UQLab User Manual – Structural Reliability (Rare Events Estimation)
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UQLAB USER MANUAL
STRUCTURAL RELIABILITY
(RARE EVENTS ESTIMATION)
S. Marelli, R. Sch¨
obi, B. Sudret
CHAIR OF RISK, SAFETY AND UNCERTAINTY QUANTIFICATION
STEFANO-FRANSCINI-PLATZ 5
CH-8093 Z¨
URICH
Risk, Safety &
Uncertainty Quantification
How to cite UQLAB
S. Marelli, and B. Sudret, UQLab: A framework for uncertainty quantification in Matlab, Proc. 2nd Int. Conf. on
Vulnerability, Risk Analysis and Management (ICVRAM2014), Liverpool, United Kingdom, 2014, 2554-2563.
How to cite this manual
S. Marelli, R. Sch¨
obi and B. Sudret, UQLab user manual – Structural Reliability, Report UQLab-V1.1-107, Chair of
Risk, Safety & Uncertainty Quantification, ETH Zurich, 2018.
BIBT
E
X entry
@TechReport{UQdoc 11 107,
author = {Marelli, S. and Sch¨
obi, R. and Sudret, B.},
title = {{UQLab user manual -- Reliability analysis}},
institution = {Chair of Risk, Safety & Uncertainty Quantification, ETH Zurich},
year = {2018},
note = {Report # UQLab-V1.1-107},
}
List of contributors:
C. Lamas: Original implementation of FORM, SORM, Monte Carlo and importance sampling methods
Document Data Sheet
Document Ref. UQLAB-V1.1-107
Title: UQLAB User Manual – Structural Reliability
(Rare Events Estimation)
Authors: S. Marelli, R. Sch¨
obi, B. Sudret
Chair of Risk. Safety and Uncertainty Quantification, ETH Zurich,
Switzerland
Date: 01/07/2018
Doc. Version Date Comments
V1.1 01/07/2018 UQLAB V1.1 release
• Updated usage and reference list for AK-MCS
V1.0 01/05/2017 UQLAB V1.0 release
• added PC-Kriging to AK-MCS
• bugfixes and display improvements
V0.92 15/02/2016 Initial release
Abstract
Structural reliability methods aim at the assessment of the probability of failure of complex
systems due to uncertainties associated to their design, manifacturing, environmental and
operating conditions. The name structural reliability comes from the emergence of such
computational methods back in the mid 70’s to evaluate the reliability of civil engineering
structures. As these probabilities are usually small (e.g. 10−2−10−8), this type of problems
is also known as rare events estimation in the recent statistics literature.
The structural reliability module of UQLAB offers a comprehensive set of techniques for the
efficient estimation of the failure probability of a wide range of systems. Classical (crude
Monte Carlo simulation, FORM/SORM, Subset Simulation) and state-of-the-art algorithms
(AK-MCS) are available and can be easily deployed in association with other UQLAB tools,
e.g. surrogate modelling or sensitivity analysis.
The structural reliability user manual is divided in three parts:
• A short introduction to the main concepts and techniques used to solve structural reli-
ability problems, with a selection of references to the relevant literature
• A detailed example-based guide, with the explanation of most of the available options
and methods
• A comprehensive reference list detailing all the available functionalities in the UQLAB
structural reliability module.
Keywords: Structural Reliability, FORM, SORM, Importance Sampling, Monte Carlo Simula-
tion, Subset Simulation, AK-MCS, UQLAB, rare event estimation
Contents
1 Theory 1
1.1 Introduction ..................................... 1
1.2 Problem statement ................................. 1
1.2.1 Limit-state function ............................. 1
1.2.2 Failure Probability ............................. 2
1.3 Strategies for the estimation of Pf........................ 3
1.3.1 First Order Reliability Method (FORM) .................. 4
1.3.2 Second Order Reliability Method (SORM) ................ 9
1.3.3 Monte Carlo Simulation .......................... 11
1.3.4 Importance Sampling ........................... 12
1.3.5 Subset Simulation ............................. 13
1.3.6 Adaptive Kriging Monte Carlo Simulation ................ 15
2 Usage 19
2.1 Reference problem: R-S .............................. 19
2.2 Problem set-up ................................... 19
2.3 Reliability analysis with different methods .................... 20
2.3.1 First Order Reliability Method (FORM) .................. 20
2.3.2 Second Order Reliability Method (SORM) ................ 23
2.3.3 Monte Carlo Sampling (MCS) ....................... 24
2.3.4 Importance Sampling ........................... 27
2.3.5 Subset Simulation ............................. 30
2.3.6 Adaptive Kriging Monte Carlo Sampling (AK-MCS) ........... 32
2.4 Advanced limit-state function options ....................... 35
2.4.1 Specify failure threshold and failure criterion .............. 35
2.4.2 Vector Output ................................ 35
2.5 Excluding parameters from the analysis ..................... 36
3 Reference List 37
3.1 Create a reliability analysis ............................ 39
3.2 Accessing the results ................................ 47
3.2.1 Monte Carlo ................................. 47
3.2.2 FORM and SORM .............................. 48
3.2.3 Importance sampling ............................ 50
3.2.4 Subset simulation .............................. 51
3.2.5 AK-MCS ................................... 52
3.3 Printing/Visualizing of the results ......................... 53
3.3.1 Printing the results: uq print ....................... 53
3.3.2 Graphically display the results: uq display ................ 54
Chapter 1
Theory
1.1 Introduction
A structural system is defined as a structure required to provide specific functionality under
well-defined safety constraints. Such constraints need to be taken into account during the
system design phase in view of the expected environmental/operating loads it will be subject
to.
In the presence of uncertainties in the physical properties of the system (e.g. due to tolerances
in the manufacturing), in the environmental loads (e.g. due to exceptional weather condi-
tions), or in the operating conditions (e.g. traffic), it can occur that the structure operates
outside of its nominal range. In such cases, the system encounters a failure.
Structural reliability analysis deals with the quantitative assessment of the probability of
occurrence of such failures (probability of failure), given a model of the uncertainty in the
structural, environmental and load parameters.
Following the formalism introduced in Sudret (2007), this chapter is intended as a brief the-
oretical introduction and literature review of the available tools in the structural reliability
module of UQLAB. Consistently with the overall design philosophy of UQLAB, all the algo-
rithms presented follow a black-box approach, i.e. they rely on the point-by-point evaluation
of a computational model, without knowledge about its inner structure.
1.2 Problem statement
1.2.1 Limit-state function
Alimit state can be defined as a state beyond which a system no longer satisfies some perfor-
mance measure (ISO Norm 2394). Regardless on the choice of the specific criterion, a state
beyond the limit state is classified as a failure of the system.
Consider a system whose state is represented by a random vector of variables X∈ DX⊂RM.
One can define two domains Ds,Df⊂ DXthat correspond to the safe and failure regions of
the state space DX, respectively. In other words, the system is failing if the current state
x∈ Dfand it is operating safely if x∈ Ds. This classification makes it possible to construct
alimit-state function g(X)(sometimes also referred to as performance function) that assumes
1
UQLAB User Manual
Figure 1: Schematic representation of the safe and failure domains Dsand Dfand the cor-
responding limit-state surface g(x)=0.
positive values in the safe domain and negative values in the failure domain:
x∈ Ds⇐⇒ g(x)>0
x∈ Df⇐⇒ g(x)≤0(1.1)
The hypersurface in Mdimensions defined by g(x)=0is known as the limit-state surface,
and it represents the boundary between safe and failure domains. A graphical representation
of Ds,Dfand the corresponding limit-state surface g(x)=0is given in Figure 1.
1.2.2 Failure Probability
If the random vector of state variables Xis described by a joint probability density function
(PDF) X∼fX(x), then one can define the failure probability Pfas:
Pf=P(g(X)≤0) .(1.2)
This is the probability that the system is in a failed state given the uncertainties of the state
parameters. The failure probability Pfis then calculated as follows:
Pf=ZDf
fX(x)dx=Z{x:g(x)≤0}
fX(x)dx.(1.3)
Note that the integration domain in Eq. (1.3) is only implicitly defined by Eq. (1.1), hence
making its direct estimation practically impossible in the general case. This limitation can be
circumvented by introducing the indicator function of the failure domain, a simple classifier
given by:
1Df(x) = (1if g(x)≤0
0if g(x)>0,x∈ DX.
In other words, 1Df(x)=1when the input parameters xcause the system to fail and
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1Df(x)=0otherwise. This function allows one to cast Eq. (1.3) as follows:
Pf=ZDX
1Df(x)fX(x)dx=E1Df(X),(1.4)
where E[·]is the expectation operator with respect to the PDF fX(x). This reduces the
calculation of Pfto the estimation of the expectation value of 1Df(X).
1.3 Strategies for the estimation of Pf
From the definition of 1Df(x)in Section 1.2.2 it is clear that determining whether a certain
state vector x∈ DXbelongs to Dsor Dfrequires the evaluation of the limit-state function
g(x). In the general case this operation can be computationally expensive, e.g. when it
entails the evaluation of a computational model on the vector x. For a detailed overview
of standard structural reliability methods and applications, see e.g. Ditlevsen and Madsen
(1996); Melchers (1999); Lemaire (2009).
In the following, three strategies are discussed for the evaluation of Pf, namely approxima-
tion, simulation and adaptive surrogate-modelling-based methods.
Approximation methods
Approximation methods are based on approximating the limit-state function locally at a
reference point (e.g. with a linear or quadratic Taylor expansion). This class of methods
can be very efficient (in that only a relatively small number of model evaluations is needed
to calculate Pf), but it tends to become unreliable in the presence of complex, non-linear
limit-state functions. Two approximation methods are currently available in UQLAB:
•FORM (First Order Reliability Method) – it is based on the combination of an iterative
gradient-based search of the so-called design point and a local linear approximation of
the limit-state function in a suitably transformed probabilistic space.
