaluates functions, gradients, and Hessians. interface, fork analysis_driver = ’../dakscilab -d -fp "exec fp.sci" -fpp "exec fpp.sci"’ parameters_file = ’r.in’ results_file = ’r.out’ deactivate active_set_vector The dakscilab executable results from compiling dakscilab.c and has the following behavior and options. The driver dakscilab launches a server. This server then facilitates communication between Dakota and Scilab via named pipes communication. The user can also use the first named pipe (${DAKSCILAB PIPE}1) to communicate with the server: echo dbg scilab_script.sce > ${DAKSCILAB_PIPE}1 echo quit > ${DAKSCILAB_PIPE}1 Dakota Version 6.7 User’s Manual generated on November 13, 2017 272 CHAPTER 16. ADVANCED SIMULATION CODE INTERFACES The first command, with the keyword ’dbg’, launches the script scilab script.sci for evaluation in Scilab. It permits to give instructions to Scilab. The second command ’quit’ stops the server. The dakscilab shim supports the following options for the driver call: 1. -s to start the server 2. -si to run an init script 3. -sf to run a final script 4. -f -fp -fpp to specify names of objective function, gradient and hessian, then load them. For the included PID example, the driver call is analysis_driver = ’../dakscilab -d -si "exec init_test_automatic.sce;" -sf "exec visualize_solution.sce;" -f "exec f_pid.sci"’ Here there is an initialization script (init test automatic.sce;) which is launched before the main computation. It initializes a specific Scilab module called xcos. A finalization script to visualize the xcos solution is also specified (visualize solution.sce). Finally, the objective function is given with the computation code called f pid.sci. Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 17 Parallel Computing 17.1 Overview This chapter describes the various parallel computing capabilities provided by Dakota. We begin with a high-level summary. Dakota has been designed to exploit a wide range of parallel computing resources such as those found in a desktop multiprocessor workstation, a network of workstations, or a massively parallel computing platform. This parallel computing capability is a critical technology for rendering real-world engineering design problems computationally tractable. Dakota employs the concept of multilevel parallelism, which takes simultaneous advantage of opportunities for parallel execution from multiple sources: Parallel Simulation Codes: Dakota works equally well with both serial and parallel simulation codes. Concurrent Execution of Analyses within a Function Evaluation: Some engineering design applications call for the use of multiple simulation code executions (different disciplinary codes, the same code for different load cases or environments, etc.) in order to evaluate a single response data set1 (e.g., objective functions and constraints) for a single set of parameters. If these simulation code executions are independent (or if coupling is enforced at a higher level), Dakota can perform them concurrently. Concurrent Execution of Function Evaluations within an Iterator: With very few exceptions, the iterative algorithms described in Chapters 3–7 all provide opportunities for the concurrent evaluation of response data sets for different parameter sets. Whenever there exists a set of function evaluations that are independent, Dakota can perform them in parallel. Concurrent Execution of Sub-Iterators within a Meta-iterator or Nested Model: The advanced methods described in Chapter 14 are examples of meta-iterators, and the advanced model recursions described in Sections 15.1–15.2 all utilize nested models. Both of these cases generate sets of iterator subproblems that can be executed concurrently. For example, the Pareto-set and multi-start strategies generate sets of optimization subproblems. Similarly, optimization under uncertainty (15.2) generates sets of uncertainty quantification subproblems. Whenever these subproblems are independent, Dakota can perform them in parallel. It is important to recognize that these four parallelism sources can be combined recursively. For example, a metaiterator can schedule and manage concurrent iterators, each of which may manage concurrent function evaluations, each of which may manage concurrent analyses, each of which may execute on multiple processors. Moreover, 1 the term “function evaluation” is used broadly to mean any individual data request from an iterative algorithm 274 CHAPTER 17. PARALLEL COMPUTING more than one source of sub-iteration concurrency can be exploited when combining meta-iteration and nested model sources. In an extreme example, defining the Pareto frontier for mixed-integer nonlinear programming under mixed aleatory-epistemic uncertainty might exploit up to four levels of nested sub-iterator concurrency in addition to available levels from function evaluation concurrency, analysis concurrency, and simulation parallelism. The majority of application scenarios, however, will employ one to two levels of parallelism. Navigating the body of this chapter: The range of capabilities is extensive and can be daunting at first; therefore, this chapter takes an incremental approach in first describing the simplest single-level parallel computing models (Section 17.2) using asynchronous local, message passing, and hybrid approaches. More advanced uses of Dakota can build on this foundation to exploit multiple levels of parallelism, as described in Section 17.3. The chapter concludes with a discussion of using Dakota with applications that run as independent MPI processes (parallel application tiling, for example on a large compute cluster). This last section is a good quick start for interfacing Dakota to your parallel (or serial) application on a cluster. 17.1.1 Categorization of parallelism To understand the parallel computing possibilities, it is instructive to first categorize the opportunities for exploiting parallelism into four main areas [39], consisting of coarse-grained and fine-grained parallelism opportunities within algorithms and their function evaluations: 1. Algorithmic coarse-grained parallelism: This parallelism involves the concurrent execution of independent function evaluations, where a “function evaluation” is defined as a data request from an algorithm (which may involve value, gradient, and Hessian data from multiple objective and constraint functions). This concept can also be extended to the concurrent execution of multiple “iterators” within a “meta-iterator.” Examples of algorithms containing coarse-grained parallelism include: • Gradient-based algorithms: finite difference gradient evaluations, speculative optimization, parallel line search. • Nongradient-based algorithms: genetic algorithms (GAs), pattern search (PS), Monte Carlo sampling. • Approximate methods: design of computer experiments for building surrogate models. • Concurrent sub-iteration: optimization under uncertainty, branch and bound, multi-start local search, Pareto set optimization, island-model GAs. 2. Algorithmic fine-grained parallelism: This involves computing the basic computational steps of an optimization algorithm (i.e., the internal linear algebra) in parallel. This is primarily of interest in large-scale optimization problems and simultaneous analysis and design (SAND). 3. Function evaluation coarse-grained parallelism: This involves concurrent computation of separable parts of a single function evaluation. This parallelism can be exploited when the evaluation of the response data set requires multiple independent simulations (e.g. multiple loading cases or operational environments) or multiple dependent analyses where the coupling is applied at the optimizer level (e.g., multiple disciplines in the individual discipline feasible formulation [26]). 4. Function evaluation fine-grained parallelism: This involves parallelization of the solution steps within a single analysis code. Support for massively parallel simulation continues to grow in areas of nonlinear mechanics, structural dynamics, heat transfer, computational fluid dynamics, shock physics, and many others. Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.1. OVERVIEW 275 By definition, coarse-grained parallelism requires very little inter-processor communication and is therefore “embarrassingly parallel,” meaning that there is little loss in parallel efficiency due to communication as the number of processors increases. However, it is often the case that there are not enough separable computations on each algorithm cycle to utilize the thousands of processors available on massively parallel machines. For example, a thermal safety application [40] demonstrated this limitation with a pattern search optimization in which the maximum speedup exploiting only coarse-grained algorithmic parallelism was shown to be limited by the size of the design problem (coordinate pattern search has at most 2n independent evaluations per cycle for n design variables). Fine-grained parallelism, on the other hand, involves much more communication among processors and care must be taken to avoid the case of inefficient machine utilization in which the communication demands among processors outstrip the amount of actual computational work to be performed. For example, a chemically-reacting flow application [39] illustrated this limitation for a simulation of fixed size in which it was shown that, while simulation run time did monotonically decrease with increasing number of processors, the relative parallel efficiency Ê of the computation for fixed model size decreased rapidly (from Ê ≈ 0.8 at 64 processors to Ê ≈ 0.4 at 512 processors). This was due to the fact that the total amount of computation was approximately fixed, whereas the communication demands were increasing rapidly with increasing numbers of processors. Therefore, there is a practical limit on the number of processors that can be employed for fine-grained parallel simulation of a particular model size, and only for extreme model sizes can thousands of processors be efficiently utilized in studies exploiting fine-grained parallelism alone. These limitations point us to the exploitation of multiple levels of parallelism, in particular the combination of coarse-grained and fine-grained approaches. This will allow us to execute fine-grained parallel simulations on sets of processors where they are most efficient and then replicate this efficiency with many coarse-grained instances involving one or more levels of nested job scheduling. 17.1.2 Parallel Dakota algorithms In Dakota, the following parallel algorithms, comprised of iterators and meta-iterators, provide support for coarsegrained algorithmic parallelism. Note that, even if a particular algorithm is serial in terms of its data request concurrency, other concurrency sources (e.g., function evaluation coarse-grained and fine-grained parallelism) may still be available. 17.1.2.1 Parallel iterators • Gradient-based optimizers: CONMIN, DOT, NLPQL, NPSOL, and OPT++ can all exploit parallelism through the use of Dakota’s native finite differencing routine (selected with method source dakota in the responses specification), which will perform concurrent evaluations for each of the parameter offsets. For n variables, forward differences result in an n + 1 concurrency and central differences result in a 2n + 1 concurrency. In addition, CONMIN, DOT, and OPT++ can use speculative gradient techniques [15] to obtain better parallel load balancing. By speculating that the gradient information associated with a given line search point will be used later and computing the gradient information in parallel at the same time as the function values, the concurrency during the gradient evaluation and line search phases can be balanced. NPSOL does not use speculative gradients since this approach is superseded by NPSOL’s gradient-based line search in user-supplied derivative mode. NLPQL also supports a distributed line search capability for generating concurrency [117]. Finally, finite-difference Newton algorithms can exploit additional concurrency in numerically evaluating Hessian matrices. • Nongradient-based optimizers: HOPSPACK, JEGA methods, and most SCOLIB methods support paralDakota Version 6.7 User’s Manual generated on November 13, 2017 276 CHAPTER 17. PARALLEL COMPUTING lelism. HOPSPACK and SCOLIB methods exploit parallelism through the use of Dakota’s concurrent function evaluations; however, there are some limitations on the levels of concurrency and asynchrony that can be exploited. These are detailed in the Dakota Reference Manual. Serial SCOLIB methods include SolisWets (coliny solis wets) and certain exploratory moves options (adaptive pattern and multi step) in pattern search (coliny pattern search). OPT++ PDS (optpp pds) and NCSU DIRECT (ncsu direct) are also currently serial due to incompatibilities in Dakota and OPT++/NCSU parallelism models. Finally, coliny pattern search and asynch pattern search support dynamic job queues managed with nonblocking synchronization. • Least squares methods: in an identical manner to the gradient-based optimizers, NL2SOL, NLSSOL, and Gauss-Newton can exploit parallelism through the use of Dakota’s native finite differencing routine. In addition, NL2SOL and Gauss-Newton can use speculative gradient techniques to obtain better parallel load balancing. NLSSOL does not use speculative gradients since this approach is superseded by NLSSOL’s gradient-based line search in user-supplied derivative mode. • Surrogate-based minimizers: surrogate based local, surrogate based global, and efficient global all support parallelism in the initial surrogate construction, but subsequent concurrency varies. In the case of efficient global, only a single point is generated for evaluation for each subsequent cycle and there is no derivatove concurrency for this point. In the case of surrogate based local, only a single point is generated per subsequent cycle, but derivative concurrency for numerical gradient or Hessian evaluations may be available. And in the case of surrogate based global, multiple points may be generated on each subsequent cycle, depending on the multipoint return capability of specific minimizers. • Parameter studies: all parameter study methods (vector, list, centered, and multidim) support parallelism. These methods avoid internal synchronization points, so all evaluations are available for concurrent execution. • Design of experiments: all dace (grid, random, oas, lhs, oa lhs, box behnken, and central composite), fsu quasi mc (halton and hammersley), fsu cvt, and psuade moat methods support parallelism. • Uncertainty quantification: all nondeterministic methods (sampling, reliability, stochastic expansion, and epistemic) support parallelism. In the case of gradient-based methods (local reliability, local interval estimation), parallelism can be exploited through the use of Dakota’s native finite differencing routine for computing gradients. In the case of many global methods (e.g., global reliability, global interval estimation, polynomial chaos), initial surrogate construction is highly parallel, but any subsequent (adaptive) refinement may have greater concurrency restrictions (including a single point per refinement cycle in some cases). 17.1.2.2 Advanced methods Certain advanced methods support concurrency in multiple iterator executions. Currently, the methods which can exploit this level of parallelism are: • Hybrid minimization: when the sequential hybrid transfers multiple solution points between methods, single-point minimizers will be executed concurrently using each of the transferred solution points. • Pareto-set optimization: a meta-iterator for multiobjective optimization using the simple weighted-sum approach for computing sets of points on the Pareto front of nondominated solutions. • Multi-start iteration: a meta-iterator for executing multiple instances of an iterator from different starting points. Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.2. SINGLE-LEVEL PARALLELISM 277 The hybrid minimization case will display varying levels of iterator concurrency based on differing support of multipoint solution input/output between iterators; however, the use of multiple parallel configurations among the iterator sequence should prevent parallel inefficiencies. On the other hand, pareto-set and multi-start have a fixed set of jobs to perform and should exhibit good load balancing. 17.1.2.3 Parallel models Parallelism support in model classes (see Chapter 8) is an important issue for advanced model recursions such as surrogate-based minimization, optimization under uncertainty, and mixed aleatory-epistemic UQ (see Chapters 14 and 15). Support is as follows: • Single model: parallelism is managed as specified in the model’s associated interface instance. • Recast model: most parallelism is forwarded on to the sub-model. An exception to this is finite differencing in the presence of variable scaling. Since it is desirable to perform offsets in the scaled space (and avoid minimum step size tolerances), this parallelism is not forwarded to the sub-model, instead being enacted at the recast level. • Data fit surrogate model: parallelism is supported in the construction of global surrogate models via the concurrent evaluation of points generated by design of experiments methods. Local and multipoint approximations evaluate only a single point at a time, so concurrency is available only from any numerical differencing required for gradient and Hessian data. Since the top-level iterator is interfaced only with the (inexpensive) surrogate, no parallelism is exploited there. Load balancing can be an important issue when performing evaluations to (adaptively) update existing surrogate models. • Hierarchical surrogate model: parallelism is supported for the low or the high fidelity models, and in some contexts, for both models at the same time. In the multifidelity optimization context, the optimizer is interfaced only with the low-fidelity model, and the high-fidelity model is used only for verifications and correction updating. For this case, the algorithmic coarse-grained parallelism supported by the optimizer is enacted on the low fidelity model and the only parallelism available for high fidelity executions arises from any numerical differencing required for high-fidelity gradient and Hessian data. In contexts that compute model discrepancies, such as multifidelity UQ, the algorithmic concurrency involves evaluation of both low and high fidelity models, so parallel schedulers can exploit simultaneous concurrency for both models. • Nested model: concurrent executions of the optional interface and concurrent executions of the sub-iterator are supported and are synchronized in succession. Currently, synchronization is blocking (all concurrent evaluations are completed before new batches are scheduled); nonblocking schedulers (see 17.2) may be supported in time. Nested model concurrency and meta-iterator concurrency (Section 17.1.2.2) may be combined within an arbitrary number of levels of recursion. Primary clients for this capability include optimization under uncertainty and mixed aleatory-epistemic UQ (see Section 8.5). 17.2 Single-level parallelism Dakota’s parallel facilities support a broad range of computing hardware, from custom massively parallel supercomputers on the high end, to clusters and networks of workstations in the middle range, to desktop multiprocessors on the low end. Given the reduced scale in the middle to low ranges, it is more common to exploit only one of the levels of parallelism; however, this can still be quite effective in reducing the time to obtain a solution. Three single-level parallelism models will be discussed, and are depicted in Figure 17.1: Dakota Version 6.7 User’s Manual generated on November 13, 2017 278 CHAPTER 17. PARALLEL COMPUTING Figure 17.1: External, internal, and hybrid job management. • asynchronous local: Dakota executes on a single processor, but launches multiple jobs concurrently using asynchronous job launching techniques. • message passing: Dakota executes in parallel using message passing to communicate between processors. A single job is launched per processor using synchronous job launching techniques. • hybrid: a combination of message passing and asynchronous local. Dakota executes in parallel across multiple processors and launches concurrent jobs on each processor. In each of these cases, jobs are executing concurrently and must be collected in some manner for return to an algorithm. Blocking and nonblocking approaches are provided for this, where the blocking approach is used in most cases: • blocking synchronization: all jobs in the queue are completed before exiting the scheduler and returning the set of results to the algorithm. The job queue fills and then empties completely, which provides a synchronization point for the algorithm. • nonblocking synchronization: the job queue is dynamic, with jobs entering and leaving continuously. There are no defined synchronization points for the algorithm, which requires specialized algorithm logic (only currently supported by coliny pattern search and asynch pattern search, which are sometimes referred to as “fully asynchronous” algorithms). Given these job management capabilities, it is worth noting that the popular term “asynchronous” can be ambiguous when used in isolation. In particular, it can be important to qualify whether one is referring to “asynchronous job launch” (synonymous with any of the three concurrent job launch approaches described above) or “asynchronous job recovery” (synonymous with the latter nonblocking job synchronization approach). Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.2. SINGLE-LEVEL PARALLELISM 17.2.1 279 Asynchronous Local Parallelism This section describes software components which manage simulation invocations local to a processor. These invocations may be either synchronous (i.e., blocking) or asynchronous (i.e., nonblocking). Synchronous evaluations proceed one at a time with the evaluation running to completion before control is returned to Dakota. Asynchronous evaluations are initiated such that control is returned to Dakota immediately, prior to evaluation completion, thereby allowing the initiation of additional evaluations which will execute concurrently. The synchronous local invocation capabilities are used in two contexts: (1) by themselves to provide serial execution on a single processor, and (2) in combination with Dakota’s message-passing schedulers to provide function evaluations local to each processor. Similarly, the asynchronous local invocation capabilities are used in two contexts: (1) by themselves to launch concurrent jobs from a single processor that rely on external means (e.g., operating system, job queues) for assignment to other processors, and (2) in combination with Dakota’s messagepassing schedulers to provide a hybrid parallelism (see Section 17.2.3). Thus, Dakota supports any of the four combinations of synchronous or asynchronous local combined with message passing or without. Asynchronous local schedulers may be used for managing concurrent function evaluations requested by an iterator or for managing concurrent analyses within each function evaluation. The former iterator/evaluation concurrency supports either blocking (all jobs in the queue must be completed by the scheduler) or nonblocking (dynamic job queue may shrink or expand) synchronization, where blocking synchronization is used by most iterators and nonblocking synchronization is used by fully asynchronous algorithms such as asynch pattern search and coliny pattern search. The latter evaluation/analysis concurrency is restricted to blocking synchronization. The “Asynchronous Local” column in Table 17.1 summarizes these capabilities. Dakota supports three local simulation invocation approaches based on the direct function, system call, and fork simulation interfaces. For each of these cases, an input filter, one or more analysis drivers, and an output filter make up the interface, as described in Section 10.4. 17.2.1.1 Direct function synchronization The direct function capability may be used synchronously. Synchronous operation of the direct function simulation interface involves a standard procedure call to the input filter, if present, followed by calls to one or more simulations, followed by a call to the output filter, if present (refer to Sections 10.2-10.4 for additional details and examples). Each of these components must be linked as functions within Dakota. Control does not return to the calling code until the evaluation is completed and the response object has been populated. Asynchronous operation will be supported in the future and will involve the use of multithreading (e.g., POSIX threads) to accomplish multiple simultaneous simulations. When spawning a thread (e.g., using pthread create), control returns to the calling code after the simulation is initiated. In this way, multiple threads can be created simultaneously. An array of responses corresponding to the multiple threads of execution would then be recovered in a synchronize operation (e.g., using pthread join). 17.2.1.2 System call synchronization The system call capability may be used synchronously or asynchronously. In both cases, the system utility from the standard C library is used. Synchronous operation of the system call simulation interface involves spawning the system call (containing the filters and analysis drivers bound together with parentheses and semi-colons) in the foreground. Control does not return to the calling code until the simulation is completed and the response file has been written. In this case, the possibility of a race condition (see below) does not exist and any errors during response recovery will cause an immediate abort of the Dakota process (note: detection of the string “fail” is not Dakota Version 6.7 User’s Manual generated on November 13, 2017 280 CHAPTER 17. PARALLEL COMPUTING a response recovery error; see Chapter 19). Asynchronous operation involves spawning the system call in the background, continuing with other tasks (e.g., spawning other system calls), periodically checking for process completion, and finally retrieving the results. An array of responses corresponding to the multiple system calls is recovered in a synchronize operation. In this synchronize operation, completion of a function evaluation is detected by testing for the existence of the evaluation’s results file using the stat utility [87]. Care must be taken when using asynchronous system calls since they are prone to the race condition in which the results file passes the existence test but the recording of the function evaluation results in the file is incomplete. In this case, the read operation performed by Dakota will result in an error due to an incomplete data set. In order to address this problem, Dakota contains exception handling which allows for a fixed number of response read failures per asynchronous system call evaluation. The number of allowed failures must have a limit, so that an actual response format error (unrelated to the race condition) will eventually abort the system. Therefore, to reduce the possibility of exceeding the limit on allowable read failures, the user’s interface should minimize the amount of time an incomplete results file exists in the directory where its status is being tested. This can be accomplished through two approaches: (1) delay the creation of the results file until the simulation computations are complete and all of the response data is ready to be written to the results file, or (2) perform the simulation computations in a subdirectory, and as a last step, move the completed results file into the main working directory where its existence is being queried. If concurrent simulations are executing in a shared disk space, then care must be taken to maintain independence of the simulations. In particular, the parameters and results files used to communicate between Dakota and the simulation, as well as any other files used by this simulation, must be protected from other files of the same name used by the other concurrent simulations. With respect to the parameters and results files, these files may be made unique through the use of the file tag option (e.g., params.in.1, results.out.1, etc.) or the default temporary file option (e.g., /var/tmp/aaa0b2Mfv, etc.). However, if additional simulation files must be protected (e.g., model.i, model.o, model.g, model.e, etc.), then an effective approach is to create a tagged working subdirectory for each simulation instance. Section 10.3 provides an example system call interface that demonstrates both the use of tagged working directories and the relocation of completed results files to avoid the race condition. 17.2.1.3 Fork synchronization The fork capability is quite similar to the system call; however, it has the advantage that asynchronous fork invocations can avoid the results file race condition that may occur with asynchronous system calls (see Section 10.2.5). The fork interface invokes the filters and analysis drivers using the fork and exec family of functions, and completion of these processes is detected using the wait family of functions. Since wait is based on a process id handle rather than a file existence test, an incomplete results file is not an issue. Depending on the platform, the fork simulation interface executes either a vfork or a fork call. These calls generate a new child process with its own UNIX process identification number, which functions as a copy of the parent process (dakota). The execvp function is then called by the child process, causing it to be replaced by the analysis driver or filter. For synchronous operation, the parent dakota process then awaits completion of the forked child process through a blocking call to waitpid. On most platforms, the fork/exec procedure is efficient since it operates in a copy-on-write mode, and no copy of the parent is actually created. Instead, the parents address space is borrowed until the exec function is called. The fork/exec behavior for asynchronous operation is similar to that for synchronous operation, the only difference being that dakota invokes multiple simulations through the fork/exec procedure prior to recovering response results for these jobs using the wait function. The combined use of fork/exec and wait functions in asynchronous mode allows the scheduling of a specified number of concurrent function evaluations and/or Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.2. SINGLE-LEVEL PARALLELISM 281 concurrent analyses. 17.2.1.4 Asynchronous Local Example The test file Dakota/test/dakota dace.in computes 49 orthogonal array samples, which may be evaluated concurrently using parallel computing. When executing Dakota with this input file on a single processor, the following execution syntax may be used: dakota -i dakota_dace.in For serial execution (the default), the interface specification within dakota dace.in would appear similar to interface, system analysis_driver = ’text_book’ which results in function evaluation output similar to the following (for output set to quiet mode): >>>>> Running dace iterator. DACE method = 12 Samples = 49 Symbols = 7 Seed (user-specified) = 5 -----------------------------Begin I1 Evaluation 1 -----------------------------text_book /tmp/fileia6gVb /tmp/filedDo5MH -----------------------------Begin I1 Evaluation 2 -----------------------------text_book /tmp/fileyfkQGd /tmp/fileAbmBAJ <<<<< Iterator dace completed. where it is evident that each function evaluation is being performed sequentially. For parallel execution using asynchronous local approaches, the Dakota execution syntax is unchanged as Dakota is still launched on a single processor. However, the interface specification is augmented to include the asynchronous keyword with optional concurrency limiter to indicate that multiple analysis driver instances will be executed concurrently: interface, system asynchronous evaluation_concurrency = 4 analysis_driver = ’text_book’ which results in output excerpts similar to the following: >>>>> Running dace iterator. DACE method = 12 Samples = 49 Symbols = 7 Seed (user-specified) = 5 Dakota Version 6.7 User’s Manual generated on November 13, 2017 282 CHAPTER 17. PARALLEL COMPUTING -----------------------------Begin I1 Evaluation 1 -----------------------------(Asynchronous job 1 added to I1 queue) -----------------------------Begin I1 Evaluation 2 -----------------------------(Asynchronous job 2 added to I1 queue) -----------------------------Begin I1 Evaluation 49 -----------------------------(Asynchronous job 49 added to I1 queue) Blocking synchronize of 49 asynchronous evaluations First pass: initiating 4 local asynchronous jobs Initiating I1 evaluation 1 text_book /tmp/fileuLcfBp /tmp/file6XIhpm & Initiating I1 evaluation 2 text_book /tmp/fileeC29dj /tmp/fileIdA22f & Initiating I1 evaluation 3 text_book /tmp/fileuhCESc /tmp/fileajLgI9 & Initiating I1 evaluation 4 text_book /tmp/filevJHMy6 /tmp/fileHFKip3 & Second pass: scheduling 45 remaining local asynchronous jobs Waiting on completed jobs I1 evaluation 1 has completed I1 evaluation 2 has completed I1 evaluation 3 has completed Initiating I1 evaluation 5 text_book /tmp/fileISsjh0 /tmp/fileSaek9W & Initiating I1 evaluation 6 text_book /tmp/filefN271T /tmp/fileSNYVUQ & Initiating I1 evaluation 7 text_book /tmp/filebAQaON /tmp/fileaMPpHK & I1 evaluation 49 has completed <<<<< Iterator dace completed. where it is evident that each of the 49 jobs is first queued and then a blocking synchronization is performed. This synchronization uses a simple scheduler that initiates 4 jobs and then replaces completing jobs with new ones until all 49 are complete. The default job concurrency for asynchronous local parallelism is all that is available from the algorithm (49 in this case), which could be too many for the computational resources or their usage policies. The concurrency level specification (4 in this case) instructs the scheduler to keep 4 jobs running concurrently, which would be appropriate for, e.g., a dual-processor dual-core workstation. In this case, it is the operating system’s responsibility to assign the concurrent text book jobs to available processors/cores. Specifying greater concurrency than that Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.2. SINGLE-LEVEL PARALLELISM 283 supported by the hardware will result in additional context switching within a multitasking operating system and will generally degrade performance. Note however that, in this example, there are a total of 5 processes running, one for Dakota and four for the concurrent function evaluations. Since the Dakota process checks periodically for job completion and sleeps in between checks, it is relatively lightweight and does not require a dedicated processor. 17.2.1.5 Local evaluation scheduling options The default behavior for asynchronous local parallelism is for Dakota to dispatch the next evaluation the local queue when one completes (and can optionally be specified by local evaluation scheduling dynamic. In some cases, the simulation code interface benefits from knowing which job number will replace a completed job. This includes some modes of application tiling with certain MPI implementations, where sending a job to the correct subset of available processors is done with relative node scheduling. The keywords local evaluation scheduling static forces this behavior, so a completed evaluation will be replaced with one congruent modulo the evaluation concurrency. For example, with 6 concurrent jobs, eval number 2 will be replaced with eval number 8. Examples of this usage can be seen in Dakota/examples/parallelism. 17.2.2 Message Passing Parallelism Dakota uses a “single program-multiple data” (SPMD) parallel programming model. It uses message-passing routines from the Message Passing Interface (MPI) standard [70], [125] to communicate data between processors. The SPMD designation simply denotes that the same Dakota executable is loaded on all processors during the parallel invocation. This differs from the MPMD model (“multiple program-multiple data”) which would have the Dakota executable on one or more processors communicating directly with simulator executables on other processors. The MPMD model has some advantages, but heterogeneous executable loads are not supported by all parallel environments. Moreover, the MPMD model requires simulation code intrusion on the same order as conversion to a subroutine, so subroutine conversion (see Section 16.2) in a direct-linked SPMD model is preferred. 17.2.2.1 Partitioning A level of message passing parallelism can use either of two processor partitioning models: • Dedicated master: a single processor is dedicated to scheduling operations and the remaining processors are split into server partitions. • Peer partition: all processors are allocated to server partitions and the loss of a processor to scheduling is avoided. These models are depicted in Figure 17.2. The peer partition is desirable since it utilizes all processors for computation; however, it requires either the use of sophisticated mechanisms for distributed scheduling or a problem for which static scheduling of concurrent work performs well (see Scheduling below). If neither of these characteristics is present, then use of the dedicated master partition supports a dynamic scheduling which assures that server idleness is minimized. Dakota Version 6.7 User’s Manual generated on November 13, 2017 284 CHAPTER 17. PARALLEL COMPUTING Figure 17.2: Communicator partitioning models. 17.2.2.2 Scheduling The following scheduling approaches are available within a level of message passing parallelism: • Dynamic scheduling: in the dedicated master model, the master processor manages a single processing queue and maintains a prescribed number of jobs (usually one) active on each slave. Once a slave server has completed a job and returned its results, the master assigns the next job to this slave. Thus, the job assignment on the master adapts to the job completion speed on the slaves. This provides a simple dynamic scheduler in that heterogeneous processor speeds and/or job durations are naturally handled, provided there are sufficient instances scheduled through the servers to balance the variation. In the case of a peer partition, dynamic schedulers can also be employed, provided that peer 1 can employ nonblocking synchronization of its local evaluations. This allows it to balance its local work with servicing job assignments and returns from the other peers. • Static scheduling: if scheduling is statically determined at start-up, then no master processor is needed to direct traffic and a peer partitioning approach is applicable. If the static schedule is a good one (ideal conditions), then this approach will have superior performance. However, heterogeneity, when not known a priori, can very quickly degrade performance since there is no mechanism to adapt. Message passing schedulers may be used for managing concurrent sub-iterator executions within a meta-iterator, concurrent evaluations within an iterator, or concurrent analyses within an evaluation. In the former and latter cases, the message passing scheduler is currently restricted to blocking synchronization, in that all jobs in the queue are completed before exiting the scheduler and returning the set of results to the algorithm. Nonblocking message-passing scheduling is supported for the iterator–evaluation concurrency level in support of fully asynchronous algorithms (e.g., asynch pattern search and coliny pattern search) that avoid synchronization points that can harm scaling. Message passing is also used within a fine-grained parallel simulation code, although this is separate from Dakota’s capabilities (Dakota may, at most, pass a communicator partition to the simulation). The “Message Passing” column in Table 17.1 summarizes these capabilities. Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.2. SINGLE-LEVEL PARALLELISM 17.2.2.3 285 Message Passing Example Revisiting the test file dakota dace.in, Dakota will now compute the 49 orthogonal array samples using a message passing approach. In this case, a parallel launch utility is used to execute Dakota across multiple processors using syntax similar to the following: mpirun -np 5 -machinefile machines dakota -i dakota_dace.in Since the asynchronous local parallelism will not be used, the interface specification does not include the asynchronous keyword and would appear similar to: interface, system analysis_driver = ’text_book’ The relevant excerpts from the Dakota output for a dedicated master partition and dynamic schedule, the default when the maximum concurrency (49) exceeds the available capacity (5), would appear similar to the following: Running MPI Dakota executable in parallel on 5 processors. ----------------------------------------------------------------------------DAKOTA parallel configuration: Level ----concurrent evaluations concurrent analyses multiprocessor analysis num_servers ----------5 1 1 procs_per_server ---------------1 1 N/A partition --------peer peer N/A Total parallelism levels = 1 (1 dakota, 0 analysis) ---------------------------------------------------------------------------->>>>> Executing environment. >>>>> Running dace iterator. DACE method = 12 Samples = 49 Symbols = 7 Seed (user-specified) = 5 -----------------------------Begin I1 Evaluation 1 -----------------------------(Asynchronous job 1 added to I1 queue) -----------------------------Begin I1 Evaluation 2 -----------------------------(Asynchronous job 2 added to I1 queue) -----------------------------Begin I1 Evaluation 49 -----------------------------(Asynchronous job 49 added to I1 queue) Dakota Version 6.7 User’s Manual generated on November 13, 2017 286 CHAPTER 17. PARALLEL COMPUTING Blocking synchronize of 49 asynchronous evaluations Peer dynamic schedule: first pass assigning 4 jobs among 4 remote peers Peer 1 assigning I1 evaluation 1 to peer 2 Peer 1 assigning I1 evaluation 2 to peer 3 Peer 1 assigning I1 evaluation 3 to peer 4 Peer 1 assigning I1 evaluation 4 to peer 5 Peer dynamic schedule: first pass launching 1 local jobs Initiating I1 evaluation 5 text_book /tmp/file5LRsBu /tmp/fileT2mS65 & Peer dynamic schedule: second pass scheduling 44 remaining jobs Initiating I1 evaluation 5 text_book /tmp/file5LRsBu /tmp/fileT2mS65 & Peer dynamic schedule: second pass scheduling 44 remaining jobs I1 evaluation 5 has completed Initiating I1 evaluation 6 text_book /tmp/fileZJaODH /tmp/filewoUJaj & I1 evaluation 2 has returned from peer server 3 Peer 1 assigning I1 evaluation 7 to peer 3 I1 evaluation 4 has returned from peer server 5 I1 evaluation 46 has returned from peer server 2 I1 evaluation 49 has returned from peer server 5 <<<<< Function evaluation summary (I1): 49 total (49 new, 0 duplicate) <<<<< Iterator dace completed. where it is evident that each of the 49 jobs is first queued and then a blocking synchronization is performed. This synchronization uses a dynamic scheduler that initiates five jobs, one on each of five evaluation servers, and then replaces completing jobs with new ones until all 49 are complete. It is important to note that job execution local to each of the four servers is synchronous. 17.2.3 Hybrid Parallelism The asynchronous local approaches described in Section 17.2.1 can be considered to rely on external scheduling mechanisms, since it is generally the operating system or some external queue/load sharing software that allocates jobs to processors. Conversely, the message-passing approaches described in Section 17.2.2 rely on internal scheduling mechanisms to distribute work among processors. These two approaches provide building blocks which can be combined in a variety of ways to manage parallelism at multiple levels. At one extreme, Dakota can execute on a single processor and rely completely on external means to map all jobs to processors (i.e., using asynchronous local approaches). At the other extreme, Dakota can execute on many processors and manage all levels of parallelism, including the parallel simulations, using completely internal approaches (i.e., using message passing at all levels as in Figure 17.4). While all-internal or all-external approaches are common cases, many additional approaches exist between the two extremes in which some parallelism is managed internally and some is managed externally. These combined approaches are referred to as hybrid parallelism, since the internal distribution of work based on message-passing is being combined with external allocation using asynchronous local approaches2 . Figure 17.1 2 The term “hybrid parallelism” is often used to describe the combination of MPI message passing and OpenMP shared memory parallelism models. This can be considered to be a special case of the meaning here, as OpenMP is based on threads, which is analagous to asynchronous Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.2. SINGLE-LEVEL PARALLELISM 287 depicts the asynchronous local, message-passing, and hybrid approaches for a dedicated-master partition. Approaches (b) and (c) both use MPI message-passing to distribute work from the master to the slaves, and approaches (a) and (c) both manage asynchronous jobs local to a processor. The hybrid approach (c) can be seen to be a combination of (a) and (b) since jobs are being internally distributed to slave servers through messagepassing and each slave server is managing multiple concurrent jobs using an asynchronous local approach. From a different perspective, one could consider (a) and (b) to be special cases within the range of configurations supported by (c). The hybrid approach is useful for supercomputers that maintain a service/compute node distinction and for supercomputers or networks of workstations that involve clusters of symmetric multiprocessors (SMPs). In the service/compute node case, concurrent multiprocessor simulations are launched into the compute nodes from the service node partition. While an asynchronous local approach from a single service node would be sufficient, spreading the application load by running Dakota in parallel across multiple service nodes results in better performance [41]. If the number of concurrent jobs to be managed in the compute partition exceeds the number of available service nodes, then hybrid parallelism is the preferred approach. In the case of a cluster of SMPs (or network of multiprocessor workstations), message-passing can be used to communicate between SMPs, and asynchronous local approaches can be used within an SMP. Hybrid parallelism can again result in improved performance, since the total number of Dakota MPI processes is reduced in comparison to a pure message-passing approach over all processors. Hybrid schedulers may be used for managing concurrent evaluations within an iterator or concurrent analyses within an evaluation. In the former case, blocking or nonblocking synchronization can be used, whereas the latter case is restricted to blocking synchronization. The “Hybrid” column in Table 17.1 summarizes these capabilities. 