•SORM (Second Order Reliability Method) – it is a second-order refinement of the solution
of FORM. The computational costs associated to this refinement increase rapidly with
the number of input random variables M.
Simulation methods
Simulation methods are based on sampling the joint distribution of the state variables X
and using sample-based estimates of the integral in Eq. (1.4). At the cost of being compu-
tationally very expensive, they generally have a well-characterized convergence behaviour
that can be exploited to calculate confidence bounds on the resulting Pfestimates. Three
sampling-based algorithms are available in UQLAB:
•Monte Carlo simulation – it is based on the direct sample-based estimation of the expec-
tation value in Eq. (1.4). The total costs increase very rapidly with decreasing values
of the probability Pfto be computed.
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•Importance Sampling – it is based on improving the efficiency of Monte Carlo simulation
by changing the sampling density so as to favour points in the failure domain Df. The
choice of the importance sampling (a.k.a. instrumental) density generally uses FORM
results.
•Subset Simulation – it is based on iteratively solving and combining a sequence of condi-
tional reliability analyses by means of Markov Chain Monte Carlo simulation (MCMC).
Metamodel-based adaptive methods
Metamodel-based adaptive methods are based on iteratively building surrogate models
that approximate the limit-state function in the direct vicinity of the limit-state surface. The
metamodels (see e.g. UQLAB User Manual – Polynomial Chaos Expansions and UQLAB User
Manual – Kriging (Gaussian process modelling) ) are adaptively refined by adding limit-state
function evaluations to their experimental designs until a suitable convergence criterion re-
lated to the accuracy of Pfis satisfied. One algorithm is currently available in UQLAB, namely
Adaptive Kriging Monte Carlo Sampling (AK-MCS). It is based on building a Kriging (aka Gaus-
sian process regression) surrogate model from a small initial sampling of the input vector X.
The surrogate is then iteratively refined close to the currently estimated limit-state surface so
as to evaluate accurately the probability of failure.
In the following, a detailed description of each of the methods is given.
1.3.1 First Order Reliability Method (FORM)
The first order reliability method aims at the approximation of the integral in Eq. (1.3) with
a three-step approach:
• An isoprobabilistic transform of the input random vector X∼fX(x)into a standard
normal vector U∼ N(0,IM)
• A search for the most likely failure point in the standard normal space (SNS), known
as the design point U∗
• A linearization of the limit-state surface at the design point U∗and the analytical com-
putation of the resulting approximation of Pf.
1.3.1.1 Isoprobabilistic transform
The first step of the FORM method is to transform the input random vector X∼fXinto
a standard normal vector U∼ N(0,IM). The corresponding isoprobabilistic transform T
reads:
X=T−1(U)(1.5)
For details about the available isoprobabilistic transforms in UQLAB, please refer to the
UQLAB User Manual – The INPUT module (Section 1.5).
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Figure 2: Graphical representation of the isoprobabilistic transform from physical to standard
normal space in Eq. (1.5). From Sudret, 2015: Lectures on structural reliability and risk
analysis.
This transform can be used to map the integral in Eq. (1.3) from the physical space of Xto
the standard normal space of U:
Pf=ZDf
fX(x)dx=Z{u∈RM:G(u)≤0}
ϕM(u)du(1.6)
where G(u) = g(T−1(u)) is the limit-state function evaluated in the standard normal space
and ϕM(u)is the standard multivariate normal PDF given by:
ϕM(u) = (2π)−M/2exp −1
2(u2
1+··· +u2
M).(1.7)
A graphical illustration of the effects of this transform for a simple 2-dimensional case is
given in Figure 2. The advantage of casting the problem in the standard normal space is that
it is a probability space equipped with the Gaussian probability measure PG:
PG(U∈A) = ZA
ϕM(u)du=ZA
(2π)−M/2exp u2
1+··· +u2
Mdu.(1.8)
This probability measure is spherically symmetric: ϕM(u)only depends on kuk2and it de-
cays exponentially as ϕM(u)∼exp −kuk2/2. Therefore, when evaluating the integral in
Eq. (1.6) in the standard normal space, most of the contributions are given by the region clos-
est to the origin. The FORM method capitalizes on this property by linearly approximating
the limit-state surface in the region closest to the origin of the standard normal space.
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Figure 3: Graphical representation of the linearization of the limit-state function around the
design point at the basis of the FORM estimation of Pf. From Sudret, 2015: Lectures on
structural reliability and risk analysis.
1.3.1.2 Search for the design point
The design point U∗is defined as the point in the failure domain closest to the origin of the
standard normal space:
U∗=argmin
u∈RM{kuk, G(u)≤0)}.(1.9)
Due to the probability measure in Eq. (1.8), U∗can be interpreted as the most likely failure
point in the standard normal space. The norm of the design point kU∗kis an important
quantity in structural reliability known as the Hasofer-Lind reliability index (Hasofer and Lind,
1974):
βHL =kU∗k.(1.10)
An important property of the βHL index is that it is directly related to the exact failure prob-
ability Pfin the case of linear limit-state function in the standard normal space:
Pf= Φ(−βHL),(1.11)
where Φis the standard normal cumulative density function. The estimation of Pfin the
FORM algorithm is based on approximating the limit-state function as the hyperplane tan-
gent to the limit-state function at the design point. Figure 3 illustrates this approximation
graphically for the two-dimensional case.
In the general non-linear case, Eq. (1.9) may be cast as a constrained optimization problem
with Lagrangian:
L(u, λ) = 1
2kuk2+λG(u)(1.12)
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where λis the Lagrange multiplier. The related optimality conditions read:
∇uL(U∗, λ∗)=0,
∂L
∂λ (U∗, λ∗)=0,(1.13)
which can be explicitly written as:
G(U∗)=0,
U∗+λ∗∇G(U∗)=0.(1.14)
The first condition in Eq. (1.14) guarantees that the design point belongs to the limit-state
surface. The second condition guarantees that the vector U∗is colinear to the limit-state
surface normal vector at U∗,i.e. ∇G(U∗). The standard iterative approach to solve this non-
linear constrained optimization problem is given by the Rackwitz-Fiessler algorithm (Rackwitz
and Fiessler,1978).
Hasofer-Lind - Rackwitz-Fiessler algorithm (HL-RF)
The rationale behind the Rackwitz-Fiessler algorithm is to iteratively solve a linearized
problem around the current point. Normally, the algorithm is started with U0=0.
At each iteration, the limit-state function is approximated as:
G(U)≈G(Uk) + ∇G|Uk·(U−Uk)(1.15)
The two optimality conditions in Eq. (1.14) read for each iteration k:
∇G|Uk·(Uk+1 −Uk) + G(Uk)=0
Uk+1 =λ∇G|Uk,(1.16)
which after some basic algebra reduce to:
Uk+1 =∇G|Uk·Uk−G(Uk)
k∇G|Ukk2∇G|Uk.(1.17)
By introducing the unit vector:
αk=−∇G|Uk
k∇G|Ukk,(1.18)
one finally obtains:
Uk+1 =αk·Uk+G(Uk)
k∇G|Ukkαk.(1.19)
The associated estimate of the reliability index βkassociated to the k-th iteration is then:
βk=αk·Uk+G(Uk)
k∇G|Ukk.(1.20)
Perfect convergence of the algorithm is obtained when G(U∗)=0, yielding βHL =α∗·U∗.
However, in practice the algorithm is iterated until some stopping criteria are satisfied, i.e.,
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Table 1: Common stopping criteria for the FORM algorithm and associated description.
Criterion Typical value Description
|βk+1 −βk| ≤ β10−3Stability of βbetween iterations
kUk+1 −Ukk ≤ U10−3Stability of Ubetween iterations
|G(Uk+1)/G(U0)| ≤ G10−6Closeness to the limit-state surface
until one or more convergence conditions are verified. The standard stopping criteria used
in FORM are reported in Table 1.
Note: In UQLAB the gradients ∇G(Uk)in Eqs. (1.13) to (1.20) are calculated numeri-
cally in the standard normal space and not in the physical space.
Improved HL-RF algorithm (iHL-RF)
The Rackwitz-Fiessler algorithm is a particular case of a wide class of iterative algorithms
generically denoted as descent direction algorithms, of the form:
Uk+1 =Uk+λkdk,(1.21)
where λkis the step size at the k-th iteration and dkis the corresponding descent direction
given by:
dk=∇G|Uk·Uk−G(Uk)
k∇G|Ukk2∇G|Uk−Uk.(1.22)
In the original HL-RF algorithm, λk= 1 ∀k.Zhang and Der Kiureghian (1995) proposed
an “improved” version of the same algorithm that takes advantage of a more sophisticated
step-size calculation based on the assumption that G(U)is differentiable everywhere. They
introduced the merit function m(U):
m(U) = 1
2kUk+c|G(U)|,(1.23)
where c > kUk
k∇G(U)kis a real penalty parameter. This function has its global minimum in the
same location as the original Eq. (1.9), as well as the same descent direction d. In addition,
it allows one to use the Armijo rule (Zhang and Der Kiureghian,1995) to determine the best
step length λkat each iteration as:
λk= max
s{bs|m(Uk+bsdk)−m(Uk)≤ −abs∇m(Uk)·dk},(1.24)
where a, b ∈(0,1) are pre-selected parameters, and s∈N.
1.3.1.3 FORM results
Once the design point U∗is identified, it can be used to extract additional important infor-
mation. According to Eq. (1.20), after the convergence of FORM the Hasofer-Lind index βHL
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is given by:
βHL =α∗·U∗,(1.25)
with associated failure probability:
Pf,FORM = Φ−1(βHL)(1.26)
The local sensitivity indices Siare defined as the fraction of the variance of the safety margin
g(X) = G(U)due to the component of the design vector Ui. It can be demonstrated that
they are given by:
Si=∂G
∂uiU∗2
/k∇G(U∗)k2.(1.27)
From Eq. (1.18) it follows that:
Si=α2
i.(1.28)
If the input variables are independent, then each coordinate in the SNS Uicorresponds to a
single input variable in the physical space Xi. Therefore, the importance factor of each Xiis
identified with α2
i.
1.3.2 Second Order Reliability Method (SORM)
The second-order reliability method (SORM) is a second-order refinement of the FORM Pf
estimate. After the design point U∗is identified by FORM, the failure probability is approxi-
mated by a tangent hyperparaboloid defined by the second order Taylor expansion of G(U∗)
given by:
G(U)≈ ∇GT
|U∗·(U−U∗) + 1
2(U−U∗)TH(U−U∗),(1.29)
where His the Hessian matrix of the second derivatives of G(U)evaluated at U∗.
The failure probability in the SORM approximation can be written as a correction factor of
the FORM estimate that depends on the curvatures of the hyper-hyperboloid in Eq. (1.29).
To estimate the curvatures, the hyperparaboloid in Eq. (1.29) is first cast in canonical form
by rotating the coordinates system such that one of its axes is the αvector. Usually the last
coordinate is chosen arbitrarily for this purpose. A rotation matrix Qcan be built by setting
αas its last row and by using the Gram-Schmidt procedure to orthogonalize the remaining
components of the basis. Qis a square matrix such that QTQ=I. The resulting vector V
satisfies:
U=QV .(1.30)
In the new coordinates system and after some basic algebra (see e.g. Breitung (1989) and
Cai and Elishakoff (1994)), one can rewrite Eq. (1.29) as:
G(V)≈ k∇G(U∗)k(β−VM) + 1
2(V−V∗)QHQT(V−V∗)(1.31)
where βis the Hasofer-Lind reliability index calculated by FORM, VM
def
=αT(QTV)and
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Figure 4: Comparison between FORM and SORM approximations of the failure domain for
a simple 2-dimensional case. From Sudret, 2015: Lectures on structural reliability and risk
analysis.
V∗={0,··· , β}Tis the design point in the new coordinates system. By dividing Eq. (1.31)
by the gradient norm k∇G(U∗)kand introducing the matrix Adef
=QHQT/k∇G(U∗)k, one
obtains:
˜
G(V)≈β−VM+1
2(V−V∗)A(V−V∗),(1.32)
where ˜
G(V) = G(V)/k∇G(U∗)k. After neglecting second-order terms in VMand diagonal-
izating the Amatrix via eigenvalue decomposition one can rewrite Eq. (1.32) explicitly in
terms of the curvatures κiof an hyper-paraboloid with axis α:
˜
G(V)≈β−VM+1
2
M−1
X
1
κiVi.(1.33)
For small curvatures κi<1, the failure probability Pfcan be approximated by the Breitung
formula (Breitung,1989):
PB
f,SORM = Φ(−βHL)
M−1
Y
i=1
(1 + βHL κi)−1
2κi<1(1.34)
Note that for small curvatures the Breitung formula approaches the FORM linear limit. The
accuracy of Eq. (1.34) decreases for larger values of κi, sometimes even if κi<1(Cai and
Elishakoff,1994). A more accurate formula is given by the Hohenbichler formula (Hohen-
bichler et al.,1987):
PH
f,SORM = Φ(−βHL)
M−1
Y
i=1 1 + ϕ(βHL)
Φ(−βHL)κi−1
2
.(1.35)
Additional methods are available in the literature for the exact computation of the failure
probability, e.g.Tvedt (1990). They are, however, outside the scope of this manual.
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1.3.3 Monte Carlo Simulation
Monte Carlo (MC) simulation is used to directly compute the integral in Eq. (1.4) by sampling
the probabilistic input model. Given a sample of size Nof the input random vector X,
X=x(1), . . . x(N), the unbiased MCS estimator of the expectation value in Eq. (1.4) is
given by:
Pf,MC
def
=b
Pf=1
N
N
X
k=1
1Df(x(k)) = Nfail
N,(1.36)
where Nfail is the number of samples such that g(x)≤0. In other words, the Monte Carlo
estimate of the failure probability is the fraction of samples that belong to the failure domain
over the total number of samples. An advantage of Monte Carlo simulation is that it provides
an error estimate for Eq. (1.36). Indeed the indicator function 1Df(x)follows by construction
a Bernoulli distribution with mean µ1Df=Pfand variance σ2
1Df=Pf(1 −Pf). For large
enough Nit can be approximated by the normal distribution:
b
Pf∼ N bµ1Df,bσ1Df,(1.37)
where bµ1Df=b
Pfand bσ1Df=qb
Pf(1 −b
Pf). Hence, the estimator of Pfhas a normal
distribution with mean b
Pfand variance given by:
bσ2
Pf=
σ2
1Df
N=b
Pf(1 −b
Pf)
N.(1.38)
Confidence intervals on b
Pfcan therefore be given as follows (Rubinstein,1981):
b
Pf∈hb
P−
f
def
=b
Pf+bσPfΦ−1(α/2),b
P+
f
def
=b
Pf+bσPfΦ−1(1 −α/2)i,(1.39)
where Φ(x)is the standard normal CDF and α∈[0,1] is a scalar such that the calculated
bounds correspond to a confidence level of 1−α. An important measure for assessing the
convergence of a MCS estimator is given by the coefficient of variation CoV defined as:
CoV =σb
Pf
b
Pf
=v
u
u
t1−b
Pf
Nb
Pf
.(1.40)
The coefficient of variation of the MCS estimate of a failure probability therefore decreases
with √Nand increases with decreasing Pf. To give an example, to estimate a Pf= 10−3
with 10% accuracy N= 105samples are needed. The CoV is often used as a convergence
criterion to adaptively increase the MC sample size until some desired CoV is reached.
An associated generalized reliability index βM CS can be defined as:
βMCS =−Φ−1(b
Pf).(1.41)
In analogy, upper and lower confidence bounds on βMCS can be directly inferred from the
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confidence bounds on b
Pfin Eq. (1.39):
β±
MCS =−Φ−1(b
P±
f).(1.42)
The MCS method is powerful, when applicable, due to its statistically sound formulation and
global convergence. However, its main drawback is the relatively slow converge rate that
depends strongly on the probability of failure.
1.3.4 Importance Sampling
Importance sampling (IS) is an extension of the FORM and MCS methods that combines the
fast convergence of FORM with the robustness of MC. The basic idea is to recast Eq. (1.4) as:
Pf=ZDX
1Df(x)fX(x)
Ψ(x)Ψ(x)dx=EΨ1Df(X)fX(X)
Ψ(X),(1.43)
where Ψ(X)is an M−dimensional sampling distribution (also referred to as importance dis-
tribution) and EΨdenotes the expectation value with respect to the same distribution. The
estimate of Pfgiven a sample X=x(1),...,x(N)drawn from Ψis therefore given by:
Pf,IS =1
N
N
X
k=1
1Df(x(k))fX(x(k))
Ψ(x(k)).(1.44)
In the standard normal space, Eq. (1.43) can be rewritten as:
Pf=EΨ1Df(T−1(U))ϕM(U)
Ψ(U).(1.45)
When the results from a previous FORM analysis are available, a particularly efficient sam-
pling distribution in the standard normal space is given by (Melchers,1999):
Ψ(u) = ϕM(u−U∗)(1.46)
where U∗is the estimated design point. Given a sample U=u(1),...,u(N)of Ψ(u), the
estimate of Pfbecomes:
Pf,IS =1
Nexp −β2
HL/2N
X
k=1
1DfT−1(u(k))exp −u(k)·U∗(1.47)
with corresponding variance:
bσ2
Pf,IS =1
N
1
N−1
N
X
k=1 1DfT−1(u(k))ϕ(u(k))
Ψ(u(k))−Pf,IS!2
.(1.48)
The coefficient of variation and the confidence bounds P±
f,IS can be calculated analogously to
Eqs. (1.40) and (1.39), respectively, and can be used as a convergence criterion to adaptively
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improve the estimation of Pf,IS. The corresponding generalized reliability index reads:
βIS =−Φ−1(b
Pf,IS),(1.49)
with upper and lower bounds:
β±
IS =−Φ−1(b
P±
f,IS).(1.50)
Note that exact convergence of FORM is not necessary to obtain accurate results, even an
approximate sampling distribution can significantly improve the convergence rate compared
to standard MC sampling.
1.3.5 Subset Simulation
Monte Carlo simulation may require a large number of limit-state function evaluations to
converge with an acceptable level of accuracy when Pfis small (see Eq. (1.40)). Subset sim-
ulation is a technique introduced by Au and Beck (2001) that aims at offsetting this limitation
by solving a series of simpler reliability problems with intermediate failure thresholds.
Consider a sequence of failure domains D1⊃ D2⊃ ··· ⊃ Dm=Dfsuch that Df=Tm
k=1 Dk.
With the conventional definition of limit-state function in Eq. (1.1), such sequence can be
built with a series of decreasing failure thresholds t1>··· > tm= 0 and the corresponding
intermediate failure domains Dk={x:g(x)≤tk}. One can then combine the probability
mass of each intermediate failure region by means of conditional probability. By introducing
the notation P(DX) = P(x∈ DX)one can write (Au and Beck,2001):
Pf=P(Dm) = P m
\
k=1
P(Dk)!=P(D1)
m−1
Y
i=1
P(Di+1|Di).(1.51)
With an appropriate choice of the intermediate thresholds t1, . . . , tm, Eq. (1.51) can be eval-
uated as a series of structural reliability problems with relatively high probabilities of failure
that are then solved with MC simulation. In practice the intermediate probability thresholds
tiare chosen on-the-fly such that they correspond to intermediate values P(Dk)≈0.1. The
convergence of each intermediate estimation is therefore much faster than the direct search
for Pfgiven in Eq. (1.36).