17.2.3.1 Hybrid Example Revisiting the test file dakota dace.in, Dakota will now compute the 49 orthogonal array samples using a hybrid approach. As for the message passing case, a parallel launch utility is used to execute Dakota across multiple processors: mpirun -np 5 -machinefile machines dakota -i dakota_dace.in Since the asynchronous local parallelism will also be used, the interface specification includes the asynchronous keyword and appears similar to interface, system asynchronous evaluation_concurrency = 2 analysis_driver = ’text_book’ In the hybrid case, the specification of the desired concurrency level must be included, since the default is no longer all available (as it is for asynchronous local parallelism). Rather the default is to employ message passing parallelism, and hybrid parallelism is only available through the specification of asynchronous concurrency greater than one. The relevant excerpts of the Dakota output for a peer partition and dynamic schedule , the default when the maximum concurrency (49) exceeds the maximum available capacity (10), would appear similar to the following: Running MPI Dakota executable in parallel on 5 processors. ----------------------------------------------------------------------------DAKOTA parallel configuration: local usage of the direct simulation interface. Dakota Version 6.7 User’s Manual generated on November 13, 2017 288 CHAPTER 17. PARALLEL COMPUTING Level num_servers procs_per_server ------------------------------concurrent evaluations 5 concurrent analyses 1 multiprocessor analysis 1 partition --------1 1 N/A peer peer N/A Total parallelism levels = 1 (1 dakota, 0 analysis) ---------------------------------------------------------------------------->>>>> Executing environment. >>>>> Running dace iterator. DACE method = 12 Samples = 49 Symbols = 7 Seed (user-specified) = 5 -----------------------------Begin I1 Evaluation 1 -----------------------------(Asynchronous job 1 added to I1 queue) -----------------------------Begin I1 Evaluation 2 -----------------------------(Asynchronous job 2 added to I1 queue) Blocking synchronize of 49 asynchronous evaluations Peer dynamic schedule: first pass assigning 8 jobs among 4 remote peers Peer 1 assigning I1 evaluation 1 to peer 2 Peer 1 assigning I1 evaluation 2 to peer 3 Peer 1 assigning I1 evaluation 3 to peer 4 Peer 1 assigning I1 evaluation 4 to peer 5 Peer 1 assigning I1 evaluation 6 to peer 2 Peer 1 assigning I1 evaluation 7 to peer 3 Peer 1 assigning I1 evaluation 8 to peer 4 Peer 1 assigning I1 evaluation 9 to peer 5 Peer dynamic schedule: first pass launching 2 local jobs Initiating I1 evaluation 5 text_book /tmp/fileJU1Ez2 /tmp/fileVGZzEX & Initiating I1 evaluation 10 text_book /tmp/fileKfUgKS /tmp/fileMgZXPN & Peer dynamic schedule: second pass scheduling 39 remaining jobs I1 evaluation 49 has completed I1 evaluation 43 has returned from peer I1 evaluation 44 has returned from peer I1 evaluation 48 has returned from peer I1 evaluation 47 has returned from peer I1 evaluation 45 has returned from peer <<<<< Function evaluation summary (I1): server 2 server 3 server 4 server 2 server 3 49 total (49 new, 0 duplicate) Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.3. MULTILEVEL PARALLELISM 289 <<<<< Iterator dace completed. where it is evident that each of the 49 jobs is first queued and then a blocking synchronization is performed. This synchronization uses a dynamic scheduler that initiates ten jobs, two on each of five evaluation servers, and then replaces completing jobs with new ones until all 49 are complete. It is important to note that job execution local to each of the four servers is asynchronous. 17.3 Multilevel parallelism Parallel computers within the Department of Energy national laboratories have achieved nearly 20 quadrillion (1015 ) floating point operations per second (20 petaFLOPS) in Linpack benchmarks. Planning for ”exascale” systems, rated at 1000 petaFLOPS, is well underway. This performance is achieved through the use of massively parallel (MP) processing using O[105 − 106 ] processors. In order to harness the power of these machines for performing design, uncertainty quantification, and other systems analyses, parallel algorithms are needed which are scalable to thousands of processors. Dakota supports an open-ended number of levels of nested parallelism which, as described in Section 17.1, can be categorized into three types of concurrent job scheduling and four types of parallelism: (a) concurrent iterators within a meta-iterator (scheduled by Dakota), (b) concurrent function evaluations within each iterator (scheduled by Dakota), (c) concurrent analyses within each function evaluation (scheduled by Dakota), and (d) multiprocessor analyses (work distributed by a parallel analysis code). In combination, these parallelism levels can minimize efficiency losses and achieve near linear scaling on MP computers. Types (a) and (b) are classified as algorithmic coarse-grained parallelism, type (c) is function evaluation coarse-grained parallelism, and type (d) is function evaluation fine-grained parallelism (see Section 17.1.1). Algorithmic fine-grained parallelism is not currently supported in Dakota, although this picture is rapidly evolving. A particular application may support one or more of these parallelism types, and Dakota provides for convenient selection and combination of multiple levels. If multiple types of parallelism can be exploited, then the question may arise as to how the amount of parallelism at each level should be selected so as to maximize the overall parallel efficiency of the study. For performance analysis of multilevel parallelism formulations and detailed discussion of these issues, refer to [41]. In many cases, the user may simply employ Dakota’s automatic parallelism configuration facilities, which implement the recommendations from the aforementioned paper. Figure 17.3 shows typical fixed-size scaling performance using a modified version of the extended text book problem (see Section 20.1). Three levels of parallelism (concurrent evaluations within an iterator, concurrent analyses within each evaluation, and multiprocessor analyses) are exercised within a modest partition of processors (circa year 2000). Despite the use of a fixed problem size and the presence of some idleness within the scheduling at multiple levels, the efficiency is still reasonably high3 . Greater efficiencies are obtainable for scaled speedup studies (or for larger problems in fixed-size studies) and for problems optimized for minimal scheduler idleness (by, e.g., managing all concurrency in as few scheduling levels as possible). Note that speedup and efficiency are measured relative to the case of a single instance of a multiprocessor analysis, since it was desired to investigate the effectiveness of the Dakota schedulers independent from the efficiency of the parallel analysis. 3 Note that overhead is reduced in these scaling studies by deactivating the evaluation cache and restart file logging. Dakota Version 6.7 User’s Manual generated on November 13, 2017 290 CHAPTER 17. PARALLEL COMPUTING (a) Relative speedup. (b) Relative efficiency. Figure 17.3: Fixed-size scaling results for three levels of parallelism. 17.3.1 Asynchronous Local Parallelism In most cases, the use of asynchronous local parallelism is the termination point for multilevel parallelism, in that any level of parallelism lower than an asynchronous local level will be serialized (see discussion in Section 17.3.3). The exception to this rule is reforking of forked processes for concurrent analyses within forked evaluations. In this case, a new process is created using fork for one of several concurrent evaluations; however, the new process is not replaced immediately using exec. Rather, the new process is reforked to create additional child processes for executing concurrent analyses within each concurrent evaluation process. This capability is not supported by system calls and provides one of the key advantages to using fork over system (see Section 10.2.5). 17.3.2 Message Passing Parallelism 17.3.2.1 Partitioning of levels Dakota uses MPI communicators to identify groups of processors. The global MPI COMM WORLD communicator provides the total set of processors allocated to the Dakota run. MPI COMM WORLD can be partitioned into new intra-communicators which each define a set of processors to be used for a multiprocessor server. Each of these servers may be further partitioned to nest one level of parallelism within the next. At the lowest parallelism level, these intra-communicators can be passed into a simulation for use as the simulation’s computational context, provided that the simulation has been designed, or can be modified, to be modular on a communicator (i.e., it does not assume ownership of MPI COMM WORLD). New intra-communicators are created with the MPI Comm split routine, and in order to send messages between these intra-communicators, new inter-communicators are created with calls to MPI Intercomm create. Multiple parallel configurations (containing a set of communicator partitions) are allocated for use in studies with multiple iterators and models (e.g., 16 servers of 64 processors each could be used for iteration on a lower fidelity model, followed by two servers of 512 processors each for subsequent iteration on a higher fidelity model), and can be alternated at run time. Each of the parallel configurations are allocated at object construction time and are reported at the beginning of the Dakota output. Each tier within Dakota’s nested parallelism hierarchy can use the dedicated master and peer partition approaches described in Section 17.2.2.1. To recursively partition the subcommunicators of Figure 17.2, COMM1/2/3 in the Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.3. MULTILEVEL PARALLELISM 291 dedicated master or peer partition case would be further subdivided using the appropriate partitioning model for the next lower level of parallelism. 17.3.2.2 Scheduling within levels Figure 17.4: Recursive partitioning for nested parallelism. Dakota is designed to allow the freedom to configure each parallelism level with either the dedicated master partition/dynamic scheduling combination or the peer partition/static scheduling combination. In addition, the iterator-evaluation level supports a peer partition/dynamic scheduling option, and certain external libraries may provide custom options. As an example, Figure 17.4 shows a case in which a branch and bound meta-iterator employs peer partition/distributed scheduling at level 1, each optimizer partition employs concurrent function evaluations in a dedicated master partition/dynamic scheduling model at level 2, and each function evaluation partition employs concurrent multiprocessor analyses in a peer partition/static scheduling model at level 3. In this case, MPI COMM WORLD is subdivided into optCOMM1/2/3/.../τ1 , each optCOMM is further subdivided into evalCOMM0 (master) and evalCOMM1/2/3/.../τ2 (slaves), and each slave evalCOMM is further subdivided into analCOMM1/2/3/.../τ3 . Logic for selecting the τi that maximize overall efficiency is discussed in [41]. 17.3.3 Hybrid Parallelism Hybrid parallelism approaches can take several forms when used in the multilevel parallel context. A conceptual boundary can be considered to exist for which all parallelism above the boundary is managed internally using message-passing and all parallelism below the boundary is managed externally using asynchronous local approaches. Hybrid parallelism approaches can then be categorized based on whether this boundary between internal and external management occurs within a parallelism level (intra-level) or between two parallelism levels (inter-level). In the intra-level case, the jobs for the parallelism level containing the boundary are scheduled using a hybrid scheduler, in which a capacity multiplier is used for the number of jobs to assign to each server. Each server is then responsible for concurrently executing its capacity of jobs using an asynchronous local approach. In the inter-level case, one level of parallelism manages its parallelism internally using a message-passing approach and the next lower level of parallelism manages its parallelism externally using an asynchronous local approach. That is, the jobs for the higher level of parallelism are scheduled using a standard message-passing scheduler, in which a single job is assigned to each server. However, each of these jobs has multiple components, as managed by the next lower level of parallelism, and each server is responsible for executing these sub-components concurrently using an asynchronous local approach. For example, consider a multiprocessor Dakota run which involves an iterator scheduling a set of concurrent Dakota Version 6.7 User’s Manual generated on November 13, 2017 292 CHAPTER 17. PARALLEL COMPUTING Table 17.1: Support of job management approaches within parallelism levels. Shown in parentheses are supported simulation interfaces and supported synchronization approaches. Parallelism Level concurrent iterators within a meta-iterator or nested model concurrent function evaluations within an iterator concurrent analyses within a function evaluation Asynchronous Local X (system, fork) (blocking, nonblocking) X (fork only) (blocking synch) fine-grained parallel analysis Message Passing X (blocking synch) X (system, fork, direct) (blocking, nonblocking) X (system, fork, direct) (blocking synch) X Hybrid X (system, fork) (blocking, nonblocking) X (fork only) (blocking synch) function evaluations across a cluster of SMPs. A hybrid parallelism approach will be applied in which messagepassing parallelism is used between SMPs and asynchronous local parallelism is used within each SMP. In the hybrid intra-level case, multiple function evaluations would be scheduled to each SMP, as dictated by the capacity of the SMPs, and each SMP would manage its own set of concurrent function evaluations using an asynchronous local approach. Any lower levels of parallelism would be serialized. In the hybrid inter-level case, the function evaluations would be scheduled one per SMP, and the analysis components within each of these evaluations would be executed concurrently using asynchronous local approaches within the SMP. Thus, the distinction can be viewed as whether the concurrent jobs on each server in Figure 17.1c reflect the same level of parallelism as that being scheduled by the master (intra-level) or one level of parallelism below that being scheduled by the master (inter-level). 17.4 Capability Summary Table 17.1 shows a matrix of the supported job management approaches for each of the parallelism levels, with supported simulation interfaces and synchronization approaches shown in parentheses. The concurrent iterator and multiprocessor analysis parallelism levels can only be managed with message-passing approaches. In the former case, this is due to the fact that a separate process or thread for an iterator is not currently supported. The latter case reflects a finer point on the definition of external parallelism management. While a multiprocessor analysis can most certainly be launched (e.g., using mpirun/yod) from one of Dakota’s analysis drivers, resulting in a parallel analysis external to Dakota (which is consistent with asynchronous local and hybrid approaches), this parallelism is not visible to Dakota and therefore does not qualify as parallelism that Dakota manages (and therefore is not included in Table 17.1). The concurrent evaluation and analysis levels can be managed either with message-passing, asynchronous local, or hybrid techniques, with the exceptions that the direct interface does not support asynchronous operations (asynchronous local or hybrid) at either of these levels and the system call interface does not support asynchronous operations (asynchronous local or hybrid) at the concurrent analysis level. The direct interface restrictions are present since multithreading in not yet supported and the system call interface restrictions result from the inability to manage concurrent analyses within a nonblocking function evaluation system call. Finally, nonblocking synchronization is only supported at the concurrent function evaluation level, although it spans asynchronous local, message passing, and hybrid parallelism options. Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.5. RUNNING A PARALLEL DAKOTA JOB 17.5 293 Running a Parallel Dakota Job Section 17.2 provides a few examples of serial and parallel execution of Dakota using asynchronous local, message passing, and hybrid approaches to single-level parallelism. The following sections provides a more complete discussion of the parallel execution syntax and available specification controls. 17.5.1 Single-processor execution The command for running Dakota on a single-processor and exploiting asynchronous local parallelism is the same as for running Dakota on a single-processor for a serial study, e.g.: dakota -i dakota.in > dakota.out See Section 2.4 for additional information on single-processor command syntax. 17.5.2 Multiprocessor execution Running a Dakota job on multiple processors requires the use of an executable loading facility such as mpirun, mpiexec, poe, or yod. On a network of workstations, the mpirun script is commonly used to initiate a parallel Dakota job, e.g.: mpirun -np 12 dakota -i dakota.in > dakota.out mpirun -machinefile machines -np 12 dakota -i dakota.in > dakota.out where both examples specify the use of 12 processors, the former selecting them from a default system resources file and the latter specifying particular machines in a machine file (see [69] for details). On a massively parallel computer, the familiar mpirun/mpiexec options may be replaced with other launch scripts as dictated by the particular software stack, e.g.: yod -sz 512 dakota -i dakota.in > dakota.out In each of these cases, MPI command line arguments are used by MPI (extracted first in the call to MPI Init) and Dakota command line arguments are used by Dakota (extracted second by Dakota’s command line handler). Finally, when running on computer resources that employ NQS/PBS batch schedulers, the single-processor dakota command syntax or the multiprocessor mpirun command syntax might be contained within an executable script file which is submitted to the batch queue. For example, a command qsub -l size=512 run_dakota could be submitted to a PBS queue for execution. The NQS syntax is similar: qsub -q snl -lP 512 -lT 6:00:00 run_dakota These commands allocate 512 compute nodes for the study, and execute the run dakota script on a service node. If this script contains a single-processor dakota command, then Dakota will execute on a single service node from which it can launch parallel simulations into the compute nodes using analysis drivers that contain yod commands (any yod executions occurring at any level underneath the run dakota script are mapped to the 512 Dakota Version 6.7 User’s Manual generated on November 13, 2017 294 CHAPTER 17. PARALLEL COMPUTING compute node allocation). If the script submitted to qsub contains a multiprocessor mpirun command, then Dakota will execute across multiple service nodes so that it can spread the application load in either a messagepassing or hybrid parallelism approach. Again, analysis drivers containing yod commands would be responsible for utilizing the 512 compute nodes. And, finally, if the script submitted to qsub contains a yod of the dakota executable, then Dakota will execute directly on the compute nodes and manage all of the parallelism internally (note that a yod of this type without a qsub would be mapped to the interactive partition, rather than to the batch partition). Not all supercomputers employ the same model for service/compute partitions or provide the same support for tiling of concurrent multiprocessor simulations within a single NQS/PBS allocation. For this reason, templates for parallel job configuration are being catalogued within Dakota/examples/script interfaces and Dakota/examples/parallelism (in the software distributions) that are intended to provide guidance for individual machine idiosyncrasies. Dakota relies on hints from the runtime environment and command line arguments to detect when it has been launched in parallel. Due to the large number of HPC vendors and MPI implementations, parallel launch is not always detected properly. A parallel launch is indicated by the status message Running MPI Dakota executable in parallel on N processors. which is written to the console near the beginning of the Dakota run. Beginning with release 6.5, if Dakota incorrectly detects a parallel launch, automatic detection can be overriden by setting the environment variable DAKOTA RUN PARALLEL. If the first character is set to 1, t, or T, Dakota will configure itself to run in parallel. If the variable exists but is set to anything else, Dakota will configure itself to run in serial mode. 17.6 Specifying Parallelism Given an allotment of processors, Dakota contains logic based on the theoretical work in [41] to automatically determine an efficient parallel configuration, consisting of partitioning and scheduling selections for each of the parallelism levels. This logic accounts for problem size, the concurrency supported by particular iterative algorithms, and any user inputs or overrides. Concurrency is pushed up for most parallelism levels. That is, available processors will be assigned to concurrency at the higher parallelism levels first as we partition from the top down. If more processors are available than needed for concurrency at a level, then the server size is increased to support concurrency in the next lower level of parallelism. This process is continued until all available processors have been assigned. These assignments can be overridden by the user by specifying a number of servers, processors per server, or both, for the concurrent iterator, evaluation, and analysis parallelism levels. For example, if it is desired to parallelize concurrent analyses within each function evaluation, then an evaluation servers = 1 override would serialize the concurrent function evaluations level and ensure processor availability for concurrent analyses. The exception to this push up of concurrency occurs for concurrent-iterator parallelism levels, since iterator executions tend to have high variability in duration whenever they utilize feedback of results. For these levels, concurrency is pushed down since it is generally best to serialize the levels with the highest job variation and exploit concurrency elsewhere. Partition type (master or peer) may also be specified for each level, and peer scheduling type (dynamic or static) may be specified at the level of evaluation concurrency. However, these selections may be overridden by Dakota if they are inconsistent with the number of user-requested servers, processors per server, and available processors. Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.6. SPECIFYING PARALLELISM 295 In the following sections, the user inputs and overrides are described, followed by specification examples for single and multi-processor Dakota executions. 17.6.1 The interface specification Specifying parallelism within an interface can involve the use of the asynchronous, evaluation concurrency, and analysis concurrency keywords to specify concurrency local to a processor (i.e., asynchronous local parallelism). This asynchronous specification has dual uses: • When running Dakota on a single-processor, the asynchronous keyword specifies the use of asynchronous invocations local to the processor (these jobs then rely on external means to be allocated to other processors). The default behavior is to simultaneously launch all function evaluations available from the iterator as well as all available analyses within each function evaluation. In some cases, the default behavior can overload a machine or violate a usage policy, resulting in the need to limit the number of concurrent jobs using the evaluation concurrency and analysis concurrency specifications. • When executing Dakota across multiple processors and managing jobs with a message-passing scheduler, the asynchronous keyword specifies the use of asynchronous invocations local to each server processor, resulting in a hybrid parallelism approach (see Section 17.2.3). In this case, the default behavior is one job per server, which must be overridden with an evaluation concurrency specification and/or an analysis concurrency specification. When a hybrid parallelism approach is specified, the capacity of the servers (used in the automatic configuration logic) is defined as the number of servers times the number of asynchronous jobs per server. In both cases, the scheduling of local evaluations is dynamic by default, but may be explicitly selected or overriden using local evaluation scheduling dynamic or static. In addition, evaluation servers, processors per evaluation, and evaluation scheduling keywords can be used to override the automatic parallel configuration for concurrent function evaluations. Evaluation scheduling may be selected to be master or peer, where the latter must be further specified to be dynamic or static. To override the automatic parallelism configuration for concurrent analyses, the analysis servers and analysis scheduling keywords may be specified, and the processors per analysis keyword can be used to override the automatic parallelism configuration for the size of multiprocessor analyses used in a direct function simulation interface. Scheduling options for this level include master or peer, where the latter is static (no dynamic peer option supported). Each of these keywords appears as part of the interface commands specification in the Dakota Reference Manual [3]. 17.6.2 The meta-iterator and nested model specifications To specify concurrency in sub-iterator executions within meta-iterators and nested models, the iterator servers, processors per iterator, and iterator scheduling keywords are used to override the automatic parallelism configuration. For this level, the available scheduling options are master or peer, where the latter is static (no dynamic peer option supported). See the method and model commands specification in the Dakota Reference Manual [3] for additional details. Dakota Version 6.7 User’s Manual generated on November 13, 2017 296 CHAPTER 17. PARALLEL COMPUTING 17.6.3 Single-processor Dakota specification Specifying a single-processor Dakota job that exploits parallelism through asynchronous local approaches (see Figure 17.1a) requires inclusion of the asynchronous keyword in the interface specification. Once the input file is defined, single-processor Dakota jobs are executed using the command syntax described previously in Section 17.5.1. 17.6.3.1 Example 1 For example, the following specification runs an NPSOL optimization which will perform asynchronous finite differencing: method, npsol_sqp variables, continuous_design = 5 initial_point 0.2 0.05 0.08 0.2 lower_bounds 0.15 0.02 0.05 0.1 upper_bounds 2.0 2.0 2.0 2.0 0.2 0.1 2.0 interface, system, asynchronous analysis_drivers = ’text_book’ responses, num_objective_functions = 1 num_nonlinear_inequality_constraints = 2 numerical_gradients interval_type central method_source dakota fd_gradient_step_size = 1.e-4 no_hessians Note that method source dakota selects Dakota’s internal finite differencing routine so that the concurrency in finite difference offsets can be exploited. In this case, central differencing has been selected and 11 function evaluations (one at the current point plus two offsets in each of five variables) can be performed simultaneously for each NPSOL response request. These 11 evaluations will be launched with system calls in the background and presumably assigned to additional processors through the operating system of a multiprocessor compute server or other comparable method. The concurrency specification may be included if it is necessary to limit the maximum number of simultaneous evaluations. For example, if a maximum of six compute processors were available, the command evaluation_concurrency = 6 could be added to the asynchronous specification within the interface keyword from the preceding example. Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.6. SPECIFYING PARALLELISM 17.6.3.2 297 Example 2 If, in addition, multiple analyses can be executed concurrently within a function evaluation (e.g., from multiple load cases or disciplinary analyses that must be evaluated to compute the response data set), then an input specification similar to the following could be used: method, npsol_sqp variables, continuous_design = 5 initial_point 0.2 0.05 0.08 0.2 lower_bounds 0.15 0.02 0.05 0.1 upper_bounds 2.0 2.0 2.0 2.0 0.2 0.1 2.0 interface, fork asynchronous evaluation_concurrency = 6 analysis_concurrency = 3 analysis_drivers = ’text_book1’ ’text_book2’ ’text_book3’ responses, num_objective_functions = 1 num_nonlinear_inequality_constraints = 2 numerical_gradients method_source dakota interval_type central fd_gradient_step_size = 1.e-4 no_hessians In this case, the default concurrency with just an asynchronous specification would be all 11 function evaluations and all 3 analyses, which can be limited by the evaluation concurrency and analysis concurrency specifications. The input file above limits the function evaluation concurrency, but not the analysis concurrency (a specification of 3 is the default in this case and could be omitted). Changing the input to evaluation concurrency = 1 would serialize the function evaluations, and changing the input to analysis concurrency = 1 would serialize the analyses. 17.6.4 Multiprocessor Dakota specification In multiprocessor executions, server evaluations are synchronous (Figure 17.1b) by default and the asynchronous keyword is only used if a hybrid parallelism approach (Figure 17.1c) is desired. Multiprocessor Dakota jobs are executed using the command syntax described previously in Section 17.5.2. 17.6.4.1 Example 3 To run Example 1 using a message-passing approach, the asynchronous keyword would be removed (since the servers will execute their evaluations synchronously), resulting in the following interface specification: interface, Dakota Version 6.7 User’s Manual generated on November 13, 2017 298 CHAPTER 17. PARALLEL COMPUTING system, analysis_drivers = ’text_book’ Running Dakota on 4 processors (syntax: mpirun -np 4 dakota -i dakota.in) would result in the following parallel configuration report from the Dakota output: ----------------------------------------------------------------------------Dakota parallel configuration: Level num_servers --------------concurrent evaluations 4 concurrent analyses 1 multiprocessor analysis 1 procs_per_server ---------------1 1 N/A partition --------peer peer N/A Total parallelism levels = 1 (1 dakota, 0 analysis) ----------------------------------------------------------------------------- In this case, a peer partition and dynamic scheduling algorithm are automatically selected for the concurrent evaluations. If a dedicated master is desired instead, then this logic could be overriden by adding evaluation scheduling master: interface, system, evaluation_scheduling master analysis_drivers = ’text_book’ Running Dakota again on 4 processors (syntax: mpirun -np 4 dakota -i dakota.in) would now result in this parallel configuration report: ----------------------------------------------------------------------------Dakota parallel configuration: Level num_servers --------------concurrent evaluations 3 concurrent analyses 1 multiprocessor analysis 1 procs_per_server ---------------1 1 N/A partition --------ded. master peer N/A Total parallelism levels = 1 (1 dakota, 0 analysis) ----------------------------------------------------------------------------- Now the 11 jobs will be dynamically distributed among 3 slave servers, under the control of 1 dedicated master. As a related example, consider the case where each of the workstations used in the parallel execution has multiple processors. In this case, a hybrid parallelism approach which combines message-passing parallelism with asynchronous local parallelism (see Figure 17.1c) would be a good choice. To specify hybrid parallelism, one uses the same asynchronous specification as was used for the single-processor examples, e.g.: interface, system asynchronous evaluation_concurrency = 3 analysis_drivers = ‘text_book’ Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.6. SPECIFYING PARALLELISM 299 With 3 function evaluations concurrent on each server, the capacity of a 4 processor Dakota execution (syntax: mpirun -np 4 dakota -i dakota.in) has increased to 12 evaluations. Since all 11 jobs can now be scheduled in a single pass, a peer static scheduler is sufficient. ----------------------------------------------------------------------------Dakota parallel configuration: Level num_servers --------------concurrent evaluations 4 concurrent analyses 1 multiprocessor analysis 1 procs_per_server ---------------1 1 N/A partition --------peer peer N/A Total parallelism levels = 1 ----------------------------------------------------------------------------- 17.6.4.2 Example 4 To run Example 2 using a message-passing approach, the asynchronous specification is again removed: interface, fork analysis_drivers = ‘text_book1’ ‘text_book2’ ‘text_book3’ Running this example on 6 processors (syntax: mpirun -np 6 dakota -i dakota.in) would result in the following parallel configuration report: ----------------------------------------------------------------------------Dakota parallel configuration: Level num_servers --------------concurrent evaluations 6 concurrent analyses 1 multiprocessor analysis 1 procs_per_server ---------------1 1 N/A partition --------peer peer N/A Total parallelism levels = 1 ----------------------------------------------------------------------------- in which all of the processors have been assigned to support evaluation concurrency due to the “push up” automatic configuration logic. To assign some of the available processors to the concurrent analysis level, the following input could be used: interface, fork analysis_drivers = ‘text_book1’ ‘text_book2’ ‘text_book3’ evaluation_scheduling peer static evaluation_servers = 2 which results in the following 2-level parallel configuration: Dakota Version 6.7 User’s Manual generated on November 13, 2017 300 CHAPTER 17. PARALLEL COMPUTING ----------------------------------------------------------------------------Dakota parallel configuration: Level num_servers --------------concurrent evaluations 2 concurrent analyses 3 multiprocessor analysis 1 procs_per_server ---------------3 1 N/A partition --------peer peer N/A Total parallelism levels = 2 ----------------------------------------------------------------------------- The six processors available have been split into two evaluation servers of three processors each, where the three processors in each evaluation server manage the three analyses, one per processor. Note that without the scheduling override, a dedicated master partition at the evaluation level would have been chosen automatically, dividing the six available processors into one evaluation server with three processors and another with two. Next, consider the following 3-level parallel case, in which text book1, text book2, and text book3 from the previous examples now execute on two processors each. In this case, the processors per analysis keyword is added and the fork interface is changed to a direct interface since the fine-grained parallelism of the three simulations is managed internally: interface, direct analysis_drivers = ‘text_book1’ ‘text_book2’ ‘text_book3’ evaluation_scheduling peer static evaluation_servers = 2 processors_per_analysis = 2 This results in the following parallel configuration for a 12 processor Dakota run (syntax: mpirun -np 12 dakota -i dakota.in): ----------------------------------------------------------------------------Dakota parallel configuration: Level num_servers --------------concurrent evaluations 2 concurrent analyses 3 multiprocessor analysis 2 procs_per_server ---------------6 2 N/A partition --------peer peer N/A Total parallelism levels = 3 (2 dakota, 1 analysis) ----------------------------------------------------------------------------- An important point to recognize is that, since each of the parallel configuration inputs has been tied to the interface specification up to this point, these parallel configurations can be reallocated for each interface in a multiiterator/multi-model study. For example, a Dakota execution on 40 processors might involve the following two interface specifications: interface, direct, id_interface = ’COARSE’ analysis_driver = ’sim1’ Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.6. SPECIFYING PARALLELISM 301 evaluation_scheduling peer dynamic processors_per_analysis = 5 interface, direct, id_interface = ’FINE’ analysis_driver = ’sim2’ evaluation_scheduling peer dynamic processors_per_analysis = 10 for which the coarse model would employ 8 evaluation servers of 5 processors each and the fine model would employ 4 evaluation servers of 10 processors each. Next, consider the following 4-level parallel case that employs the Pareto set optimization meta-iterator. In this case, iterator servers and iterator scheduling peer requests are included in the method specification: method, pareto_set iterator_servers = 2 iterator_scheduling peer opt_method_pointer = ’NLP’ random_weight_sets = 4 Adding this pareto set method specification to the input file from the previous 12 processor example results in the following parallel configuration for a 24 processor Dakota run (syntax: mpirun -np 24 dakota -i dakota.in): ----------------------------------------------------------------------------Dakota parallel configuration: Level num_servers --------------concurrent iterators 2 concurrent evaluations 2 concurrent analyses 3 multiprocessor analysis 2 procs_per_server ---------------12 6 2 N/A partition --------peer peer peer N/A Total parallelism levels = 4 (3 dakota, 1 analysis) ----------------------------------------------------------------------------- Note that for this example, the parallel configuration is written to the file dakota.out.1 because of the use of concurrent iterators. 17.6.4.3 Example 5 As a final example, consider a multi-start optimization conducted on 384 processors. A job of this size must be submitted to the batch queue, using syntax similar to: qsub -q snl -lP 384 -lT 6:00:00 run_dakota where the run dakota script appears as Dakota Version 6.7 User’s Manual generated on November 13, 2017 302 CHAPTER 17. PARALLEL COMPUTING #!/bin/sh cd /scratch/ yod -sz 384 dakota -i dakota.in > dakota.out the interface specifications from the dakota.in input file appears as interface, direct, analysis_drivers = ’text_book1’ ’text_book2’ ’text_book3’ evaluation_servers = 8 evaluation_scheduling peer dynamic processors_per_analysis = 2 and finally, an additional method section is added method, multi_start method_pointer = ’CPS’ iterator_servers = 8 random_starts = 8 The resulting parallel configuration is reported as ----------------------------------------------------------------------------Dakota parallel configuration: Level num_servers --------------concurrent iterators 8 concurrent evaluations 8 concurrent analyses 3 multiprocessor analysis 2 procs_per_server ---------------48 6 2 N/A partition --------peer peer peer N/A Total parallelism levels = 4 (3 dakota, 1 analysis) ----------------------------------------------------------------------------- Since the concurrency at each of the nested levels has a multiplicative effect on the number of processors that can be utilized, it is easy to see how large numbers of processors can be put to effective use in reducing the time to reach a solution, even when, as in this example, the concurrency per level is relatively low. 17.7 Application Parallelism Use Cases This section describes several common use cases for running Dakota on parallel computing clusters with various combinations of Dakota and application parallelism. In three of the four cases addressed, the application launched by Dakota is assumed MPI-enabled and run as an independent parallel process. The examples/parallelism folder in the Dakota installation includes examples of the use cases. In all four, Dakota performs a vector parameter on the ”textbook” test function described in Section 20.1. The application executed for serial demonstration is the text book example driver, and for parallel execution, a modified version named text book simple par. Both are located in Dakota’s test folder. Dakota uses its fork interface to Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.7. APPLICATION PARALLELISM USE CASES 303 Table 17.2: Cases for Dakota and application-level parallelism with M available processors and each application job requiring N processors. Cases 1–3 assume that Dakota and any application runs will execute wholly within a single scheduled job, whereas Case 4 is relevant when analysis jobs must be individually submitted to a scheduler. Case 1 2 3 4 Name Massively Serial Sequential Parallel Evaluation Tiling Evaluation Submission Notes M simultaneous application instances, each N = 1 processor Dakota Application parallel serial serial parallel 1 simultaneous application instance on N processors serial parallel M/N simultaneous N processor jobs serial parallel submit expensive N processor application jobs to a scheduler (e.g., qsub) launch interface scripts written either in Bash or Python, which include mock pre-processing to prepare application input, application execution in serial or parallel, and post-processing of application results to return to Dakota. The combinations of Dakota and application parallelism are summarized in Table 17.2. In each case, M denotes the total number of processors (or MPI tasks) allocated and N denotes the number of processors used by a single application analysis. For most scenarios, Cases 1–3, where Dakota and the application jobs run within a single cluster processor allocation (queued job), are preferred. However for particularly long-running or large jobs, or platforms that not supporting the first scheduling modes, Case 4 may be most appropriate. Relevant example files for each case are included in directories Dakota/examples/parallelism/ with the Dakota distribution. These typically include a PBS or SLURM job submission script to launch the Dakota study, a Dakota input file, and a driver script. 