1.3.5.1 Sampling
To estimate Pffrom Eq. (1.51) one thus needs to estimate the intermediate probabilities
P(D1)and conditional probabilities {P(Di+1|Di), i = 1, . . . , m −1}. Given an initial thresh-
old t1,P(D1)can be readily estimated from a sample of size NSof the input distribution
X=x(1),...,x(N)with Eq. (1.36):
P(D1)≈b
P1=1
NS
NS
X
k=1
1D1(x(k)).(1.52)
The remaining conditional probabilities can be estimated similarly, but an efficient sampling
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algorithm is needed for the underlying conditional distributions. The latter can be efficiently
accomplished by using the modified Metropolis-Hastings Markov Chain Monte Carlo sam-
pling (MCMC) introduced by Au and Beck (2001).
1.3.5.2 Intermediate failure thresholds
The efficiency of the subset-simulation method depends on the choice of the intermediate
failure thresholds tk. If the thresholds are too large the MCS convergence in each subset
would be very good, but the number of subsets needed would increase. Vice-versa, too small
intermediate thresholds would correspond to fewer subsets with inaccurate estimates of the
underying P(Dk). A strategy to deal with this problem comes by sampling each subset Dkand
determining each threshold tkas the empirical quantile that correspond to a predetermined
failure probability, typically P(Dk)≈P0= 0.1. Note that for practical reasons, P0is normally
limited to 0< P0≤0.5. For each subset, the samples falling below the calculated threshold
are used as MCS seeds for the next subset (Au and Beck,2001).
1.3.5.3 Subset simulation algorithm
The subset simulation algorithm can be summarized in the following steps:
1. Sample the original space with standard MC sampling (see UQLAB User Manual – The
INPUT module for efficient sampling strategies)
2. Calculate the empirical quantile tkin the current subset such that b
Pk≈P0
3. Using the samples below the identified quantile as the seeds of parallel MCMC chains,
sample Dk+1|Dkuntil a predetermined number of samples is available
4. Repeat Steps 2 and 3 until the identified quantile tm<0
5. Calculate the failure probability of the last subset b
Pmby setting tm= 0
6. Combine the intermediate calculated failure probabilities into the final estimate of b
Pf.
The last step of the algorithm consists simply in evaluating Eq. (1.51) with the current esti-
mates of the conditional probabilities Pi:
b
Pf,SS =
m
Y
i=1 b
Pi=Pm−1
0b
Pm.(1.53)
1.3.5.4 Error estimation
Due to the intrinsic correlation of the samples drawn from each subset resulting from the
MCMC sampling strategy, the estimation of a CoV for the Pfestimate in Eq. (1.53) is non-
trivial. Au and Beck (2001) and Papaioannou et al. (2015) derived an estimate for the CoV
of b
Pf:
CoVf≈
m
X
i=1
δ2
i,(1.54)
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where mis the number of subsets and δiis defined as:
δi=r1−Pi
NPi
(1 + γi),(1.55)
with
γi= 2
N/Ns
X
k=1 1−kNs
Nρi(k),(1.56)
where NSis the number of seeds, Piis the conditional failure probability of the i−th subset
and ρi(k)is the average k-lag auto-correlation coefficient of the Markov Chain samples in
the i−th subset. By assuming normally distributed errors, confidence bounds P±
f,SS can be
given on Pf,SS based on the calculated CoVfin analogy with Eq. (1.39). The corresponding
generalized reliability index reads:
βSS =−Φ−1(b
Pf,SS),(1.57)
with upper and lower bounds:
β±
SS =−Φ−1(b
P±
f,SS).(1.58)
1.3.6 Adaptive Kriging Monte Carlo Simulation
Adaptive Kriging Monte Carlo Simulation (AK-MCS) combines Monte Carlo simulation with
adaptively built Kriging (a.k.a. Gaussian process modelling) metamodels. In cases where
the evaluation of the limit-state function is costly, Monte Carlo simulation and its variants
may become intractable due to the large number of limit-state function evaluations they
require. In AK-MCS, a Kriging metamodel surrogates the limit-state function to reduce the
total computational costs of the Monte Carlo simulation.
Kriging metamodels (see UQLAB User Manual – Kriging (Gaussian process modelling) ) pre-
dict the value of the limit-state function most accurately in the vicinity of the experimental
design samples X=x(1),...,x(n). These samples, however, are generally not optimal to
estimate the failure probability. Thus, an adaptive experimental design algorithm is intro-
duced to increase the accuracy of the surrogate model in the vicinity the limit-state function.
This is achieved by adding carefully selected samples to the experimental design of the Krig-
ing metamodel based on the current estimate of the limit-state surface (g(x)=0).
The adaptive experimental design algorithm is summarized as follows (Echard et al.,2011;
Sch¨
obi et al.,2016):
1. Generate a small initial experimental design X=x(1),...,x(N0)and evaluate the
corresponding limit-state function responses Y=y(1), . . . , y(N0)=g(x(1)), . . . , g(x(N0))
2. Train a Kriging metamodel bgbased on the experimental design {X,Y}
3. Generate a large set of NMC candidate samples S=s(1),...,s(NM C )and predict the
corresponding metamodel responses bg(s(1)),...,bg(s(NM C ))
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4. Choose the best next sample s∗to be added to the experimental design Xbased on an
appropriate learning function
5. Check whether some convergence criterion is met. If it is, skip to Step 7, otherwise
continue with Step 6
6. Add s∗and the corresponding limit-state function response y∗=g(s∗)to the experi-
mental design of the metamodel. Return to Step 2
7. Estimate the failure probability through Monte Carlo simulation with the final limit-
state function surrogate bg(x).
1.3.6.1 Selection of the best next candidate sample
A learning function is a measure of the attractiveness of a candidate sample xwith respect to
improving the estimate of the failure probability when it is added to the experimental design
X. A variety of learning functions are available in the literature (Bichon et al.,2008;Dani
et al.,2008;Echard et al.,2011;Srinivas et al.,2012;Ginsbourger et al.,2013;Dubourg,
2011), amongst which the U-function (Echard et al.,2011). The U-function is based on the
concept of misclassification and is defined for a Gaussian process as follows:
U(x) = µbg(x)
σbg(x),(1.59)
where µbg(x)and σbg(x)are the prediction mean and standard deviation of bg. A misclassifica-
tion happens when the sign of the surrogate model and the sign of the underlying limit-state
function do not match. The corresponding probability of misclassification is then:
Pm(x) = Φ (−U(x)) ,
where Φis the CDF of a standard Gaussian variable.
The next candidate sample from the set S=s(1),...,s(NMC )is chosen as the one that
maximizes the probability of misclassification or, in other words, as the one most likely to
have been misclassified as safe/failed by the surrogate limit-state function bg(x):
s∗= arg min
s∈S U(s)≡arg max
s∈S Pm(s).(1.60)
Another popular learning function is the expected feasibility function (EFF) (Bichon et al.,
2008):
EF F (x) = µbg(x)2Φ −µbg(x)
σbg(x)−Φ−−µbg(x)
σbg(x)−Φ−µbg(x)
σbg(x)
−σbg(x)2ϕ−µbg(x)
σbg(x)−ϕ−−µbg(x)
σbg(x)−ϕ−µbg(x)
σbg(x)
+Φ−µbg(x)
σbg(x)−Φ−−µbg(x)
σbg(x),(1.61)
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where = 2σbg(x)and ϕis the PDF value of a standard normal Gaussian variable. The next
candidate sample is then chosen by:
s∗= arg max
s∈S EF F (s).(1.62)
1.3.6.2 Convergence criteria
The convergence criterion terminates the addition of samples to the experimental design of
the Kriging metamodel and thus terminates the improvement in the accuracy of the fail-
ure probability estimate. The standard convergence criterion related to the U-function is
defined as follows (Echard et al.,2011): the iterations stop when miniU(s(i))>2where
i= 1, . . . , NM C .Sch¨
obi et al. (2016) demonstrated that this criterion is very conservative
and that an alternative stopping criterion, related to the uncertainty in the estimate of the
failure probability itself, is often more efficient in the context of structural reliability. It is
given by the following condition:
b
P+
f−b
P−
f
b
P0
f≤b
Pf,(1.63)
where b
Pf= 10% and the three failure probabilities are defined as:
b
P0
f=Pµbg(x)≤0,(1.64)
b
P±
f=Pµbg(x)∓kσbg(x)≤0,(1.65)
where k= Φ−1(1 −α/2) sets the confidence level (1 −α), typically k= Φ−1(97.5%) = 1.96.
A similar convergence criterion can be defined for the reliability index:
b
β+−b
β−
b
β0≤b
β,(1.66)
where the threshold b
βf= 5% and the three reliability indices correspond to the aforemen-
tioned failure probabilities:
b
β0=−Φ−1(b
P0
f),(1.67)
b
β±=−Φ−1(b
P∓
f).(1.68)
1.3.6.3 AK-MCS with a PC-Kriging metamodel
As originally proposed by Echard et al. (2011), AK-MCS uses an ordinary Kriging model
for approximating the limit-state function. As demonstrated by Sch¨
obi et al. (2016) replac-
ing the ordinary Kriging metamodel with a Polynomial-Chaos-Kriging (PC-Kriging) one (see
also UQLAB User Manual – PC-Kriging ) can significantly improve the convergence of the
algorithm. The corresponding reliability methods is called Adaptive PC-Kriging Monte Carlo
Simulation (APCK-MCS) and is available in UQLAB as an advanced option of AK-MCS (see
Section 2.3.6.2).
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Chapter 2
Usage
In this section, a reference problem will be set up to showcase how each of the techniques in
Chapter 1can be deployed in UQLAB.
2.1 Reference problem: R-S
The benchmark of choice to showcase the methods described in Section 1.3 is a basic problem
in structural reliability, namely the R-S case. It is one of the simplest possible abstract setting
consisting of only two input state variables: a resistance Rand a stress S. The system fails
when the stress is higher than the resistance, leading to the following limit-state function:
X={R, S}g(X) = R−S;(2.1)
The two-dimensional probabilistic input model consists of independent variables distributed
according to Table 2.
Table 2: Distributions of the input parameters of the R−Smodel in Eq. (2.1).
Name Distributions Parameters Description
R Gaussian [5, 0.8] Resistance of the system
S Gaussian [2, 0.6] Stress applied to the system
An example UQLAB script that showcases how to deploy all of the algorithms available in the
structural reliability module can be found in the example file:
Examples/Reliability/uq_Examples_Reliability_01_RS.m
2.2 Problem set-up
Solving a structural reliability problem in UQLAB requires the definition of three basic com-
ponents:
• a MODEL object that describes the limit-state function
• an INPUT object that describes the probabilistic model of the random vector X
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• a reliability ANALYSIS object.
The UQLAB framework is first initialized with the following command:
uqlab
The model in Eq. (2.1) can be added as a MODEL object directly with a MATLAB vectorized
string as follows:
MOpts.mString = 'X(:,1) - X(:,2)';% R−S
MOpts.isVectorized = 1;
myModel = uq_createModel(MOpts);
For more details on the available options to create a model object in UQLAB, please refer to
the UQLAB User Manual – The MODEL module .
Correspondingly, an INPUT object with independent Gaussian variables as specified in Table 2
can be created as:
IOpts.Marginals(1).Name = 'R';% R e s i s t a n c e
IOpts.Marginals(1).Type = 'Gaussian';
IOpts.Marginals(1).Moments = [5, 0.8];
IOpts.Marginals(2).Name = 'S';% S t r e s s
IOpts.Marginals(2).Type = 'Gaussian';
IOpts.Marginals(2).Moments = [2, 0.6];
myInput = uq_createInput(IOpts);
For more details about the configuration options available for an INPUT object, please refer
to the UQLAB User Manual – The INPUT module .
2.3 Reliability analysis with different methods
This section showcases how all the methods introduced in Section 1.3, can be deployed in
UQLAB. In addition, visualization and advanced options are also described in detail. The
following methods are showcased in this section:
• FORM: Section 2.3.1
• SORM: Section 2.3.2
• Monte Carlo Simulation: Section 2.3.3
• Importance Sampling: Section 2.3.4
• Subset Simulation: Section 2.3.5
• AK-MCS: Section 2.3.6
2.3.1 First Order Reliability Method (FORM)
Running a FORM analysis on the specified UQLAB MODEL and INPUT objects does not require
any specific configuration. The following minimum syntax is required:
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FORMOpts.Type = 'Reliability';
FORMOpts.Method = 'FORM';
FORMAnalysis = uq_createAnalysis(FORMOpts);
Once the analysis is performed, a report with the FORM results can be printed on screen by:
uq_print(FORMAnalysis)
which produces the following:
-------------------------------------------
FORM
-------------------------------------------
Pf 1.3499e-03
BetaHL 3.0000
ModelEvaluations 8
-------------------------------------------
Variables R S
Ustar -2.400000 1.800000
Xstar 3.08e+00 3.08e+00
Importance 0.640000 0.360000
-------------------------------------------
The results can be visualized graphically as follows:
uq_display(FORMAnalysis)
which produces the images in Figure 5. Note that the graphical representation of the FORM
iterations (right panel of Figure 5) is only produced for the 2-dimensional case.
FORM - Convergence
1 2
number of iterations
0
0.5
1
1.5
2
2.5
3
3.5
βHL
FORM - Design point, failure plane
-5 -4 -3 -2 -1 0
u1
0
0.5
1
1.5
2
2.5
3
3.5
u2
Iterations
FORM limit state surface
Figure 5: Graphical visualization of the results of the FORM analysis in Section 2.3.1.
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Note: In the preceding example no specifications are provided. If not further specified
the FORMruns with the following defaults:
–Algorithm to find the design point: 'iHLRF';
–Starting point for the Rackwitz-Fiessler (RW) algorithm: (0, . . . , 0);
–Tolerance value for the RW algorithm on the design point: 10−4;
–Tolerance value for the RW algorithm on the limit-state function: 10−4;
–Maximum number of iterations for the RW algorithm: 100;
–Failure is defined for: limit-state g(x)≤0.
Since FORM is a gradient-based method, the gradient of the limit-state function
needs to be computed. This is done using finite differences with the following
defaults:
–Type of finite difference scheme: 'forward';
–Value of the difference scheme: 10−3.
2.3.1.1 Accessing the results
The analysis results can be accessed in the FORMAnalysis.Results structure:
FORMAnalysis.Results
ans =
BetaHL: 3
Pf: 0.0013
ModelEvaluations: 8
Ustar: [-2.4000 1.8000]
Xstar: [3.0800 3.0800]
Importance: [0.6400 0.3600]
Iterations: 2
History: [1x1 struct]
In the Results structure, Pf is the estimate of Pfaccording to Eq. (1.26), BetaHL the
corresponding Hasofer-Lindt reliability index, Ustar the design point U∗in the standard
normal space, Xstar the correspondingly transformed design point in the original space
X∗=T−1(U∗),Importance the importance factors Siin Eq. (1.28), Iterations the num-
ber of FORM iterations needed to converge, ModelEvaluations the total number of eval-
uations of the limit-state function, and History a set of additional information about the
convergence behaviour of the algorithm.
2.3.1.2 Advanced options
Several advanced options are available for the FORM method to tweak which algorithm is
used to calculate the solution. They can be specified by adding a FORM field to the FORM
options structure FORMOpts. In the following, the most common advanced options for FORM
are specified:
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•Specify the FORM algorihm: by default, the iHL-RF algorithm is used (see page 8).
The original HL-RF algorithm (see page 7) can be enforced by adding:
FORMOpts.FORM.Algorithm = 'HLRF';
•Specify a starting point: by default the search of the design point is started in the
SNS at U0=0. It is possible to specify an alternative starting point (useful, e.g., when
multiple design points are expected) as:
FORMOpts.FORM.StartingPoint = [u1,...,uM];
where [u1,...,uM] are the desired coordinates of the starting point in the SNS.
•Numerical calculation of the gradient: advanced options related to the numerical
calculation of the gradient can be specified by using the FORMOpts.Gradient structure.
As an example, to specify a gradient relative step-size h= 0.001 one can write:
FORMOpts.Gradient.h = 0.001;
Details on the gradient computation options are given in Table 7, page 43.
For a comprehensive list of the advanced options available to the FORM method, please see
Table 6, page 42.
2.3.2 Second Order Reliability Method (SORM)
A SORM analysis is set up very similarly to its FORM counterpart:
SORMOpts.Type = 'Reliability';
SORMOpts.Method = 'SORM';
SORMAnalysis = uq_createAnalysis(SORMOpts);
Once the analysis is performed, a report with the FORM+SORM results can be printed by:
uq_print(SORMAnalysis)
which produces the following:
-------------------------------------------
FORM/SORM
-------------------------------------------
Pf 1.3499e-03
BetaHL 3.0000
PfFORM 1.3499e-03
PfSORM 1.3499e-03
PfSORMBreitung 1.3499e-03
ModelEvaluations 20
-------------------------------------------
Variables R S
Ustar -2.400000 1.800000
Xstar 3.08e+00 3.08e+00
Importance 0.640000 0.360000
-------------------------------------------
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The results can be visualized graphically as follows:
uq_display(SORMAnalysis)
which produces the same images as in FORM (Figure 5), as SORM is only a refinement of
the final FORM Pfestimate.
Note: In the preceding example no specifications are provided. If not further specified
the SORM runs with the same defaults as FORM.
2.3.2.1 Accessing the results
The analysis results can be accessed in the SORMAnalysis.Results structure:
SORMAnalysis.Results
ans =
Pf: 0.0013
BetaHL: 3
ModelEvaluations: 20
PfFORM: 0.0013
PfSORM: 0.0013
PfSORMBreitung: 0.0013
Ustar: [-2.4000 1.8000]
Xstar: [3.0800 3.0800]
Importance: [0.6400 0.3600]
Iterations: 2
History: [1x1 struct]
The Results structure contains the same fields as FORM (see Section 2.3.1) (when necessary
with a FORM or SORM suffix for clarity). In addition, the two PfSORMBreitung and PfSORM
fields provide the Pf,SORM and PH
f,SORM given in Eqs. (1.34) and (1.35), respectively.
2.3.2.2 Advanced options
The SORM method shares the same advanced options as the FORM method, described in
Section 2.3.1.2.
2.3.3 Monte Carlo Sampling (MCS)
The Monte Carlo Sampling algorithm only requires the user to specify the maximum number
of limit-state function evaluations, corresponding to Nin Eq. (1.36), if different from the
default value N= 105. As an example, to run a reliability analysis with N= 106samples one
can write:
MCOpts.Type = 'Reliability';
MCOpts.Method = 'MCS';
MCOpts.Simulation.MaxSampleSize = 1e6;
MCAnalysis = uq_createAnalysis(MCOpts);
Once the analysis is performed, a report with the Monte-Carlo sampling results can be printed
on screen by:
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uq_print(MCAnalysis)
which produces the following:
-------------------------------------------
Monte Carlo simulation
-------------------------------------------
Pf 1.3110e-03
Beta 3.0089
CoV 0.0276
ModelEvaluations 1000000
PfCI [1.2401e-03 1.381919e-03]
BetaCI [2.9929e+00 3.025751e+00]
-------------------------------------------
The results can be visualized graphically as follows:
uq_display(MCAnalysis)
which produces the convergence images in Figure 6 and the Monte Carlo samples image in
Figure 7. Note that in uq_display, the maximum number of samples plotted is n= 105to
limit the size of the figure.