17.7.1 Case 1: Massively Serial — Multiple serial analysis jobs In this case, Dakota will launch multiple simultaneous single processor application runs (an embarrassingly parallel model). Dakota is run in parallel, making this example an elaboration of the message-passing single-level parallel mode described in Section 17.2. Specifically in this example, Dakota is run in parallel with M = 6 processors (pbs submission): mpiexec -n 6 dakota dakota_pstudy.in and will launch M simultaneous analysis jobs, and as each job completes, another will be launched, until all jobs are complete. • If the analysis is extremely fast, performance may be improved by launching multiple evaluation jobs local to each Dakota MPI process, specifying asynchronous evaluation_concurrency = [2 or more] As discussed in Section 17.2.3, combining MPI and local (asynchronous) parallelism in this way is an example of hybrid parallelism. • Conversely, if the analysis has large memory requirements, Dakota may be launched on fewer than the total number of available cores, which has the effect of increasing the memory available to each MPI task. This is Dakota Version 6.7 User’s Manual generated on November 13, 2017 304 CHAPTER 17. PARALLEL COMPUTING known as undersubscription. In this case, the simulation may still be able to take advantage of thread-based parallelism technologies such as OpenMP. Users are advised to consult their HPC’s documentation or user support to determine how to control the number of MPI tasks launched per compute node. • Hybrid parallelism is another way to reduce Dakota’s memory footprint. Dakota may be launched in parallel using one MPI task per node and configured to run multiple evaluations concurrently on each node using local parallelism. Suppose it is desired to run 160 concurrent evaluations, and the compute nodes each have 16 processors. The job script should reserve 10 nodes, assign one MPI task per node, and to run Dakota using 10 tasks. The interface section of the Dakota input file should contain: asynchronous evaluation_concurrency = 16 Note: The MPI standard does not support nested calls to MPI Init. Although some MPI implementations are tolerant of nested calls and work as naively expected, it is not possible generally to launch an MPI-enabled user simulation in parallel beneath Dakota running in parallel. This restriction includes launching parallelized user simulations on one core (i.e. mpiexec -n 1). 17.7.2 Case 2: Sequential Parallel — One parallel analysis job at a time This case is relevant for multi-processor analysis jobs, typically where the analysis is expensive (i.e., is longrunning or sufficient processors are not available to run more than one simultaneous analysis). Note that for extremely long-running parallel jobs, Case 4 (Evaluation Submission) below may be more appropriate. In this case, Dakota runs in serial dakota dakota_pstudy.in and the driver script launches the application with mpiexec -n K, where K ≤ M , to launch the application code within the processor allocation: mpiexec -n 6 text_book_par application.in application.out 17.7.3 Case 3: Evaluation Tiling — Multiple simultaneous parallel analysis jobs In this case, the nodes or processors (or MPI tasks) of a single job are partitioned into equally-sized tiles. The number of MPI tasks in each tile is N , the number needed to run the parallel application, and so there are a total of M/N tiles, where M is the total number of MPI tasks in the allocation. Dakota, which is run serially by the job script, asynchronously launches evaluations, each of which runs a parallel application on an available tile. It is up to the user to ensure consistency among the number of nodes in the allocation, the number of processors (or MPI tasks) per node, Dakota’s evaluation concurrency, and the number of processors (or MPI tasks) per parallel application run. For instance, suppose it is desired to perform 10 concurrent runs of a parallel application, each requiring 32 processors. The compute nodes each have 16 processors. The job script must reserve 2 nodes per application run (32/16) for a total of 2 · 10 = 20 nodes. Dakota’s evaluation concurrency must be set to 10. Under ideal circumstances, as Dakota concurrently launches evaluations of the user’s parallel application, the cluster workload manager (e.g. SLURM, PBS) performs load balancing and ensures that the runs ”land” on idle resources. In this situation, the Dakota-application interface script is relatively simple; in the execution phase, the application is run using the appropriate parallel launcher (e.g. srun), specifying the number of MPI tasks to use. Dakota Version 6.7 User’s Manual generated on November 13, 2017 17.7. APPLICATION PARALLELISM USE CASES 305 However, if load balancing is not automatically handled by the workload manager, and the user does nothing to manage tiling, then all the evaluations may land on the first few nodes, leaving the rest idle and severly degrading performance. Clearly, care must be taken to ensure that evaluations are tiled correctly. Whether correct evaluation tiling occurs automatically can depend intimately on how the HPC adminstrators configured the workload manager and MPI. Users are advised to perform small-scale experiments to determine whether performance is as expected, and/or to contact their system administrator for guidance. Dakota provides a few examples and tools to help users orchestrate placement of parallel applications on available resources when the resource manager does not. They are explained in the following sections. A related consideration is the memory usage of Dakota itself. If the user’s application is memory intensive, it may be desirable to reserve a node or a portion of a node for Dakota to prevent it from degrading the performance of evaluations. It is necessary in this case to determine where the job script, and hence Dakota, is run. Consulting the workload manager’s documenation or the HPC’s system administrator is advised. 17.7.3.1 Mpiexec server mode Mpiexec (http://www.osc.edu/ pw/mpiexec/) works in concert with MPICH implementations, extending mpirun to run jobs in a PBS environment with additional features. It offers a background server option which can be used to tile multiple MPI jobs within a single parallel resource allocation. (Note that with MPICH, there is a difference between mpirun and mpiexec, unlike with OpenMPI, where both are typically aliases for orterun.) See the example in Case3-EvaluationTiling/MPICH. In this case, an mpiexec server process is started and backgrounded to service application requests for processors; Dakota runs in serial (pbs submission): mpiexec -server & dakota dakota_pstudy.in and asynchronously launches M/N = 3 evaluations (dakota pstudy.in): interface fork asynchronous evaluation_concurrency = 3 analysis_driver = ’text_book_par_driver’ The simulator script calls mpiexec -n 2 to run the analysis in parallel and the mpiexec server assigns a subset of the available processors to the particular MPI task (text book par): mpiexec -n 2 text_book_simple_par application.in application.out An error will result if more application tasks are launched than the processor allocation permits. An error may result if the application does not exit cleanly. At present similar capability is not supported by OpenMPI, although a daemon mode similar to Mpiexec has been proposed. 17.7.3.2 Relative node scheduling This Evaluation Tiling variant uses OpenMPI 1.3.3 or newer. It leverages Dakota’s local evaluation scheduling static option together with integer arithmetic to schedule each evaluation on the right subset of the processor alDakota Version 6.7 User’s Manual generated on November 13, 2017 306 CHAPTER 17. PARALLEL COMPUTING location. A Bash-based example is provided in Case3-EvaluationTiling/OpenMPI. Similar approaches work with some AIX/POE installations as well. The mpitile utility, released with Dakota 6.6, transparently manages construction of relative node lists when using the OpenMPI command mpirun and the SLURM workload manager. mpitile resides in the Dakota bin/ folder and is a wrapper for mpirun. It uses a file locking mechanism to support dynamic scheduling of evaluations but also has a --static option. Using the --dedicated-master option, either an entire NODE or a TILE can be reserved for Dakota. Running mpitile with the --help option provides a basic description of its options. The script text book mpitile dynamic.sh in the OpenMPI example folder demonstrates usage of mpitile. mpitile is based on the Python module dakota.interfacing.parallel, also released with Dakota 6.6. Interface scripts written in Python may benefit from using its API directly. An example is located at Case3-EvaluationTiling/OpenMPI/text book di dynamic.py. The dakota Python package is located in share/dakota/Python/, which users should add to the environment variable PYTHONPATH. 17.7.3.3 Machinefile management This Evaluation Tiling variant applies when the application must be compiled with OpenMPI or another MPI implementation that does not support a server mode for job tiling, but does support the use of machine files specifying the resources on which to run the application job. A set of scripts are used to manage the partitioning of the M processor allocation into tiles contain N processors. Each tile has an associated machines file consisting of a unique subset of the assigned resources. Note that this will not work with early OpenMPI versions with some resource managers (e.g., OpenMPI 1.2 with Torque), where machinefiles, even if a proper subset of $PBS NODEFILE, are ignored. This will however work with OpenMPI 1.3 and newer. See the example in Case3-EvaluationTiling/MachinefileMgmt. In this case the pbs submission script defines variables specifying how to create a separate node file for each job and sets up a set of nodefiles for use by each evaluation. As when using relative node lists, Dakota runs in serial and uses asynchronous evaluation concurrency to launch the jobs. The interface script text book par driver contains logic to lock a node file for the application run and return it when complete. As each job completes, the next is scheduled. 17.7.4 Case 4: Evaluation Submission — Parallel analysis jobs submitted to a queue This case describes running Dakota to submit parallel jobs to a batch queue. This option is likely only useful when the cost of an individual analysis evaluation is high (such that the job requires far too many processors or hours to run all the evaluations) and there is no feedback to Dakota required to generate subsequent evaluation points. So this scenario is likely more relevant for sensitivity analysis and uncertainty quantification than optimization. In the first pass, Dakota runs (likely interactively) in serial on a login node or other node capable of job submission: dakota dakota_pstudy.in For each evaluation, the simulator script (text book par driver) will generate a pbs submission script and submit it to the scheduler. Dummy results are returned to Dakota which will exit when all jobs have been scheduled. In the second pass, when analysis is complete, the analysis driver is changed to post process and Dakota is executed on a login node to collect the results of the study. Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 18 Restart Capabilities and Utilities 18.1 Restart Management Dakota was developed for solving problems that require multiple calls to computationally expensive simulation codes. In some cases you may want to conduct the same optimization, but to a tighter final convergence tolerance. This would be costly if the entire optimization analysis had to be repeated. Interruptions imposed by computer usage policies, power outages, and system failures could also result in costly delays. However, Dakota automatically records the variable and response data from all function evaluations so that new executions of Dakota can pick up where previous executions left off. The Dakota restart file (e.g., dakota.rst) is written in a binary format, leveraging the Boost.Serialization library. While the cross-platform portability may NOT be as general as, say, the XDR standard, experience has shown it to be a sufficiently portable format to meet most users needs. Caution should be exercised to ensure consistent endianness of the computer architectures involved when attempting to leverage the restart capability in a multi-host environment. For example, if a little endian host is used to create the restart file, it can only be reliably ported and read on a host that is also little endian. As shown in Section 2.4, the primary restart commands for Dakota are -read restart, -write restart, and -stop restart. To write a restart file using a particular name, the -write restart command line input (may be abbreviated as -w) is used: dakota -i dakota.in -write_restart my_restart_file If no -write restart specification is used, then Dakota will still write a restart file, but using the default name dakota.rst instead of a user-specified name. To turn restart recording off, the user may select deactivate restart file in the interface specification (refer to the Interface Commands chapter in the Dakota Reference Manual [3] for additional information). This can increase execution speed and reduce disk storage requirements, but at the expense of a loss in the ability to recover and continue a run that terminates prematurely. Obviously, this option is not recommended when function evaluations are costly or prone to failure. Please note that using the deactivate restart file specification will result in a zero length restart file with the default name dakota.rst. To restart Dakota from a restart file, the -read restart command line input (may be abbreviated as -r) is used: dakota -i dakota.in -read_restart my_restart_file 308 CHAPTER 18. RESTART CAPABILITIES AND UTILITIES If no -read restart specification is used, then Dakota will not read restart information from any file (i.e., the default is no restart processing). A new Dakota feature (as of version 6.0) is an input file specification block providing users with additional control in the management of the function evaluation cache, duplicate evaluation detection, and restart data file entries. In the interfaces analysis driver definition, it is possible to provide additional deactivate parameters in the specification block (e.g., deactivate strict cache equality. It should be noted that, by default, Dakota’s evaluation cache and restart capabilities are based on strict binary equality. This provides a performance advantage, as it permits a hash-based data structure to be used to search the evaluation cache. The use of the deactivate strict cache equality keywords may prevent cache misses, which can occur when attempting to use a restart file on a machine different from the one on which it was generated. Specifying those keywords in the Dakota input file when performing a restart analysis should be considered judiciously, on a caseby-case basis, since there will be a performance penalty for the non-hashed evaluation cache lookups for detection of duplicates. That said, there are situations in which it is desirable to accept the performance hit of the slower cache lookups (for example a computationally expensive analysis driver). If the -write restart and -read restart specifications identify the same file (including the case where -write restart is not specified and -read restart identifies dakota.rst), then new evaluations will be appended to the existing restart file. If the -write restart and -read restart specifications identify different files, then the evaluations read from the file identified by -read restart are first written to the -write restart file. Any new evaluations are then appended to the -write restart file. In this way, restart operations can be chained together indefinitely with the assurance that all of the relevant evaluations are present in the latest restart file. To read in only a portion of a restart file, the -stop restart control (may be abbreviated as -s) is used to specify the number of entries to be read from the database. Note that this integer value corresponds to the restart record processing counter (as can be seen when using the print utility; see Section 18.2.1 below), which may differ from the evaluation numbers used in the previous run if, for example, any duplicates were detected (since these duplicates are not recorded in the restart file). In the case of a -stop restart specification, it is usually desirable to specify a new restart file using -write restart so as to remove the records of erroneous or corrupted function evaluations. For example, to read in the first 50 evaluations from dakota.rst: dakota -i dakota.in -r dakota.rst -s 50 -w dakota_new.rst The dakota new.rst file will contain the 50 processed evaluations from dakota.rst as well as any new evaluations. All evaluations following the 50th in dakota.rst have been removed from the latest restart record. Dakota’s restart algorithm relies on its duplicate detection capabilities. Processing a restart file populates the list of function evaluations that have been performed. Then, when the study is restarted, it is started from the beginning (not a “warm” start) and many of the function evaluations requested by the iterator are intercepted by the duplicate detection code. This approach has the primary advantage of restoring the complete state of the iteration (including the ability to correctly detect subsequent duplicates) for all iterators and multi-iterator methods without the need for iterator-specific restart code. However, the possibility exists for numerical round-off error to cause a divergence between the evaluations performed in the previous and restarted studies. This has been extremely rare to date. 18.2 The Dakota Restart Utility The Dakota restart utility program provides a variety of facilities for managing restart files from Dakota executions. The executable program name is dakota restart util and it has the following options, as shown by the usage message returned when executing the utility without any options: Dakota Version 6.7 User’s Manual generated on November 13, 2017 18.2. THE DAKOTA RESTART UTILITY 309 Usage: dakota_restart_util command [ ...] --options dakota_restart_util print dakota_restart_util to_neutral dakota_restart_util from_neutral dakota_restart_util to_tabular [--custom_annotated [header] [eval_id] [interface_id]] [--output_precision ] dakota_restart_util remove dakota_restart_util remove_ids ... dakota_restart_util cat ... options: --help show dakota_restart_util help message --custom_annotated arg tabular file options: header, eval_id, interface_id --freeform tabular file: freeform format --output_precision arg (=10) set tabular output precision Several of these functions involve format conversions. In particular, the binary format used for restart files can be converted to ASCII text and printed to the screen, converted to and from a neutral file format, or converted to a tabular format for importing into 3rd-party graphics programs. In addition, a restart file with corrupted data can be repaired by value or id, and multiple restart files can be combined to create a master database. 18.2.1 Print The print option outputs the contents of a particular restart file in human-readable format, since the binary format is not convenient for direct inspection. The restart data is printed in full precision, so that (near-)exact matching of points is possible for restarted runs or corrupted data removals. For example, the following command dakota_restart_util print dakota.rst results in output similar to the following (from the example in Section 20.5): -----------------------------------------Restart record 1 (evaluation id 1): -----------------------------------------Parameters: 1.8000000000000000e+00 intake_dia 1.0000000000000000e+00 flatness Active response data: Active set vector = { 3 3 3 3 } -2.4355973813420619e+00 obj_fn -4.7428486677140930e-01 nln_ineq_con_1 -4.5000000000000001e-01 nln_ineq_con_2 1.3971143170299741e-01 nln_ineq_con_3 [ -4.3644298963447897e-01 1.4999999999999999e-01 ] obj_fn gradient [ 1.3855136437818300e-01 0.0000000000000000e+00 ] nln_ineq_con_1 gradient [ 0.0000000000000000e+00 1.4999999999999999e-01 ] nln_ineq_con_2 gradient [ 0.0000000000000000e+00 -1.9485571585149869e-01 ] nln_ineq_con_3 gradient -----------------------------------------Restart record 2 (evaluation id 2): -----------------------------------------Dakota Version 6.7 User’s Manual generated on November 13, 2017 310 CHAPTER 18. RESTART CAPABILITIES AND UTILITIES Parameters: 2.1640000000000001e+00 intake_dia 1.7169994018008317e+00 flatness Active response data: Active set vector = { 3 3 3 3 } -2.4869127192988878e+00 obj_fn 6.9256958799989843e-01 nln_ineq_con_1 -3.4245008972987528e-01 nln_ineq_con_2 8.7142207937157910e-03 nln_ineq_con_3 [ -4.3644298963447897e-01 1.4999999999999999e-01 ] obj_fn gradient [ 2.9814239699997572e+01 0.0000000000000000e+00 ] nln_ineq_con_1 gradient [ 0.0000000000000000e+00 1.4999999999999999e-01 ] nln_ineq_con_2 gradient [ 0.0000000000000000e+00 -1.6998301774282701e-01 ] nln_ineq_con_3 gradient ... ... Restart file processing completed: 11 evaluations retrieved. 18.2.2 To/From Neutral File Format A Dakota restart file can be converted to a neutral file format using a command like the following: dakota_restart_util to_neutral dakota.rst dakota.neu which results in a report similar to the following: Writing neutral file dakota.neu Restart file processing completed: 11 evaluations retrieved. Similarly, a neutral file can be returned to binary format using a command like the following: dakota_restart_util from_neutral dakota.neu dakota.rst which results in a report similar to the following: Reading neutral file dakota.neu Writing new restart file dakota.rst Neutral file processing completed: 11 evaluations retrieved. The contents of the generated neutral file are similar to the following (from the first two records for the example in Section 20.5): 6 7 2 1.8000000000000000e+00 intake_dia 1.0000000000000000e+00 flatness 0 0 0 0 NULL 4 2 1 0 3 3 3 3 1 2 obj_fn nln_ineq_con_1 nln_ineq_con_2 nln_ineq_con_3 -2.4355973813420619e+00 -4.7428486677140930e-01 -4.5000000000000001e-01 1.3971143170299741e-01 -4.3644298963447897e-01 1.4999999999999999e-01 1.3855136437818300e-01 0.0000000000000000e+00 0.0000000000000000e+00 1.4999999999999999e-01 0.0000000000000000e+00 -1.9485571585149869e-01 1 6 7 2 2.1640000000000001e+00 intake_dia 1.7169994018008317e+00 flatness 0 0 0 0 NULL 4 2 1 0 3 3 3 3 1 2 obj_fn nln_ineq_con_1 nln_ineq_con_2 nln_ineq_con_3 Dakota Version 6.7 User’s Manual generated on November 13, 2017 18.2. THE DAKOTA RESTART UTILITY 311 -2.4869127192988878e+00 6.9256958799989843e-01 -3.4245008972987528e-01 8.7142207937157910e-03 -4.3644298963447897e-01 1.4999999999999999e-01 2.9814239699997572e+01 0.0000000000000000e+00 0.0000000000000000e+00 1.4999999999999999e-01 0.0000000000000000e+00 -1.6998301774282701e-01 2 This format is not intended for direct viewing (print should be used for this purpose). Rather, the neutral file capability has been used in the past for managing portability of restart data across platforms of dissimilar endianness of the computer architectures (e.g. creator of the file was little endian but the need exists to run dakota with restart on a big endian host. The neutral file format has also been shown to be useful for for advanced repair of restart records (in cases where the techniques of Section 18.2.5 were insufficient). 18.2.3 To Tabular Format Conversion of a binary restart file to a tabular format enables convenient import of this data into 3rd-party postprocessing tools such as Matlab, TECplot, Excel, etc. This facility is similar to the tabular data option in the Dakota input file specification (described in Section 13.3), but with two important differences: 1. No function evaluations are suppressed as they are with tabular data (i.e., any internal finite difference evaluations are included). 2. The conversion can be performed after Dakota completion, i.e., for Dakota runs executed previously. An example command for converting a restart file to tabular format is: dakota_restart_util to_tabular dakota.rst dakota.m which results in a report similar to the following: Writing tabular text file dakota.m Restart file processing completed: 10 evaluations tabulated. The contents of the generated tabular file are similar to the following (from the example in Section 20.1.1). Note that while evaluations resulting from numerical derivative offsets would be reported (as described above), derivatives returned as part of the evaluations are not reported (since they do not readily fit within a compact tabular format): %eval_id interface 1 NO_ID 2 NO_ID 3 NO_ID 4 NO_ID 5 NO_ID 6 NO_ID 7 NO_ID 8 NO_ID 9 NO_ID 10 NO_ID ... x1 x2 0.9 1.1 0.58256179 0.4772224441 0.5 0.4318131566 0.5 0.3695495062 0.5 0.3757758727 0.5 0.3695495062 0.5005468682 -0.5204065326 0.5000092554 0.4156974409 0.500000919 0.4302129149 0.50037519 -0.2214765079 obj_fn 0.0002 0.1050555937 0.1667232695 0.2204806721 0.2143316122 0.2204806721 5.405888123 0.1790558059 0.1679019175 2.288391116 nln_ineq_con_1 nln_ineq_con_2 0.26 0.76 0.1007670171 -0.06353963386 0.03409342169 -0.06353739777 0.06522524692 -0.1134331625 0.06211206365 -0.1087924935 0.06522524692 -0.1134331625 0.5107504335 0.02054952507 0.04216053506 -0.07720026537 0.0348944616 -0.0649173074 0.3611135847 -0.2011357515 Controlling tabular format: The command-line options --freeform and --custom annotated give control of headers in the resulting tabular file. give control of headers in the resulting tabular file. Freeform will generate a tabular file with no leading row nor columns (variable and response values only). Custom annotated format accepts any or all of the options: Dakota Version 6.7 User’s Manual generated on November 13, 2017 312 CHAPTER 18. RESTART CAPABILITIES AND UTILITIES • header: include %-commented header row with labels • eval id: include leading column with evaluation ID • interface id: include leading column with interface ID For example, to recover Dakota 6.0 tabular format, which contained a header row, leading column with evaluation ID, but no interface ID: dakota_restart_util to_tabular dakota.rst dakota.m --custom_annotated header eval_id Resulting in %eval_id 1 2 3 ... x1 0.9 0.90009 0.89991 x2 obj_fn nln_ineq_con_1 nln_ineq_con_2 1.1 0.0002 0.26 0.76 1.1 0.0001996404857 0.2601620081 0.759955 1.1 0.0002003604863 0.2598380081 0.760045 Finally, --output precision will generate tabular output with the specified integer digits of precision. 18.2.4 Concatenation of Multiple Restart Files In some instances, it is useful to combine restart files into a single master function evaluation database. For example, when constructing a data fit surrogate model, data from previous studies can be pulled in and reused to create a combined data set for the surrogate fit. An example command for concatenating multiple restart files is: dakota_restart_util cat dakota.rst.1 dakota.rst.2 dakota.rst.3 dakota.rst.all which results in a report similar to the following: Writing new restart file dakota.rst.all dakota.rst.1 processing completed: 10 evaluations retrieved. dakota.rst.2 processing completed: 110 evaluations retrieved. dakota.rst.3 processing completed: 65 evaluations retrieved. The dakota.rst.all database now contains 185 evaluations and can be read in for use in a subsequent Dakota study using the -read restart option to the dakota executable (see Section 18.1). 18.2.5 Removal of Corrupted Data On occasion, a simulation or computer system failure may cause a corruption of the Dakota restart file. For example, a simulation crash may result in failure of a post-processor to retrieve meaningful data. If 0’s (or other erroneous data) are returned from the user’s analysis driver, then this bad data will get recorded in the restart file. If there is a clear demarcation of where corruption initiated (typical in a process with feedback, such as gradient-based optimization), then use of the -stop restart option for the dakota executable can be effective in continuing the study from the point immediately prior to the introduction of bad data. If, however, there are interspersed corruptions throughout the restart database (typical in a process without feedback, such as sampling), then the remove and remove ids options of dakota restart util can be useful. An example of the command syntax for the remove option is: Dakota Version 6.7 User’s Manual generated on November 13, 2017 18.2. THE DAKOTA RESTART UTILITY 313 dakota_restart_util remove 2.e-04 dakota.rst dakota.rst.repaired which results in a report similar to the following: Writing new restart file dakota.rst.repaired Restart repair completed: 65 evaluations retrieved, 2 removed, 63 saved. where any evaluations in dakota.rst having an active response function value that matches 2.e-04 within machine precision are discarded when creating dakota.rst.repaired. An example of the command syntax for the remove ids option is: dakota_restart_util remove_ids 12 15 23 44 57 dakota.rst dakota.rst.repaired which results in a report similar to the following: Writing new restart file dakota.rst.repaired Restart repair completed: 65 evaluations retrieved, 5 removed, 60 saved. where evaluation ids 12, 15, 23, 44, and 57 have been discarded when creating dakota.rst.repaired. An important detail is that, unlike the -stop restart option which operates on restart record numbers (see Section 18.1)), the remove ids option operates on evaluation ids. Thus, removal is not necessarily based on the order of appearance in the restart file. This distinction is important when removing restart records for a run that contained either asynchronous or duplicate evaluations, since the restart insertion order and evaluation ids may not correspond in these cases (asynchronous evaluations have ids assigned in the order of job creation but are inserted in the restart file in the order of job completion, and duplicate evaluations are not recorded which introduces offsets between evaluation id and record number). This can also be important if removing records from a concatenated restart file, since the same evaluation id could appear more than once. In this case, all evaluation records with ids matching the remove ids list will be removed. If neither of these removal options is sufficient to handle a particular restart repair need, then the fallback position is to resort to direct editing of a neutral file (refer to Section 18.2.2) to perform the necessary modifications. Dakota Version 6.7 User’s Manual generated on November 13, 2017 314 CHAPTER 18. RESTART CAPABILITIES AND UTILITIES Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 19 Simulation Failure Capturing Dakota provides the capability to manage failures in simulation codes within its system call, fork, and direct simulation interfaces (see Section 10.2 for simulation interface descriptions). Failure capturing consists of three operations: failure detection, failure communication, and failure mitigation. 19.1 Failure detection Since the symptoms of a simulation failure are highly code and application dependent, it is the user’s responsibility to detect failures within their analysis driver, input filter, or output filter. One popular example of simulation monitoring is to rely on a simulation’s internal detection of errors. In this case, the UNIX grep utility can be used within a user’s driver/filter script to detect strings in output files which indicate analysis failure. For example, the following simple C shell script excerpt grep ERROR analysis.out > /dev/null if ( $status == 0 ) echo "FAIL" > results.out endif will pass the if test and communicate simulation failure to Dakota if the grep command finds the string ERROR anywhere in the analysis.out file. The /dev/null device file is called the “bit bucket” and the grep command output is discarded by redirecting it to this destination. The $status shell variable contains the exit status of the last command executed [6], which is the exit status of grep in this case (0 if successful in finding the error string, nonzero otherwise). For Bourne shells [13], the $? shell variable serves the same purpose as $status for C shells. In a related approach, if the return code from a simulation can be used directly for failure detection purposes, then $status or $? could be queried immediately following the simulation execution using an if test like that shown above. If the simulation code is not returning error codes or providing direct error diagnostic information, then failure detection may require monitoring of simulation results for sanity (e.g., is the mesh distorting excessively?) or potentially monitoring for continued process existence to detect a simulation segmentation fault or core dump. While this can get complicated, the flexibility of Dakota’s interfaces allows for a wide variety of user-defined monitoring approaches. 316 19.2 CHAPTER 19. SIMULATION FAILURE CAPTURING Failure communication Once a failure is detected, it must be communicated so that Dakota can take the appropriate corrective action. The form of this communication depends on the type of simulation interface in use. In the system call and fork simulation interfaces, a detected simulation failure is communicated to Dakota through the results file. Instead of returning the standard results file data, the string “fail” should appear at the beginning of the results file. Any data appearing after the fail string will be ignored. Also, Dakota’s detection of this string is case insensitive, so “FAIL”, “Fail”, etc., are equally valid. In the direct simulation interface case, a detected simulation failure is communicated to Dakota through the return code provided by the user’s analysis driver, input filter, or output filter. As shown in Section 16.2.1, the prototype for simulations linked within the direct interface includes an integer return code. This code has the following meanings: zero (false) indicates that all is normal and nonzero (true) indicates an exception (i.e., a simulation failure). 19.3 Failure mitigation Once the analysis failure has been communicated, Dakota will attempt to recover from the failure using one of the following four mechanisms, as governed by the interface specification in the user’s input file (see the Interface Commands chapter in the Dakota Reference Manual [3] for additional information). 19.3.1 Abort (default) If the abort option is active (the default), then Dakota will terminate upon detecting a failure. Note that if the problem causing the failure can be corrected, Dakota’s restart capability (see Chapter 18) can be used to continue the study. 19.3.2 Retry If the retry option is specified, then Dakota will re-invoke the failed simulation up to the specified number of retries. If the simulation continues to fail on each of these retries, Dakota will terminate. The retry option is appropriate for those cases in which simulation failures may be resulting from transient computing environment issues, such as shared disk space, software license access, or networking problems. 19.3.3 Recover If the recover option is specified, then Dakota will not attempt the failed simulation again. Rather, it will return a “dummy” set of function values as the results of the function evaluation. The dummy function values to be returned are specified by the user. Any gradient or Hessian data requested in the active set vector will be zero. This option is appropriate for those cases in which a failed simulation may indicate a region of the design space to be avoided and the dummy values can be used to return a large objective function or constraint violation which will discourage an optimizer from further investigating the region. Dakota Version 6.7 User’s Manual generated on November 13, 2017 19.4. SPECIAL VALUES 19.3.4 317 Continuation If the continuation option is specified, then Dakota will attempt to step towards the failing “target” simulation from a nearby “source” simulation through the use of a continuation algorithm. This option is appropriate for those cases in which a failed simulation may be caused by an inadequate initial guess. If the “distance” between the source and target can be divided into smaller steps in which information from one step provides an adequate initial guess for the next step, then the continuation method can step towards the target in increments sufficiently small to allow for convergence of the simulations. When the failure occurs, the interval between the last successful evaluation (the source point) and the current target point is halved and the evaluation is retried. This halving is repeated until a successful evaluation occurs. The algorithm then marches towards the target point using the last interval as a step size. If a failure occurs while marching forward, the interval will be halved again. Each invocation of the continuation algorithm is allowed a total of ten failures (ten halvings result in up to 1024 evaluations from source to target) prior to aborting the Dakota process. While Dakota manages the interval halving and function evaluation invocations, the user is responsible for managing the initial guess for the simulation program. For example, in a GOMA input file [118], the user specifies the files to be used for reading initial guess data and writing solution data. When using the last successful evaluation in the continuation algorithm, the translation of initial guess data can be accomplished by simply copying the solution data file leftover from the last evaluation to the initial guess file for the current evaluation (and in fact this is useful for all evaluations, not just continuation). However, a more general approach would use the closest successful evaluation (rather than the last successful evaluation) as the source point in the continuation algorithm. This will be especially important for nonlocal methods (e.g., genetic algorithms) in which the last successful evaluation may not necessarily be in the vicinity of the current evaluation. This approach will require the user to save and manipulate previous solutions (likely tagged with evaluation number) so that the results from a particular simulation (specified by Dakota after internal identification of the closest point) can be used as the current simulation’s initial guess. This more general approach is not yet supported in Dakota. 19.4 Special values In IEEE arithmetic, “NaN” indicates “not a number” and ±“Inf” or ±“Infinity” indicates positive or negative infinity. These special values may be returned directly in function evaluation results from a simulation interface or they may be specified in a user’s input file within the recover specification described in Section 19.3.3. There is a key difference between these two cases. In the former case of direct simulation return, failure mitigation can be managed on a per response function basis. When using recover, however, the failure applies to the complete set of simulation results. In both of these cases, the handling of NaN or Inf is managed using iterator-specific approaches. Currently, nondeterministic sampling methods (see Section 5.2), polynomial chaos expansions using either regression approaches or spectral projection with random sampling (see Section 5.4), and the NL2SOL method for nonlinear least squares (see §7.4.3) are the only methods with special numerical exception handling: the sampling methods simply omit any samples that are not finite from the statistics generation, the polynomial chaos methods omit any samples that are not finite from the coefficient estimation, and NL2SOL treats NaN or Infinity in a residual vector (i.e., values in a results file for a function evaluation) computed for a trial step as an indication that the trial step was too long and violates an unstated constraint; NL2SOL responds by trying a shorter step. Dakota Version 6.7 User’s Manual generated on November 13, 2017 318 CHAPTER 19. SIMULATION FAILURE CAPTURING Dakota Version 6.7 User’s Manual generated on November 13, 2017 Chapter 20 Additional Examples This chapter contains additional examples of available methods being applied to several test problems. The examples are organized by the test problem being used. See Section 1.5 for help in finding the files referenced here. Many of these examples are also used as code verification tests. The examples are run periodically and the results are checked against known solutions. This ensures that the algorithms are correctly implemented. 20.1 Textbook The two-variable version of the “textbook” test problem provides a nonlinearly constrained optimization test case. It is formulated as: minimize subject to f = (x1 − 1)4 + (x2 − 1)4 x2 g1 = x21 − ≤0 2 x 1 ≤0 g2 = x22 − 2 0.5 ≤ x1 ≤ 5.8 (20.1) −2.9 ≤ x2 ≤ 2.9 Contours of this test problem are illustrated in Figure 20.1(a), with a close-up view of the feasible region given in Figure 20.1(b). For the textbook test problem, the unconstrained minimum occurs at (x1 , x2 ) = (1, 1). However, the inclusion of the constraints moves the minimum to (x1 , x2 ) = (0.5, 0.5). Equation 20.1 presents the 2-dimensional form of 320 CHAPTER 20. ADDITIONAL EXAMPLES (a) (b) Figure 20.1: Contours of the textbook problem (a) on the [−3, 4] × [−3, 4] domain and (b) zoomed into an area containing the constrained optimum point (x1 , x2 ) = (0.5, 0.5). The feasible region lies at the intersection of the two constraints g1 (solid) and g2 (dashed). the textbook problem. An extended formulation is stated as f= minimize n X (xi − 1)4 i=1 x2 ≤0 2 x 1 g2 = x22 − ≤0 2 0.5 ≤ x1 ≤ 5.8 g1 = x21 − subject to (20.2) −2.9 ≤ x2 ≤ 2.9 where n is the number of design variables. The objective function is designed to accommodate an arbitrary number of design variables in order to allow flexible testing of a variety of data sets. Contour plots for the n = 2 case have been shown previously in Figure 20.1. For the optimization problem given in Equation 20.2, the unconstrained solution (num nonlinear inequality constraints set to zero) for two design variables is: x1 = 1.0 x2 = 1.0 f∗ = 0.0 with The solution for the optimization problem constrained by g1 (num nonlinear inequality constraints set to one) is: x1 = 0.763 x2 = 1.16 Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.1. TEXTBOOK 321 with f∗ = 0.00388 g1∗ = 0.0 (active) The solution for the optimization problem constrained by g1 and g2 (num nonlinear inequality constraints set to two) is: x1 = 0.500 x2 = 0.500 with f∗ = 0.125 g1∗ = 0.0 (active) g2∗ = 0.0 (active) Note that as constraints are added, the design freedom is restricted (the additional constraints are active at the solution) and an increase in the optimal objective function is observed. 20.1.1 Gradient-based Constrained Optimization This example demonstrates the use of a gradient-based optimization algorithm on a nonlinearly constrained problem. The Dakota input file for this example is shown in Figure 20.2. This input file is similar to the input file for the unconstrained gradient-based optimization example involving the Rosenbrock function, seen in Section 2.3.3. Note the addition of commands in the responses block of the input file that identify the number and type of constraints, along with the upper bounds on these constraints. The commands direct and analysis driver = ’text book’ specify that Dakota will use its internal version of the textbook problem. The conmin mfd keyword in Figure 20.2 tells Dakota to use the CONMIN package’s implementation of the Method of Feasible Directions (see Section 6.2.1.1 for more details). A significantly faster alternative is the DOT package’s Modified Method of Feasible Directions, i.e. dot mmfd (see Section 6.2.1.1 for more details). However, DOT is licensed software that may not be available on a particular system. If it is installed on your system and Dakota has been compiled without the --without-dot flag, you may use it by commenting out the line with conmin mfd and uncommenting the line with dot mmfd. The results of the optimization example are listed at the end of the output file (see Section 2.1.3. This information shows that the optimizer stopped at the point (x1 , x2 ) = (0.5, 0.5), where both constraints are approximately satisfied, and where the objective function value is 0.128. The progress of the optimization algorithm is shown in Figure 20.3(b) where the dots correspond to the end points of each iteration in the algorithm. The starting point is (x1 , x2 ) = (0.9, 1.1), where both constraints are violated. The optimizer takes a sequence of steps to minimize the objective function while reducing the infeasibility of the constraints. The optimization graphics are also shown in Figure 20.3(a). Dakota Version 6.7 User’s Manual generated on November 13, 2017 322 CHAPTER 20. ADDITIONAL EXAMPLES # Dakota Input File: textbook_opt_conmin.in environment graphics tabular_data tabular_data_file = ’textbook_opt_conmin.dat’ method # dot_mmfd #DOT performs better but may not be available conmin_mfd max_iterations = 50 convergence_tolerance = 1e-4 variables continuous_design = 2 initial_point 0.9 upper_bounds 5.8 lower_bounds 0.5 descriptors ’x1’ interface analysis_drivers = direct 1.1 2.9 -2.9 ’x2’ ’text_book’ responses objective_functions = 1 nonlinear_inequality_constraints = 2 descriptors ’f’ ’c1’ ’c2’ numerical_gradients method_source dakota interval_type central fd_step_size = 1.e-4 no_hessians Figure 20.2: Textbook gradient-based constrained optimization example: Dakota/examples/users/textbook opt conmin.in the Dakota input file – see Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.1. TEXTBOOK 323 (a) (b) Figure 20.3: Textbook gradient-based constrained optimization example: (a) screen capture of the Dakota graphics shows how the objective function was reduced during the search for a feasible design point and (b) iteration history (iterations marked by solid dots). Dakota Version 6.7 User’s Manual generated on November 13, 2017 324 20.2 CHAPTER 20. ADDITIONAL EXAMPLES Rosenbrock The Rosenbrock function [57] is a well known test problem for optimization algorithms. The standard formulation includes two design variables, and computes a single objective function. This problem can also be posed as a least-squares optimization problem with two residuals to be minimzed because the objective function is the sum of squared terms. Standard Formulation The standard two-dimensional formulation can be stated as minimize f = 100(x2 − x21 )2 + (1 − x1 )2 (20.3) Surface and contour plots for this function have been shown previously in Figure 2.5. The optimal solution is: x1 = 1.0 x2 = 1.0 f∗ = 0.0 with A discussion of gradient based optimization to minimize this function is in Section 2.3.3. A Least-Squares Optimization Formulation This test problem may also be used to exercise least-squares solution methods by recasting the standard problem formulation into: minimize f = (f1 )2 + (f2 )2 (20.4) where f1 = 10(x2 − x21 ) (20.5) f2 = 1 − x1 (20.6) and are residual terms. The included analysis driver can handle both formulations. In the Dakota/test directory, the rosenbrock executable (compiled from Dakota Source/test/rosenbrock.cpp) checks the number of response functions passed in the parameters file and returns either an objective function (as computed from Equation 20.3) for use with optimization methods or two least squares terms (as computed from Equations 20.5-20.6) for use with least squares methods. Both cases support analytic gradients of the function set with respect to the design variables. See Figure 2.7 (std formulation) and Figure 20.4 (least squares formulation) for examples of each usage. 