Figure 6: Graphical visualization of the convergence of the Monte Carlo Sampling analysis in
Section 2.3.3.
Note: In the preceding example only the maximum sample size for the analysis is pro-
vided. If not further specified the Monte Carlo Sampling runs with the following
defaults:
–Confidence level: 0.05;
–Number of samples evaluated per batch: 104;
–Failure is defined for: limit-state g(x)≤0.
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MCS - Samples
2468
x1
-1
0
1
2
3
4
5
x2
g(X)≤0
g(X)>0
Figure 7: Graphical visualization of the samples of the Monte Carlo Sampling analysis in
Section 2.3.3.
2.3.3.1 Accessing the results
The analysis results can be accessed in the MCAnalysis.Results structure:
MCAnalysis.Results
ans =
Pf: 0.0013
Beta: 3.0013
CoV: 0.0273
ModelEvaluations: 1000000
PfCI: [0.0013 0.0014]
BetaCI: [2.9855 3.0180]
History: [1x1 struct]
The Results structure contains the following fields: Pf, the estimated b
Pf,MC as in Eq. (1.36);
Beta, the corresponding generalized reliability index in Eq. (1.41); CoV, the coefficient of
variation calculated with Eq. (1.40); ModelEvaluations, the total number of limit-state
function evaluations; PfCI, the confidence intervals calculated with Eq. (1.39); BetaCI, the
corresponding confidence intervals on βHL calculated with Eq. (1.42); History, a structure
containing the convergence of Pf,CoV and the corresponding confidence intervals calcu-
lated at preset sample batches (by default once every 104samples). The content of the
History structure is used to produce the convergence plot shown in Figure 6.
2.3.3.2 Advanced options
The advanced options available in Monte Carlo sampling are related to the convergence
criterion of the algorithm and to the definition of the confidence bounds reported in the
MCAnalysis.Results structure. In the following, a list of the most commonly used param-
eters for a MC analysis are given:
•Specify a target CoV and a corresponding batch size: in addition to specifying the
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MaxSampleSize option, one can specify a target CoV . The algorithm will sequentially
add batches of points to the current sample and stop as soon as the current CoV is be-
low the specify threshold. To specify a target CoV = 0.01 and batches of size NB= 104,
one can write:
MCOpts.Simulation.TargetCoV = 0.01;
MCOpts.Simulation.BatchSize = 1e4;
Note that the two options are independent from each other. The BatchSize option is
also used to set the breakpoints for the MCAnalysis.Results.History structure.
•Specify αfor the confidence intervals: the αin Eq. (1.39) can also be specified. To
set α= 0.1, one can write:
MCOpts.Simulation.Alpha = 0.1;
For a comprehensive list of the advanced options available for Monte Carlo simulation, please
refer to Table 5, page 41.
2.3.4 Importance Sampling
Importance sampling shares configuration options from both the FORM and the Monte Carlo
simulation methods. A basic IS analysis can be setup with the following code:
ISOpts.Type = 'Reliability';
ISOpts.Method = 'IS';
ISAnalysis = uq_createAnalysis(ISOpts);
Using this minimal setup the analysis will run FORM first with the default options as in Sec-
tion 2.3.1 to determine the design point U∗. Then the standard normal importance density
centred at the obtained design point is used. Sampling is carried out with the following
default options: N= 1000, batch size NB= 100.
Once the analysis is performed, a report with the importance sampling results can be printed
on screen by:
uq_print(ISAnalysis)
which produces the following:
-------------------------------------------
Importance Sampling
-------------------------------------------
Pf 1.3132e-03
Beta 3.0084
CoV 0.0599
ModelEvaluations 1008
PfCI [1.1591e-03 1.467421e-03]
BetaCI [2.9745e+00 3.046121e+00]
-------------------------------------------
The results can be visualized graphically as follows:
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uq_display(ISAnalysis)
which produces the convergence image in Figure 8. As with FORM, the second panel in
Figure 8 is produced only in the 2D case.
IS - FORM Design point and failure plane
-6 -4 -2 0
u1
-1
0
1
2
3
4
5
u2
FORM iterations
FORM limit state surface
g(X)≤0
g(X)>0
Figure 8: Graphical visualization of the convergence of the importance sampling analysis in
Section 2.3.4.
Note: In the preceding example no specifications are provided. The Importance Sam-
pling shares the same defaults values as FORM and MCS. The exceptions are:
–Maximum number of evaluated samples: 103;
–Number of samples evaluated per batch: 102.
2.3.4.1 Accessing the results
The results of the importance sampling analysis can be accessed with the ISAnalysis.Results
structure:
ISAnalysis.Results
ans =
Pf: 0.0013
Beta: 3.0095
CoV: 0.0593
ModelEvaluations: 100008
PfCI: [0.0012 0.0015]
BetaCI: [2.9759 3.0468]
History: [1x1 struct]
FORM: [1x1 struct]
The basic structure of Results closely resembles that of Monte Carlo sampling (see Sec-
tion 2.3.3.1). However, an additional structure Results.FORM is available:
ISAnalysis.Results.FORM
ans =
BetaHL: 3
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(Rare Events Estimation)
Pf: 0.0013
ModelEvaluations: 8
Ustar: [-2.4000 1.8000]
Xstar: [3.0800 3.0800]
Importance: [0.6400 0.3600]
Iterations: 2
History: [1x1 struct]
This structure is identical to the FORM results given in Section 2.3.1.1.
2.3.4.2 Advanced options
The importance sampling algorithm accepts all of the options specific to both FORM and
Monte Carlo sampling, described in Section 2.3.1.2 and Section 2.3.3.2. In addition, two
additional options can be specified for importance sampling:
•Specify existing FORM results: by default, importance sampling first runs FORM to
determine the design point, followed by sampling around this design point to calculate
Pf,IS. If the results of a previous FORM analysis are already available, they can be
specified so as to avoid running FORM again. If the results are stored in a FORMResults
structure with the same format as described in Section 2.3.1.1, one can write:
ISOpts.IS.FORM = FORMResults;
Alternatively, one can also directly specify a pre-existing UQLAB FORM or SORM anal-
ysis, say FORMAnalysis:
ISOpts.IS.FORM = FORMAnalysis;
•Specify a custom sampling distribution:alternatively, one can directly specify a cus-
tom sampling distribution. This can be achieved by providing the marginals and copula
structure of the desired distribution, e.g.IOpts in Section 2.2, as follows:
ISOpts.IS.Instrumental = IOpts;
Alternatively, a pre-existent UQLAB INPUT object, say myISInput, can also be specified:
ISOpts.IS.Instrumental = myISInput;
In case the model has multiple outputs Nout, it might be desirable to specify a custom
sampling distribution for each one of them. This can be done by either providing the
IOpts as a 1×Nout structure or the pre-existing inputs myISInputs as a 1×Nout
uq_input object. Please note, that custom distributions should be specified either for
all outputs or for none.
For a complete overview of the available options specific to the importance sampling algo-
rithm, see Table 8.
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2.3.5 Subset Simulation
The subset simulation algorithm can be used with the default options P0= 0.1 and NS= 103
by specifying:
SSOpts.Type = 'Reliability';
SSOpts.Method = 'Subset';
SSimAnalysis = uq_createAnalysis(SSOpts);
Once the analysis is performed, a report with the subset-simulation results can be printed on
screen by:
uq_print(SSimAnalysis)
which produces the following:
-------------------------------------------
Subset simulation
-------------------------------------------
Pf 1.0600e-03
Beta 3.0729
CoV 0.2433
ModelEvaluations 2680
PfCI [5.5463e-04 1.565373e-03]
BetaCI [2.9546e+00 3.261242e+00]
-------------------------------------------
The results can be visualized graphically as follows:
uq_display(SSimAnalysis)
which illustrates the samples of each subset in Figure 9 (applicable only for one and two-
dimensional problems).
SubsetSim - Samples in each subset
02468
x1
0
1
2
3
4
5
x2
Figure 9: Graphical visualization of the convergence of the subset simulation analysis in
Section 2.3.5.
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Structural Reliability
(Rare Events Estimation)
Note: In the preceding example no specifications are provided. The Importance Sam-
pling shares the same defaults values as FORM and MCS. Additionally, there are
the following default values:
–Target conditional failure probability of auxiliary limit-staates: 0.1;
–Maximum number of subsets: 20;
–Type of the proposal distribution in the Markov Chain: 'uniform';
–Parameter (standard deviation / halfwidth) of the proposal distribution: 1.
2.3.5.1 Accessing the results
The results of subset simulation are stored in the SSimAnalysis.Results structure:
SSimAnalysis.Results
ans =
Pf: 0.0016
Beta: 2.9402
CoV: 0.2426
ModelEvaluations: 2597
PfCI: [8.4859e-04 0.0024]
BetaCI: [2.8160 3.1387]
NumberSubsets: 3
History: [1x1 struct]
The fields in the Results structure have the same meaning as their counterparts in impor-
tance sampling and Monte-Carlo sampling. Further, the field NumberSubsets denotes the
number of subsets. Note that the ModelEvaluations field does not contain exactly the ex-
pected N=NS∗m∗(1 −P0) = 27000 limit-state function evaluations, but a slightly smaller
N= 26095. This discrepancy is due to the modified Metropolis-Hastings MCMC acceptance
criterion described in Au and Beck (2001), which in some uncommon cases can reject sam-
ples without the need of evaluating the limit-state function.
2.3.5.2 Advanced options
Subset simulation uses the same advanced options as Monte-Carlo sampling described in
Section 2.3.3.2, as well as some additional options. The most important are summarized in
the following:
•Specify P0:the value of P0in Eq. (1.53) can be specified in 0< P0≤0.5. One can set
e.g. P0= 0.2as follows:
SSOpts.SubsetSim.p0 = 0.2;
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•Specify the number of samples in each subset: the number of samples in each subset
NScan be specified by using the .Simulation.BatchSize field. To set it to NS= 1000
one can write:
SSOpts.Simulation.BatchSize = 1000;
For a comprehensive overview of the available options specific to subset simulation see Ta-
ble 9, page 44.
2.3.6 Adaptive Kriging Monte Carlo Sampling (AK-MCS)
Adaptive Kriging Monte Carlo Sampling method with default values (see Table 11 and the
related linked tables for details on the defaults) can be deployed in UQLAB with the following
code:
AKOpts.Type = 'Reliability';
AKOpts.Method = 'AKMCS';
AKAnalysis = uq_createAnalysis(AKOpts);
Once the analysis is complete, a report with the AK-MCS results can be printed on screen by:
uq_print(AKAnalysis)
which produces the following:
-------------------------------------------
AK-MCS
-------------------------------------------
Pf 1.5200e-03
Beta 2.9637
CoV 0.0810
ModelEvaluations 18
PfCI [1.2785e-03 1.761457e-03]
BetaCI [3.0165e+00 2.917994e+00]
PfMinus/Plus [1.5200e-03 1.520000e-03]
-------------------------------------------
The results can be visualized graphically as follows:
uq_display(AKAnalysis)
which produces the images in Figure 10. Note that the plot on the right of Figure 10 is
only available when the input is two-dimensional. Additionally, if the verbosity is set to
.Display ≥5the AK-MCS analysis will plot the convergence of Pfand βwhile the analysis
is running.
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Structural Reliability
(Rare Events Estimation)
AK-MCS - Experimental design
2 3 4 5 6
1.5
2
2.5
3
3.5
4
Figure 10: Graphical visualization of the convergence of the AK-MCS analysis in Sec-
tion 2.3.6.
Note: In the preceding example no specifications are provided. If not further specified
the Monte Carlo Sampling runs with the following defaults:
–Confidence level: 0.05;
–Maximum number of evaluated samples: 105;
–Number of samples evaluated per batch: 104;
–Failure is defined for: limit-state g(x)≤0;
–Type of metamodel: 'Kriging';
–Learning function to determine the best next sample(s): 'U';
–Convergence criterion for the adaptive ED algorithm: 'stopU';
–Number of samples added to the ED for the metamodel: 103;
–Number of samples in the initial ED: Nini = max(10,2M)
–Initial ED sampling strategy: 'LHS'.
2.3.6.1 Accessing the results
The results from the AK-MCS algorithm are stored in the AKAnalysis.Results structure:
AKAnalysis.Results
ans =
Pf: 0.0015
Beta: 2.9637
CoV: 0.0810
ModelEvaluations: 25
PfCI: [0.0013 0.0018]
BetaCI: [2.9180 3.0165]
Kriging: [1x1 uq_model]
History: [1x1 struct]
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The fields in the Results structure have the same meaning as their counterparts in Monte-
Carlo sampling. Further, the field Kriging contains the final Kriging metamodel used to
estimate the failure probability. This metamodel can be reused within UQLAB for any other
purpose, see UQLAB User Manual – Kriging (Gaussian process modelling) for details.
2.3.6.2 Advanced options
AK-MCS uses the same advanced options as Monte-Carlo simulation described in Sec-
tion 2.3.3.2, as well as some additional options. The most important are summarized in
the following:
•Learning function: The learning function can be set to a custom function handle,
which points to an existing MATLAB function file. For instance, for a learning function
called EFF, a function called uq_LF_EFF must be available in the MATLAB path. It is
then used in AK-MCS as:
AKOpts.AKMCS.LearningFunction = 'EFF';
•Convergence criterion: There are three different convergence criteria mentioned in
Section 1.3.6.2. They are all available and can be specified e.g.criterion on failure
probability:
AKOpts.AKMCS.Convergence = 'stopPf';
•Specify the Kriging metamodel: The specifications for the Kriging metamodel (see
also UQLAB User Manual – Kriging (Gaussian process modelling) ) can be set in the
field .AKMCS.Kriging,e.g.for an ordinary Kriging model:
AKOpts.AKMCS.Kriging.Trend = 'ordinary';
•Specify the initial experimental design: Apart from specifying a number of points
and a sampling strategy, the initial experimental design can be specified by providing
X=x(1),...,x(N0)in the matrix Xand the corresponding limit-state function values
g(x(1),...,x(N0)in G:
AKOpts.AKMCS.IExpDesign.X = X;
AKOpts.AKMCS.IExpDesign.G = G;
•Specify the number of added experimental design points: The maximum number of
samples added to the experimental design of the Kriging metamodel can be specified
to e.g.100:
AKOpts.AKMCS.MaxAddedED = 100;
Note that the total number of runs of the limit-state function then is at most the initial
ED size plus the above number.
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(Rare Events Estimation)
•Use a PC-Kriging metamodel: Instead of Kriging, a PC-Kriging model (see also
UQLAB User Manual – PC-Kriging ) can be used as a surrogate model in AK-MCS.
AKOpts.AKMCS.MetaModel = 'PCK';
Specific options of the PCK model can be added in the field .AKMCS.PCK. As an exam-
ple, a Gaussian correlation function in PC-Kriging is set as:
AKOpts.AKMCS.PCK.Kriging.Corr.Family = 'Gaussian';
For an overview of the advanced options available for the AK-MCS method, refer to Table 11,
page 45.
2.4 Advanced limit-state function options
2.4.1 Specify failure threshold and failure criterion
While it is normally good practice to define the limit-state function directly as a UQLAB
MODEL object as in Section 2.2, in some cases it can be useful to be able to create one from
small modifications of existing MODEL objects. A typical scenario where this is apparent is
when the same objective function needs to be tested against a set of different failure thresh-
olds, e.g. for a parametric study. In this case, the limit-state specifications can be modified.
As an example, when g(x)≤T= 5 defines the failure criterion, one can use the following
syntax:
MCOpts.LimitState.Threshold = 5;
MCOpts.LimitState.CompOp = '<=';
UQLAB offers several possibilities to create simple (or arbitrarily complex) objective functions
from existing MODEL objects (see also UQLAB User Manual – The INPUT module ).
For an overview of the advanced options for the limit-state function, refer to Table 4, page
41.
2.4.2 Vector Output
In case the limit-state function g(x)results in a vector rather than a scalar value, the struc-
tural reliability module estimates the failure probability for each component independently.
Note: There is no system-type reasoning implemented to combine the failure probabil-
ities of each component.
However, the implemented methods make use of evaluations of the limit-state function if
available, as follows:
•Monte Carlo simulation: The enrichment of the sample size is increased until the
convergence criteria are fulfilled for all components.
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•Subset Simulation: The first batch of samples (MCS) is reused for every output com-
ponent limit-state.
•AK-MCS: The initial experimental design for the Kriging model of output component i
consists of the final experimental design of component i−1.
2.5 Excluding parameters from the analysis
In various usage scenarios (e.g. parametric studies) one or more input variables may be set
to fixed constant values. This can have important consequences for many of the methods
available in UQLAB e.g. FORM/SORM and AK-MCS, whose costs increase significantly with
the number of input variables. Whenever applicable, UQLAB will appropriately account for
the set of constant input parameters and exclude them from the analysis so as to avoid
unnecessary costs. This process is transparent to the user as the analysis results will still
show the excluded variables, but they will not be included in the calculations.
To set a parameter to constant, the following command can be used when the probabilistic
input is defined (See UQLAB User Manual – The INPUT module ):
inputOpts.Marginals.Type = 'Constant' ;
inputOpts.Marginals.Parameters = value;
Furthermore, when the standard deviation of a parameter equals zero, UQLAB treats it as a
Constant. For example, the following uniformly distributed variable whose upper and lower
bounds are identical is automatically set to a constant with value 1:
inputOpts.Marginals.Type = 'Uniform' ;
inputOpts.Marginals.Parameters = [1 1];
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Chapter 3
Reference List
How to read the reference list
Structures play an important role throughout the UQLAB syntax. They offer a natural way
to group configuration options and output quantities semantically. Due to the complexity of
the algorithms implemented, it is not uncommon to employ nested structures to fine-tune
inputs/outputs. Throughout this reference guide, we adopt a table-based description of the
configuration structures.