20.2.1 Least-Squares Optimization Least squares methods are often used for calibration or parameter estimation, that is, to seek parameters maximizing agreement of models with experimental data. The least-squares formulation was described in the previous section. When using a least-squares approach to minimize a function, each of the least-squares terms f1 , f2 , . . . is driven toward zero. This formulation permits the use of specialized algorithms that can be more efficient than general Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.2. ROSENBROCK 325 purpose optimization algorithms. See Chapter 7 for more detail on the algorithms used for least-squares minimization, as well as a discussion on the types of engineering design problems (e.g., parameter estimation) that can make use of the least-squares approach. Figure 20.4 is a listing of the Dakota input file rosen opt nls.in. This differs from the input file shown in Figure 2.7 in several key areas. The responses block of the input file uses the keyword calibration terms = 2 instead of objective functions = 1. The method block of the input file shows that the NL2SOL algorithm [25] (nl2sol) is used in this example. (The Gauss-Newton, NL2SOL, and NLSSOL SQP algorithms are currently available for exploiting the special mathematical structure of least squares minimization problems). # Dakota Input File: rosen_opt_nls.in environment graphics tabular_data tabular_data_file = ’rosen_opt_nls.dat’ method nl2sol convergence_tolerance = 1e-4 max_iterations = 100 model single variables continuous_design = 2 initial_point -1.2 lower_bounds -2.0 upper_bounds 2.0 descriptors ’x1’ 1.0 -2.0 2.0 "x2" interface analysis_drivers = ’rosenbrock’ direct responses calibration_terms = 2 analytic_gradients no_hessians Figure 20.4: Rosenbrock nonlinear least squares Dakota/examples/users/rosen opt nls.in example: the Dakota input file – see The optimization results at the end of the output file show that the least squares minimization approach has found the same optimum design point, (x1, x2) = (1.0, 1.0), as was found using the conventional gradient-based optimization approach. The iteration history of the least squares minimization is given in Figure 20.5, and shows that 14 function evaluations were needed for convergence. In this example the least squares approach required about half the number of function evaluations as did conventional gradient-based optimization. In many cases a good least squares algorithm will converge more rapidly in the vicinity of the solution. Dakota Version 6.7 User’s Manual generated on November 13, 2017 326 CHAPTER 20. ADDITIONAL EXAMPLES Figure 20.5: Rosenbrock nonlinear least squares example: iteration history for least squares terms f1 and f2 . Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.3. HERBIE, SMOOTH HERBIE, AND SHUBERT 20.3 327 Herbie, Smooth Herbie, and Shubert Lee, et al. [94] developed the Herbie function as a 2D test problem for surrogate-based optimization. However, since it is separable and each dimension is identical it is easily generalized to an arbitrary number of dimensions. The generalized (to M dimensions) Herbie function is herb(x) = − M Y wherb (xk ) k=1 where wherb (xk ) = exp(−(xk − 1)2 ) + exp(−0.8(xk + 1)2 ) − 0.05 sin (8 (xk + 0.1)) . The Herbie function’s high frequency sine component creates a large number of local minima and maxima, making it a significantly more challenging test problem. However, when testing a method’s ability to exploit smoothness in the true response, it is desirable to have a less oscillatory function. For this reason, the “smooth Herbie” test function omits the high frequency sine term but is otherwise identical to the Herbie function. The formula for smooth Herbie is M Y wsm (xk ) herbsm (x) = − k=1 where wsm (xk ) = exp(−(xk − 1)2 ) + exp(−0.8(xk + 1)2 ). Two dimensional versions of the herbie and smooth herbie test functions are plotted in Figure 20.6. Figure 20.6: Plots of the herbie (left) and smooth herbie (right) test functions in 2 dimensions. They can accept an arbitrary number of inputs. The direction of the z-axis has been reversed (negative is up) to better view the functions’ minima. Dakota Version 6.7 User’s Manual generated on November 13, 2017 328 CHAPTER 20. ADDITIONAL EXAMPLES Shubert is another separable (and therefore arbitrary dimensional) test function. Its analytical formula is shu(x) = M Y wshu (xk ) k=1 wshu (xk ) = 5 X i cos((i + 1)xk + i) i=1 The 2D version of the shubert function is shown in Figure 20.8. 20.3.1 Efficient Global Optimization The Dakota input file herbie shubert opt ego.in shows how to use efficient global optimization (ego) to minimize the 5D version of any of these 3 separable functions. The input file is shown in Figure 20.7. Note that in the variables section the 5* preceding the values -2.0 and 2.0 for the lower bounds and upper bounds, respectively, tells Dakota to repeat them 5 times. The “interesting” region for each of these functions is −2 ≤ xk ≤ 2 for all dimensions. 20.4 Sobol and Ishigami Functions These functions are often used to test sensitivity analysis methods. These are documented in [127]. The first is the Sobol rational function, given by the equation: f (x) = (x2 + 0.5)4 (x1 + 0.5)4 (20.7) This function is monotonic across each of the inputs. However, there is substantial interaction between x1 and x2 which makes sensitivity analysis difficult. This function in shown in Figure 20.9. The Ishigami test problem [127] is a smooth C ∞ function: f (x) = sin(2πx1 − π) + 7 sin2 (2πx2 − π) + 0.1(2πx3 − π)4 sin(2πx1 − π) (20.8) where the distributions for x1 , x2 , and x3 are iid uniform on [0,1]. This function was created as a test for global sensitivity analysis methods, but is challenging for any method based on low-order structured grids (e.g., due to term cancellation at midpoints and bounds). This function in shown in Figure 20.10. At the opposite end of the smoothness spectrum, Sobol’s g-function [127] is C 0 with the absolute value contributing a slope discontinuity at the center of the domain: f (x) = 2 5 Y |4xj − 2| + aj ; a = [0, 1, 2, 4, 8] 1 + aj j=1 (20.9) The distributions for xj for j = 1, 2, 3, 4, 5 are iid uniform on [0,1]. This function in shown in Figure 20.11. Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.5. CYLINDER HEAD 329 # Dakota Input File: herbie_shubert_opt_ego.in # Example of using EGO to find the minumum of the 5 dimensional version # of the abitrary-dimensional/separable ’herbie’ OR ’smooth_herbie’ OR # ’shubert’ test functions environment tabular_data tabular_data_file = ’herbie_shubert_opt_ego.dat’ method efficient_global initial_samples 50 seed = 123456 variables continuous_design = 5 lower_bounds 5*-2.0 upper_bounds 5*2.0 # EGO Efficient Global Optimization # 5 dimensions # use 5 copies of -2.0 for bound # use 5 copies of 2.0 for bound interface # analysis_drivers = ’herbie’ # use this for herbie analysis_drivers = ’smooth_herbie’ # use this for smooth_herbie # analysis_drivers = ’shubert’ # use this for shubert direct responses objective_functions = 1 no_gradients no_hessians Figure 20.7: Herbie/Shubert examples: the Dakota/examples/users/herbie shubert opt ego.in 20.5 Dakota input file – see Cylinder Head The cylinder head test problem is stated as: minimize subject to horsepower warranty f = −1 + 250 100000 σmax ≤ 0.5σyield (20.10) warranty ≥ 100000 timecycle ≤ 60 1.5 ≤ dintake ≤ 2.164 0.0 ≤ flatness ≤ 4.0 This formulation seeks to simultaneously maximize normalized engine horsepower and engine warranty over variables of valve intake diameter (dintake ) in inches and overall head flatness (flatness) in thousandths of an inch subject to inequality constraints that the maximum stress cannot exceed half of yield, that warranty must be at least 100000 miles, and that manufacturing cycle time must be less than 60 seconds. Since the constraints Dakota Version 6.7 User’s Manual generated on November 13, 2017 330 CHAPTER 20. ADDITIONAL EXAMPLES Figure 20.8: Plot of the shubert test function in 2 dimensions. It can accept an arbitrary number of inputs. Figure 20.9: Plot of the sobol rational test function in 2 dimensions. Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.5. CYLINDER HEAD Figure 20.10: Plot of the sobol ishigami test function as a function of x1 and x3. Dakota Version 6.7 User’s Manual generated on November 13, 2017 331 332 CHAPTER 20. ADDITIONAL EXAMPLES Figure 20.11: Plot of the sobol g function test function. Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.5. CYLINDER HEAD 333 involve different scales, they should be nondimensionalized (note: the nonlinear constraint scaling described in Section 6.3.2 can now do this automatically). In addition, they can be converted to the standard 1-sided form g(x) ≤ 0 as follows: 2σmax −1≤0 σyield warranty ≤0 g2 = 1 − 100000 timecycle g3 = −1≤0 60 g1 = (20.11) The objective function and constraints are related analytically to the design variables according to the following simple expressions: warranty = 100000 + 15000(4 − flatness) 45 + 4.5(4 − flatness)1.5 dintake −1 horsepower = 250 + 200 1.833 1 σmax = 750 + (twall )2.5 timecycle twall = = offsetintake − offsetexhaust − (20.12) (dintake + dexhaust ) 2 where the constants in Equation 20.11 and Equation 20.12 assume the following values: σyield = 3000, offsetintake = 3.25, offsetexhaust = 1.34, and dexhaust = 1.556. 20.5.1 Constrained Gradient Based Optimization An example using the cylinder head test problem is shown below: The interface keyword specifies use of the cyl head executable (compiled from Dakota Source/test/cyl head.cpp) as the simulator. The variables and responses keywords specify the data sets to be used in the iteration by providing the initial point, descriptors, and upper and lower bounds for two continuous design variables and by specifying the use of one objective function, three inequality constraints, and numerical gradients in the problem. The method keyword specifies the use of the npsol sqp method to solve this constrained optimization problem. No environment keyword is specified, so the default single method approach is used. The solution for the constrained optimization problem is: intake dia = 2.122 flatness = 1.769 with f∗ = −2.461 g1∗ g2∗ g3∗ = 0.0 (active) = −0.3347 (inactive) = Dakota Version 6.7 User’s Manual generated on November 13, 2017 0.0 (active) 334 CHAPTER 20. ADDITIONAL EXAMPLES # Dakota Input File: cylhead_opt_npsol.in method npsol_sqp convergence_tolerance = 1.e-8 variables continuous_design = 2 initial_point 1.8 1.0 upper_bounds 2.164 4.0 lower_bounds 1.5 0.0 descriptors ’intake_dia’ ’flatness’ interface analysis_drivers = ’cyl_head’ fork asynchronous responses objective_functions = 1 nonlinear_inequality_constraints = 3 numerical_gradients method_source dakota interval_type central fd_step_size = 1.e-4 no_hessians Figure 20.12: Cylinder Head Example: Dakota/examples/users/cylhead opt npsol.in the Dakota input file – see which corresponds to the following optimal response quantities: warranty = 133472 cycle time = 60 wall thickness = 0.0707906 horse power = 281.579 max stress = 1500 The final report from the Dakota output is as follows: <<<<< Iterator npsol_sqp completed. <<<<< Function evaluation summary: 55 total (55 new, 0 duplicate) <<<<< Best parameters = 2.1224188322e+00 intake_dia 1.7685568331e+00 flatness <<<<< Best objective function = -2.4610312954e+00 <<<<< Best constraint values = 1.8407497748e-13 -3.3471647504e-01 Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.5. CYLINDER HEAD 0.0000000000e+00 <<<<< Best data captured at function evaluation 51 <<<<< Environment execution completed. Dakota execution time in seconds: Total CPU = 0.04 [parent = 0.031995, child = Total wall clock = 0.232134 Dakota Version 6.7 User’s Manual generated on November 13, 2017 335 0.008005] 336 20.6 CHAPTER 20. ADDITIONAL EXAMPLES Container For this example, suppose that a high-volume manufacturer of light weight steel containers wants to minimize the amount of raw sheet material that must be used to manufacture a 1.1 quart cylindrical-shaped can, including waste material. Material for the container walls and end caps is stamped from stock sheet material of constant thickness. The seal between the end caps and container wall is manufactured by a press forming operation on the end caps. The end caps can then be attached to the container wall forming a seal through a crimping operation. Figure 20.13: Container wall-to-end-cap seal For preliminary design purposes, the extra material that would normally go into the container end cap seals is approximated by increasing the cut dimensions of the end cap diameters by 12% and the height of the container wall by 5%, and waste associated with stamping the end caps in a specialized pattern from sheet stock is estimated as 15% of the cap area. The equation for the area of the container materials including waste is end cap end cap waste seal A=2× material × material factor factor container nominal nominal wall seal container × end cap + material × wall area area factor or A = 2(1.15)(1.12)π D2 + (1.05)πDH 4 (20.13) where D and H are the diameter and height of the finished product in units of inches, respectively. The volume of the finished product is specified to be V =π D2 H = (1.1qt)(57.75in3 /qt) 4 (20.14) The equation for area is the objective function for this problem; it is to be minimized. The equation for volume is an equality constraint; it must be satisfied at the conclusion of the optimization problem. Any combination of D and H that satisfies the volume constraint is a feasible solution (although not necessarily the optimal solution) to the area minimization problem, and any combination that does not satisfy the volume constraint is an infeasible solution. The area that is a minimum subject to the volume constraint is the optimal area, and the corresponding values for the parameters D and H are the optimal parameter values. It is important that the equations supplied to a numerical optimization code be limited to generating only physically realizable values, since an optimizer will not have the capability to differentiate between meaningful and Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.6. CONTAINER 337 Figure 20.14: A graphical representation of the container optimization problem. nonphysical parameter values. It is often up to the engineer to supply these limits, usually in the form of parameter bound constraints. For example, by observing the equations for the area objective function and the volume constraint, it can be seen that by allowing the diameter, D, to become negative, it is algebraically possible to generate relatively small values for the area that also satisfy the volume constraint. Negative values for D are of course physically meaningless. Therefore, to ensure that the numerically-solved optimization problem remains meaningful, a bound constraint of −D ≤ 0 must be included in the optimization problem statement. A positive value for H is implied since the volume constraint could never be satisfied if H were negative. However, a bound constraint of −H ≤ 0 can be added to the optimization problem if desired. The optimization problem can then be stated in a standardized form as minimize subject to 2(1.15)(1.12)π D2 + (1.05)2 πDH 4 D2 H = (1.1qt)(57.75in3 /qt) 4 −D ≤ 0, − H ≤ 0 π (20.15) A graphical view of the container optimization test problem appears in Figure 20.14. The 3-D surface defines the area, A, as a function of diameter and height. The curved line that extends across the surface defines the areas that satisfy the volume equality constraint, V . Graphically, the container optimization problem can be viewed as one of finding the point along the constraint line with the smallest 3-D surface height in Figure 20.14. This point corresponds to the optimal values for diameter and height of the final product. 20.6.1 Constrained Gradient Based Optimization The input file for this example is named container opt npsol.in. The solution to this example problem is (H, D) = (4.99, 4.03), with a minimum area of 98.43 in2 . The final report from the Dakota output is as follows: <<<<< Iterator npsol_sqp completed. Dakota Version 6.7 User’s Manual generated on November 13, 2017 338 CHAPTER 20. ADDITIONAL EXAMPLES <<<<< Function evaluation summary: 40 total (40 new, 0 duplicate) <<<<< Best parameters = 4.9873894231e+00 H 4.0270846274e+00 D <<<<< Best objective function = 9.8432498116e+01 <<<<< Best constraint values = -9.6301439045e-12 <<<<< Best data captured at function evaluation 36 <<<<< Environment execution completed. Dakota execution time in seconds: Total CPU = 0.18 [parent = 0.18, child = Total wall clock = 0.809126 20.7 0] Cantilever This test problem is adapted from the reliability-based design optimization literature [129], [150] and involves a simple uniform cantilever beam as shown in Figure 20.15. Figure 20.15: Cantilever beam test problem. The design problem is to minimize the weight (or, equivalently, the cross-sectional area) of the beam subject to a displacement constraint and a stress constraint. Random variables in the problem include the yield stress R of the beam material, the Young’s modulus E of the material, and the horizontal and vertical loads, X and Y , which are modeled with normal distributions using N (40000, 2000), N (2.9E7, 1.45E6), N (500, 100), and N (1000, 100), respectively. Problem constants include L = 100in and D0 = 2.2535in. The constraints have the following analytic form: stress = displacement = 600 600 Y + 2 X≤R 2 wt s w t 2 2 4L3 Y X + ≤ D0 Ewt t2 w2 (20.16) or when scaled: gS = gD = stress −1≤0 R displacement −1≤0 D0 (20.17) (20.18) Deterministic Formulation If the random variables E, R, X, and Y are fixed at their means, the resulting deterministic design problem can Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.7. CANTILEVER 339 be formulated as minimize f = wt subject to gS ≤ 0 (20.19) gD ≤ 0 1.0 ≤ w ≤ 4.0 1.0 ≤ t ≤ 4.0 Stochastic Formulation If the normal distributions for the random variables E, R, X, and Y are included, a stochastic design problem can be formulated as minimize subject to f = wt βD ≥ 3 (20.20) βS ≥ 3 1.0 ≤ w ≤ 4.0 1.0 ≤ t ≤ 4.0 where a 3-sigma reliability level (probability of failure = 0.00135 if responses are normally-distributed) is being sought on the scaled constraints. 20.7.1 Constrained Gradient Based Optimization The test problem is solved using cantilever opt npsol.in: The deterministic solution is (w, t) = (2.35, 3.33) with an objective function of 7.82. The final report from the Dakota output is as follows: <<<<< Iterator npsol_sqp completed. <<<<< Function evaluation summary: 33 total (33 new, 0 duplicate) <<<<< Best parameters = 2.3520341271e+00 beam_width 3.3262784077e+00 beam_thickness 4.0000000000e+04 R 2.9000000000e+07 E 5.0000000000e+02 X 1.0000000000e+03 Y <<<<< Best objective function = 7.8235203313e+00 <<<<< Best constraint values = -1.6009000260e-02 -3.7083558446e-11 <<<<< Best data captured at function evaluation 31 <<<<< Environment execution completed. Dakota execution time in seconds: Total CPU = 0.03 [parent = 0.027995, child = 0.002005] Total wall clock = 0.281375 Dakota Version 6.7 User’s Manual generated on November 13, 2017 340 CHAPTER 20. ADDITIONAL EXAMPLES # Dakota Input File: cantilever_opt_npsol.in method npsol_sqp convergence_tolerance = 1.e-8 variables continuous_design = 2 initial_point 4.0 4.0 upper_bounds 10.0 10.0 lower_bounds 1.0 1.0 descriptors ’w’ ’t’ continuous_state = 4 initial_state 40000. 29.E+6 500. 1000. descriptors ’R’ ’E’ ’X’ ’Y’ interface analysis_drivers = ’cantilever’ fork asynchronous evaluation_concurrency = 2 responses objective_functions = 1 nonlinear_inequality_constraints = 2 numerical_gradients method_source dakota interval_type forward fd_step_size = 1.e-4 no_hessians Figure 20.16: Cantilever Example: the Dakota input file – see Dakota/examples/users/cantilever opt npsol.in 20.7.2 Optimization Under Uncertainty Optimization under uncertainty solutions to the stochastic problem are described in [38, 35, 36], for which the solution is (w, t) = (2.45, 3.88) with an objective function of 9.52. This demonstrates that a more conservative design is needed to satisfy the probabilistic constraints. 20.8 Multiobjective Test Problems Multiobjective optimization means that there are two or more objective functions that you wish to optimize simultaneously. Often these are conflicting objectives, such as cost and performance. The answer to a multi-objective problem is usually not a single point. Rather, it is a set of points called the Pareto front. Each point on the Pareto front satisfies the Pareto optimality criterion, i.e., locally there exists no other feasible vector that would improve some objective without causing a simultaneous worsening in at least one other objective. Thus a feasible point X 0 from which small moves improve one or more objectives without worsening any others is not Pareto optimal: it is said to be “dominated” and the points along the Pareto front are said to be “non-dominated”. Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.8. MULTIOBJECTIVE TEST PROBLEMS 341 Often multi-objective problems are addressed by simply assigning weights to the individual objectives, summing the weighted objectives, and turning the problem into a single-objective one which can be solved with a variety of optimization techniques. While this approach provides a useful “first cut” analysis (and is supported within Dakota—see Section 6.3.1), this approach has many limitations. The major limitation is that a local solver with a weighted sum objective will only find one point on the Pareto front; if one wants to understand the effects of changing weights, this method can be computationally expensive. Since each optimization of a single weighted objective will find only one point on the Pareto front, many optimizations must be performed to get a good parametric understanding of the influence of the weights and to achieve a good sampling of the entire Pareto frontier. There are three examples that are taken from a multiobjective evolutionary algorithm (MOEA) test suite described by Van Veldhuizen et. al. in [17]. These three examples illustrate the different forms that the Pareto set may take. For each problem, we describe the Dakota input and show a graph of the Pareto front. These problems are all solved with the moga method. The first example is discussed in Section 6.3.1. The next two are discussed below. Section 6.3.1 provide more information on multiobjective optimization. 20.8.1 Multiobjective Test Problem 2 The second test problem is a case where both Ptrue and PFtrue are disconnected. PFtrue has four separate Pareto curves. The problem is to simultaneously optimize f1 and f2 given two input variables, x1 and x2 , where the inputs are bounded by 0 ≤ xi ≤ 1, and: f1 (x) = x1 " f2 (x) = (1 + 10x2 ) × 1 − x1 1 + 10x2 2 x1 − sin(8πx1 ) 1 + 10x2 # The input file for this example is shown in Figure 20.17, which references the mogatest2 executable (compiled from Dakota Source/test/mogatest2.cpp) as the simulator. The Pareto front is shown in Figure 20.18. Note the discontinuous nature of the front in this example. 20.8.2 Multiobjective Test Problem 3 The third test problem is a case where Ptrue is disconnected but PFtrue is connected. This problem also has two nonlinear constraints. The problem is to simultaneously optimize f1 and f2 given two input variables, x1 and x2 , where the inputs are bounded by −20 ≤ xi ≤ 20, and: f1 (x) = (x1 − 2)2 + (x2 − 1)2 + 2 f2 (x) = 9x1 − (x2 − 1)2 0 ≤ x21 + x22 − 225 0 ≤ x1 − 3x2 + 10 The constraints are: The input file for this example is shown in Figure 20.19. It differs from Figure 20.17 in the variables and responses specifications, in the use of the mogatest3 executable (compiled from Dakota Source/test/mogatest3.cpp) as the simulator, and in the max function evaluations and mutation type MOGA controls. The Pareto set is shown in Figure 20.20. Note the discontinuous nature of the Pareto set (in the design space) in this example. The Pareto front is shown in Figure 20.21. Dakota Version 6.7 User’s Manual generated on November 13, 2017 342 CHAPTER 20. ADDITIONAL EXAMPLES # Dakota Input File: mogatest2.in environment graphics tabular_data tabular_data_file = ’mogatest2.dat’ method moga seed = 10983 max_function_evaluations = 3000 initialization_type unique_random crossover_type multi_point_parameterized_binary = 2 crossover_rate = 0.8 mutation_type replace_uniform mutation_rate = 0.1 fitness_type domination_count replacement_type below_limit = 6 shrinkage_fraction = 0.9 convergence_type metric_tracker percent_change = 0.05 num_generations = 10 final_solutions = 3 output silent variables continuous_design = 2 initial_point 0.5 upper_bounds 1 lower_bounds 0 descriptors ’x1’ 0.5 1 0 ’x2’ interface analysis_drivers = ’mogatest2’ direct responses objective_functions = 2 no_gradients no_hessians Figure 20.17: Dakota input file specifying Dakota/examples/users/mogatest2.in the use of MOGA on mogatest2 – see Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.8. MULTIOBJECTIVE TEST PROBLEMS Figure 20.18: Pareto Front showing Tradeoffs between Function F1 and Function F2 for mogatest2 Dakota Version 6.7 User’s Manual generated on November 13, 2017 343 344 CHAPTER 20. ADDITIONAL EXAMPLES # Dakota Input File: mogatest3.in environment graphics tabular_data tabular_data_file = ’mogatest3.dat’ method moga seed = 10983 max_function_evaluations = 2000 initialization_type unique_random crossover_type multi_point_parameterized_binary = 2 crossover_rate = 0.8 mutation_type offset_normal mutation_scale = 0.1 fitness_type domination_count replacement_type below_limit = 6 shrinkage_fraction = 0.9 convergence_type metric_tracker percent_change = 0.05 num_generations = 10 final_solutions = 3 output silent variables continuous_design = 2 descriptors ’x1’ initial_point 0 upper_bounds 20 lower_bounds -20 ’x2’ 0 20 -20 interface analysis_drivers = ’mogatest3’ direct responses objective_functions = 2 nonlinear_inequality_constraints = 2 upper_bounds = 0.0 0.0 no_gradients no_hessians Figure 20.19: Dakota input file specifying Dakota/examples/users/mogatest3.in the use of MOGA on mogatest3 – see Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.8. MULTIOBJECTIVE TEST PROBLEMS Figure 20.20: Pareto Set of Design Variables corresponding to the Pareto front for mogatest3 Dakota Version 6.7 User’s Manual generated on November 13, 2017 345 346 CHAPTER 20. ADDITIONAL EXAMPLES Figure 20.21: Pareto Front showing Tradeoffs between Function F1 and Function F2 for mogatest3 Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.9. MORRIS 20.9 347 Morris Morris [103] includes a screening design test problem with a single-output analytical test function. The output depends on 20 inputs with first- through fourth-order interaction terms, some having large fixed coefficients and others small random coefficients. Thus the function values generated depend on the random number generator employed in the function evaluator. The computational model is: y = β0 + 20 X βi wi + i=1 + 20 X 20 X βi,j wi wj + i >>>> Executing environment. >>>>> Running psuade_moat iterator. PSUADE DACE method = psuade_moat Samples = 84 Seed (user-specified) = 500 Partitions = 3 (Levels = 4) Figure 20.22: Dakota initialization output for the PSUADE MOAT method on the Morris test problem showing the study parameters. >>>>>> PSUADE MOAT output for function 0: ************************************************************* *********************** MOAT Analysis *********************** ------------------------------------------------------------Input 1 (mod. mean & std) = 9.5329e+01 9.0823e+01 Input 2 (mod. mean & std) = 6.7297e+01 9.5242e+01 Input 3 (mod. mean & std) = 1.0648e+02 1.5479e+02 Input 4 (mod. mean & std) = 6.6231e+01 7.5895e+01 Input 5 (mod. mean & std) = 9.5717e+01 1.2733e+02 Input 6 (mod. mean & std) = 8.0394e+01 9.9959e+01 Input 7 (mod. mean & std) = 3.2722e+01 2.7947e+01 Input 8 (mod. mean & std) = 4.2013e+01 7.6090e+00 Input 9 (mod. mean & std) = 4.1965e+01 7.8535e+00 Input 10 (mod. mean & std) = 3.6809e+01 3.6151e+00 Input 11 (mod. mean & std) = 8.2655e+00 1.0311e+01 Input 12 (mod. mean & std) = 4.9299e+00 7.0591e+00 Input 13 (mod. mean & std) = 3.5455e+00 4.4025e+00 Input 14 (mod. mean & std) = 3.4151e+00 2.4905e+00 Input 15 (mod. mean & std) = 2.5143e+00 5.5168e-01 Input 16 (mod. mean & std) = 9.0344e+00 1.0115e+01 Input 17 (mod. mean & std) = 6.4357e+00 8.3820e+00 Input 18 (mod. mean & std) = 9.1886e+00 2.5373e+00 Input 19 (mod. mean & std) = 2.4105e+00 3.1102e+00 Input 20 (mod. mean & std) = 5.8234e+00 7.2403e+00 <<<<< Function evaluation summary: 84 total (84 new, 0 duplicate) Figure 20.23: Dakota analysis output for the PSUADE MOAT method on the Morris problem showing the modified mean and standard deviation of the elementary effect corresponding to each input factor. Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.9. MORRIS 349 Figure 20.24: Standard deviation of elementary effects plotted against modified mean for Morris for each of 20 inputs. Red circles 1–7 correspond to inputs having interactions or nonlinear effects, blue squares 8–10 indicate those with mainly linear effects, and black Xs denote insignificant inputs. Dakota Version 6.7 User’s Manual generated on November 13, 2017 350 20.10 CHAPTER 20. ADDITIONAL EXAMPLES Test Problems for Reliability Analyses This section includes several test problems and examples related to reliability analyses. These are NOT included in the Dakota/examples/users directory, but are in the Dakota/test directory. 20.10.1 Log Ratio This test problem, mentioned previously in Section 5.3.3, has a limit state function defined by the ratio of two lognormally-distributed random variables. x1 g(x) = (20.21) x2 The distributions for both x1 and x2 are Lognormal(1, 0.5) with a correlation coefficient between the two variables of 0.3. Reliability Analyses First-order and second-order reliability analysis (FORM and SORM) are performed in the logratio uq reliability.in in the Dakota/examples/users directory and dakota logratio taylor2.in in directory Dakota/test. For the reliability index approach (RIA), 24 response levels (.4, .5, .55, .6, .65, .7, .75, .8, .85, .9, 1, 1.05, 1.15, 1.2, 1.25, 1.3, 1.35, 1.4, 1.5, 1.55, 1.6, 1.65, 1.7, and 1.75) are mapped into the corresponding cumulative probability levels. For performance measure approach (PMA), these 24 probability levels (the fully converged results from RIA FORM) are mapped back into the original response levels. Figure 20.25 overlays the computed CDF values for a number of first-order reliability method variants as well as a Latin Hypercube reference solution of 106 samples. (a) RIA methods (b) PMA methods Figure 20.25: Lognormal ratio cumulative distribution function, RIA/PMA methods. Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.10. TEST PROBLEMS FOR RELIABILITY ANALYSES 20.10.2 351 Steel Section This test problem is used extensively in [73]. It involves a W16x31 steel block of A36 steel that must carry an applied deterministic bending moment of 1140 kip-in. For Dakota, it has been used as a code verification test for second-order integrations in reliability methods. The limit state function is defined as: g(x) = Fy Z − 1140 (20.22) where Fy is Lognormal(38., 3.8), Z is Normal(54., 2.7), and the variables are uncorrelated. The Dakota/test/dakota steel section.in input file computes a first-order CDF probability of p(g ≤ 0.) = 1.297e-07 and a second-order CDF probability of p(g ≤ 0.) = 1.375e-07. This second-order result differs from that reported in [73], since Dakota uses the Nataf nonlinear transformation to u-space (see MPP Search Methods block in Reliability Methods chapter of Dakota Theory Manual [4]) and [73] uses a linearized transformation. 20.10.3 Portal Frame This test problem is taken from [138, 80]. It involves a plastic collapse mechanism of a simple portal frame. It also has been used as a verification test for second-order integrations in reliability methods. The limit state function is defined as: g(x) = x1 + 2x2 + 2x3 + x4 − 5x5 − 5x6 (20.23) where x1 − x4 are Lognormal(120., 12.), x5 is Lognormal(50., 15.), x6 is Lognormal(40., 12.), and the variables are uncorrelated. While the limit state is linear in x-space, the nonlinear transformation of lognormals to u-space induces curvature. The Dakota/test/dakota portal frame.in input file computes a first-order CDF probability of p(g ≤ 0.) = 9.433e-03 and a second-order CDF probability of p(g ≤ 0.) = 1.201e-02. These results agree with the published results from the literature. 20.10.4 Short Column This test problem involves the plastic analysis and design of a short column with rectangular cross section (width b and depth h) having uncertain material properties (yield stress Y ) and subject to uncertain loads (bending moment M and axial force P ) [91]. The limit state function is defined as: g(x) = 1 − 4M P2 − 2 2 2 2 bh Y b h Y (20.24) The distributions for P , M , and Y are Normal(500, 100), Normal(2000, 400), and Lognormal(5, 0.5), respectively, with a correlation coefficient of 0.5 between P and M (uncorrelated otherwise). The nominal values for b and h are 5 and 15, respectively. Reliability Analyses First-order and second-order reliability analysis are performed in the dakota short column.in and dakota short column taylor2.in input files in Dakota/test. For RIA, 43 response levels (-9.0, 8.75, -8.5, -8.0, -7.75, -7.5, -7.25, -7.0, -6.5, -6.0, -5.5, -5.0, -4.5, -4.0, -3.5, -3.0, -2.5, -2.0, -1.9, -1.8, -1.7, -1.6, -1.5, -1.4, -1.3, -1.2, -1.1, -1.0, -0.9, -0.8, -0.7, -0.6, -0.5, -0.4, -0.3, -0.2, -0.1, 0.0, 0.05, 0.1, 0.15, 0.2, 0.25) Dakota Version 6.7 User’s Manual generated on November 13, 2017 352 CHAPTER 20. ADDITIONAL EXAMPLES (a) RIA methods (b) PMA methods Figure 20.26: Short column cumulative distribution function, RIA/PMA methods. are mapped into the corresponding cumulative probability levels. For PMA, these 43 probability levels (the fully converged results from RIA FORM) are mapped back into the original response levels. Figure 20.26 overlays the computed CDF values for several first-order reliability method variants as well as a Latin Hypercube reference solution of 106 samples. Reliability-Based Design Optimization The short column test problem is also amenable to Reliability-Based Design Optimization (RBDO). An objective function of cross-sectional area and a target reliability index of 2.5 (cumulative failure probability p(g ≤ 0) ≤ 0.00621) are used in the design problem: min s.t. bh β ≥ 2.5 5.0 ≤ b ≤ 15.0 15.0 ≤ h ≤ 25.0 (20.25) As is evident from the UQ results shown in Figure 20.26, the initial design of (b, h) = (5, 15) is infeasible and the optimization must add material to obtain the target reliability at the optimal design (b, h) = (8.68, 25.0). Simple bi-level, fully analytic bi-level, and sequential RBDO methods are explored in inputs files dakota rbdo short column.in, dakota rbdo short column analytic.in, and dakota rbdo short column trsb.in, with results as described in [35, 36]. These files are located in Dakota/test. 20.10.5 Steel Column This test problem involves the trade-off between cost and reliability for a steel column [91]. The cost is defined as Cost = bd + 5h (20.26) Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.11. TEST PROBLEMS FOR FORWARD UNCERTAINTY QUANTIFICATION 353 where b, d, and h are the means of the flange breadth, flange thickness, and profile height, respectively. Nine uncorrelated random variables are used in the problem to define the yield stress Fs (lognormal with µ/σ = 400/35 MPa), dead weight load P1 (normal with µ/σ = 500000/50000 N), variable load P2 (gumbel with µ/σ = 600000/90000 N), variable load P3 (gumbel with µ/σ = 600000/90000 N), flange breadth B (lognormal with µ/σ = b/3 mm), flange thickness D (lognormal with µ/σ = d/2 mm), profile height H (lognormal with µ/σ = h/5 mm), initial deflection F0 (normal with µ/σ = 30/10 mm), and Young’s modulus E (Weibull with µ/σ = 21000/4200 MPa). The limit state has the following analytic form: F0 Eb 1 + (20.27) g = Fs − P 2BD BDH Eb − P where P = P1 + P2 + P3 π 2 EBDH 2 = 2L2 Eb (20.28) (20.29) and the column length L is 7500 mm. This design problem (dakota rbdo steel column.in in Dakota/test) demonstrates design variable insertion into random variable distribution parameters through the design of the mean flange breadth, flange thickness, and profile height. The RBDO formulation maximizes the reliability subject to a cost constraint: maximize β subjectto Cost ≤ 4000. 200.0 ≤ b ≤ 400.0 (20.30) 10.0 ≤ d ≤ 30.0 100.0 ≤ h ≤ 500.0 which has the solution (b, d, h) = (200.0, 17.50, 100.0) with a maximal reliability of 3.132. 20.11 Test Problems for Forward Uncertainty Quantification This section includes several test problems and examples related to forward uncertainty quantification. These are NOT included in the Dakota/examples/users directory, but are in the Dakota/test directory. 20.11.1 Genz functions The Genz functions have traditionally been used to test quadrature methods, however more recently they hav also been used to test forward UQ methdos. Here we consider the oscilatory and corner-peak test functions, respectively given by ! d X fOS (ξ) = cos − ci ξi , ξ ∈ [0, 1]d i=1 fCP (ξ) = 1+ d X i=1 Dakota Version 6.7 User’s Manual generated on November 13, 2017 !−(d+1) ci ξ i , ξ ∈ [0, 1]d 354 CHAPTER 20. ADDITIONAL EXAMPLES The coefficients ck can be used to control the effective dimensionality and the variability of these functions. In dakota we support three choices of = (c1 , . . . , cd )T , specifically k − 12 k log(10−8 ) 1 (3) (1) (2) ck = , ck = 2 and ck = exp , k = 1, . . . , d d k d Pd Pd normalizing such that for the oscillatory function k=1 ck = 4.5 and for the corner peak function k=1 ck = 0.25. The different decay rates of the coefficients represent increasing levels of anisotropy and decreasing effective dimensionality. Anisotropy refers to the dependence of the function variability. In Dakota the genz function can be set by specifying analysis driver=’genz’. The type of coefficients decay can be set by sepcifying os1, cp1 which uses c(1) , os2, cp2 which uses c(2) , and os3, cp3 which uses c(3) , where os is to be used with the oscillatory function and cp for the corner peak function 20.11.2 Elliptic Partial differential equation with uncertain coefficients Consider the following problem with d ≥ 1 random dimensions: du d a(x, ξ) (x, ξ) = 10, (x, ξ) ∈ (0, 1) × Iξ − dx dx (20.31) subject to the physical boundary conditions u(0, ξ) = 0, u(1, ξ) = 0. (20.32) We are interested in quantifying the uncertainty in the solution u at a number of locations x ∈ [0, 1] which can be specified by the user. This function can be run using dakota by specifying analysis driver=’steady state diffusion 1d’. We represent the random diffusivity field a using two types of expansions. The first option is to use represent the logarithm of the diffusivity field as Karhunen Loéve expansion log(a(x, ξ)) = ā + σa d p X λk φk (x)ξk (20.33) k=1 where {λk }dk=1 and {φk (x)}dk=1 are, respectively, the eigenvalues and eigenfunctions of the covariance kernel (x1 − x2 )p Ca (x1 , x2 ) = exp − . (20.34) lc for some power p = 1, 2. The variability of the diffusivity field (20.33) is controlled by σa and the correlation length lc which determines the decay of the eigenvalues λk . Dakota sets σa = 1. but allows the number of random variables d, the order of the kernel p and the correlation length lc to vary. The random variables can be bounded or unbounded. The second expansion option is to represent the diffusivity by a(x, ξ) = 1 + σ d X k=1 1 cos(2πkx)ξ k k2 π2 (20.35) where ξ k ∈ [−1, 1], k = 1, . . . , d are bounded random variables. The form of (20.35) is similar to that obtained from a Karhunen-Loève expansion. If the variables are i.i.d. uniform in [-1,1] the the diffusivity satisfies satisfies the auxiliary properties σ σ (20.36) E[a(x, ξ)] = 1 and 1 − < a(x, ξ) < 1 + . 6 6 This is the same test case used in [151]. Dakota Version 6.7 User’s Manual generated on November 13, 2017 20.11. TEST PROBLEMS FOR FORWARD UNCERTAINTY QUANTIFICATION 20.11.3 355 Damped Oscillator Consider a damped linear oscillator subject to external forcing with six unknown parameters ξ = (γ, k, f, ω, x0 , x1 ) dx d2 x (t, ξ) + γ + kx = f cos(ωt), 2 dt dt (20.37) subject to the initial conditions x(0) = x0 , ẋ(0) = x1 , (20.38) Here we assume the damping coefficient γ, spring constant k, forcing amplitude f and frequency ω, and the initial conditions x0 and x1 are all uncertain. This test function can be specified with analysis driver=’damped oscillator’. This function only works with the uncertain variables defined over certain ranges. These ranges are γk ∈ [0.08, 0.12], k ∈ [0.03, 0.04], f ∈ [0.08, 0.12], ω ∈ [0.8, 1.2], x0 ∈ [0.45, 0.55], x1 ∈ [−0.05, 0.05]. Do not use this function with unbounded variables, however any bounded variables such as Beta and trucated Normal variables that satisfy the aforementioned variable bounds are fine. 20.11.4 Non-linear coupled system of ODEs Consider the non-linear system of ordinary differential equations governing a competitive LotkaVolterra model of the population dynamics of species competing for some common resource. The model is given by ( P3 dui = r u 1 − α u , t ∈ (0, 10], i i ij j j=1 dt , (20.39) ui (0) = ui,0 for i = 1, 2, 3. The initial condition, ui,0 , and the self-interacting terms, αii , are given, but the remaining interaction parameters, αij with i 6= j as well as the re-productivity parameters, ri , are unknown. We approximate the solution to (20.39) in time using a Backward Euler method. This test function can be specified with analysis driver=’predator prey’. We assume that these parameters are bounded. We have tested this model with all 9 parameters ξi ∈ [0.3, 0.7]. However larger bounds are probably valid. When using the aforemetioned ranges and 1000 time steps with ∆t = 0.01 the average (over the random parameters) deterministic error is approximately 1.00 × 10−4 . Dakota returns the population of each species ui (T ) at the final time T . The final time and the number of timesteps can be changed. 20.11.5 Experimental Design This example tests the Bayesian experimental design algorithm described in Section 5.8.9, in which a low-fidelity model is calibrated to optimally-selected data points of a high-fidelity model. The analysis driver for the the low and high-fidelity models implement the steady state heat example given in [95]. The high-fidelity model is the analytic solution to the example described therein, T (x) = c1 exp(−γx) + c2 exp(γx) + Tamb , Dakota Version 6.7 User’s Manual generated on November 13, 2017 (20.40) 356 CHAPTER 20. ADDITIONAL EXAMPLES where c1 = c2 = γ = Φ exp(γL)(h + Kγ) − , Kγ exp(−γL)(h − Kγ) + exp(γL)(h + Kγ) Φ + c1 , Kγ r 2(a + b)h . abK The ambient room temperature Tamb , thermal conductivity coefficient K, convective heat transfer coefficient h, source heat flux Φ, and system dimenions a, b, and L are all set to constant values. The experimental design variable x ∈ [10, 70], is called a configuration variable in Dakota and is the only variable that also appears in the low-fidelity model, y = Ax2 + Bx + C. (20.41) The goal of this example is to calibrate the low-fidelity model parameters A, B, and C with the Bayesian experimental design algorithm. Refer to Section 5.8.9 for further details regarding the Dakota implementation of this example. 20.11.6 Bayes linear This is a simple model that is only available by using the direct interface with the analysis driver=’bayes linear’. The model is discussed extensively in [5] both on pages 93–102 and in Appendix A. The model is simply the sum of the d input parameters: d X y= xi (20.42) i=1 where the input varaibles xi can be any uncertain variable type but are typically considered as uniform or normal uncertain inputs. Dakota Version 6.7 User’s Manual generated on November 13, 2017 Bibliography [1] M.A. Abramson, C. Audet, G. Couture, J.E. Dennis, Jr., S. Le Digabel, and C. Tribes. The NOMAD project. Software available at http://www.gerad.ca/nomad. 125 [2] B. M. Adams, W. J. Bohnhoff, K. R. Dalbey, J. P. Eddy, M. S. Ebeida, M. S. Eldred, J. R. Frye, G Gerarci, R. W. Hooper, P. D. Hough, K. T. Hu, J. D. Jakeman, M. Khalil, K. A. Maupin, J. A. Monschke, E. M. Ridgway, A. Rushdi, J. A. 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