The simplest case is given when a field of the structure is a simple value/array of values:
Table X: Input
.Name String A description of the field is put here
which corresponds to the following syntax
Input.Name = 'My Input';
The columns correspond to name, data type and a brief description of each field. At the
beginning of each row a symbol is given to inform as to whether the corresponding field is
mandatory, optional, mutually exclusive, etc. The comprehensive list of symbols is given in
the following table:
Mandatory
Optional
⊕Mandatory, mutually exclusive (only one of
the fields can be set)
Optional, mutually exclusive (one of them
can be set, if at least one of the group is set,
otherwise none is necessary)
When one of the fields of a structure is a nested structure, we provide a link to a table that
describes the available options, as in the case of the Options field in the following example:
37
UQLAB User Manual
Table X: Input
.Name String Description
.Options Table Y Description of the Options
structure
Table Y: Input.Options
.Field1 String Description of Field1
.Field2 Double Description of Field2
In some cases an option value gives the possibility to define further options related to that
value. The general syntax would be
Input.Option1 = 'VALUE1' ;
Input.VALUE1.Val1Opt1 = ...;
Input.VALUE1.Val1Opt2 = ...;
This is illustrated as follows:
Table X: Input
.Option1 String Short description
'VALUE1' Description of 'VALUE1'
'VALUE2' Description of 'VALUE2'
.VALUE1 Table Y Options for 'VALUE1'
.VALUE2 Table Z Options for 'VALUE2'
Table Y: Input.VALUE1
.Val1Opt1 String Description
.Val1Opt2 Double Description
Table Z: Input.VALUE2
.Val2Opt1 String Description
.Val2Opt2 Double Description
Note: In the sequel, double/doubles mean a real number represented in double pre-
cision (resp. a set of such real numbers).
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Structural Reliability
(Rare Events Estimation)
3.1 Create a reliability analysis
Syntax
myAnalysis = uq_createAnalysis(ROpts)
Input
All the parameters required to determine the analysis are to be given as fields of the struc-
ture ROpts. Each method has its own options, that will be reviewed in different tables. The
options described in Table 3 are common to all methods.
Table 3: ROpts
.Type 'uq_reliability' Identifier of the module. The options
corresponding to other types are in
the corresponding guides.
.Method String Type of structural reliability method.
The available options are listed
below:
'MCS' Monte Carlo simulation.
'FORM', First order reliability method.
'SORM', Second order reliability method.
'IS' Importance sampling.
'Subset' Subset simulation.
'AKMCS' Adaptive Kriging Monte Carlo
Simulation (AK-MCS).
.Name String Name of the module. If not set by
the user, a unique string is
automatically assigned to it.
.Input INPUT object INPUT object used in the analysis. If
not specified, the currently selected
one is used.
.Model MODEL object MODEL object used in the analysis. If
not specified, the currently selected
one is used.
.LimitState See Table 4 Specification of the limit-state
function.
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.Display String
default: 'standard'
Level of information displayed by the
methods.
'quiet' Minimum display level, displays
nothing or very few information.
'standard' Default display level, shows the most
important information.
'verbose' Maximum display level, shows all the
information on runtime, like updates
on iterations, etc.
.Simulation See Table 5 Options field for the simulation
methods. Only applies when
ROpts.Method is 'MCS','IS',
'SS', or 'AKMCS'.
.FORM See Table 6 Options field for the FORM
algorithm methods. Only applies
when ROpts.Method is 'FORM',
'SORM', or 'IS'.
.Gradient See Table 7 Options field for computing the
gradient. It applies to the methods
that use FORM, namely, when
ROpts.Method is 'FORM',
'SORM', or 'IS'.
.IS See Table 8 Options field for importance
sampling. It applies only when
ROpts.Method is 'IS'.
.Subset See Table 9 Options field for subset simulation. It
applies only when ROpts.Method is
'Subset'.
.AKMCS See Table 11 Options field for the adaptive
experimental design algorithm in
AK-MCS. This applies when
ROpts.Method is 'AKMCS'.
.SaveEvaluations Logical
default: true
Storage or not of performed
evaluations of the limit-state
function.
true Store the evaluations.
false Do not store the evaluations.
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Structural Reliability
(Rare Events Estimation)
In order to perform a structural reliability analysis, the limit-state function g(x)is compared
to a threshold value T(by default T= 0). In analogy with Eq. (1.1), failure is defined as
g(x)≤T. Alternatively, failure can be specified as g(x)≥Tby adjustment of the field
ROpts.LimitState.CompOp to '>='. The relevant options are summarized in Table 4:
Table 4: ROpts.LimitState
.Threshold Double
default: 0
Threshold T, compared to the
limit-state function g(x).
.CompOp String
default: '<='
Comparison operator for the
limit-state function.
'<','<=' Failure is defined by g(x)< T .
'>','>=' Failure is defined by g(x)> T .
The available methods to perform structural reliability analysis are Monte Carlo simulation,
importance sampling, subset simulation, AK-MCS, FORM, and SORM. The first four methods
share the simulation options. FORM and SORM are gradient-based, so they allow the user to
specify the finite difference options as well as the algorithm options.
In Table 5, the options for the simulation methods (Monte Carlo, importance sampling, subset
simulation and AK-MCS) are shown:
Table 5: ROpts.Simulation
.Alpha Double
default: 0.05
Confidence level α. For the Monte
Carlo estimators, a confidence
interval is constructed with
confidence level 1−α.
.MaxSampleSize Integer
default: 103for 'IS';
105otherwise
Maximum number of samples to be
evaluated. If there is no target
coefficient of variation (CoV), this is
the total number of samples to be
evaluated. If the target CoV is
present, the method will run until
TargetCoV or MaxSampleSize is
reached. In this case, the default
value of MaxSampleSize, if not
specified in the options, is Inf,
i.e.the method will run until the
target CoV is achieved.
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UQLAB User Manual
.TargetCoV Double Target coefficient of variation. If
present, the method will run until
the estimate of the CoV (Eq. (1.40))
is below TargetCoV or until
MaxSampleSize function
evaluations are performed. The
value of the coefficient of variation of
the estimator is checked after each
BatchSize evaluations. By default
this option is disabled. Note: this
option has no effect in
Method = 'Subset' and
'AKMCS'.
.BatchSize Integer
default: 104for 'MCS'
and 'AKMCS';
103for 'Subset';
102for 'IS'
Number of samples that will be
evaluated at once. Note that this
option has no effect in
Method = 'AKMCS'.
Note: In order to use importance sampling after an already computed FORM analysis,
one can provide these results to the analysis options in order to avoid repeating
FORM. If FORMResults is a structure containing the results of a FORM analysis,
the syntax reads:
ISOpts.Type = 'Reliability';
ISOpts.Method = 'IS';
ISOpts.FORM = FORMResults;
ISAnalysis = uq_createAnalysis(ISOpts);
The FORM algorithm has special parameters that can be tuned in the subfield FORM of the
options. These parameters also affect the methods that depend on FORM, namely importance
sampling and SORM. These are listed in Table 6.
Table 6: ROpts.FORM
.Algorithm String
default: 'iHLRF'
Algorithm used to find the design
point.
'iHLRF' Improved HLRF.
'HLRF' HLRF.
.StartingPoint 1×MDouble
default: zeros(1,M)
Starting point for the
Rackwitz-Fiessler algorithm.
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(Rare Events Estimation)
.StopU Double
default: 10−4
Tolerance value for the
Rackwitz-Fiessler algorithm on the
design point. The algorithm will stop
when |Uk+1 −Uk|< StopU.
.StopG Double
default: 10−6
Tolerance value for the
Rackwitz-Fiessler algorithm on the
limit-state function value. The
algorithm will stop when
|G(Uk)
G(U0)|< StopG.
.MaxIterations Integer
default: 100
Maximum number of iterations
allowed in the Rackwitz-Fiessler
algorithm. If this property should be
ignored, it can be set to Inf.
Since FORM is a gradient-based method, the gradient of the limit-state function needs to be
computed. This is done using finite differences. The options for the differentiation are listed
in Table 7.
Table 7: ROpts.Gradient
.h Double
default: 10−3
Value of the difference for the
scheme.
.Method String
default: 'forward'
Specifies the type of finite differences
scheme to be used.
'forward' Forward finite differences. ∂g
∂xiis
approximated using g(x)and
g(x+hei).
'backward' ∂g
∂xiis approximated using g(x)and
g(x−hei).
'centered' ∂g
∂xiis approximated using g(x+hei)
and g(x−hei). (More accurate and
more costly.)
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UQLAB User Manual
The options specifically set for the importance sampling are presented in Table 8. Note that
the options of .Simulation and .FORM are also processed in the case of importance sampling
due to the nature of the MCS, FORM and IS.
Table 8: ROpts.IS
.Instrumental 1×Nout INPUT object or
Struct
Instrumental distribution defined as
either a structure of input marginals
and copula or an INPUT object (refer
to UQLAB User Manual – The INPUT
module for details).
.FORM FORM ANALYSIS object or
FORMAnalysis.Results
Struct
FORM results computed previously.
See Section 2.3.4.2 for details.
The options specifically set for subset simulation are presented in Table 9. Note that the
options of .Simulation are also processed in the case of subset simulation due to the similar
nature of Monte Carlo simulation and subset simulation.
Table 9: ROpts.Subset
.p0 Double
default: 0.1
Target conditional failure probability
of auxiliary limit-states
(0<p0≤0.5).
.Proposal See Table 10 Description of the proposal
distribution in the Markov Chain.
.MaxSubsets Integer
default: 20
Maximum number of subsets. In the
subset simulation algorithm, the
maximum number of subsets is set to
the minimum of MaxSubsets and
MaxSampleSize
BatchSize·(1−p0).
The settings of the Markov Chain Monte Carlo simulation in subset simulation are summa-
rized in Table 10. Note that the default values are taken from Au and Beck (2001).
Table 10: ROpts.Subset.Proposal (Proposal distributions)
.Type String
default: 'Uniform'
Type of proposal distribution (in the
standard normal space).
'Gaussian' Gaussian distribution.
'Uniform' Uniform distribution.
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Structural Reliability
(Rare Events Estimation)
.Parameters Double
default: 1
Parameter of the proposal
distribution. Corresponds to the
standard deviation for a Gaussian
distribution and the half-width for
the uniform distribution.
AK-MCS is a combination of Kriging metamodels and Monte Carlo simulation. The options
for the Kriging metamodel and the adaptive experimental design algorithm are listed here.
Table 11: ROpts.AKMCS
.MetaModel String
default: 'Kriging'
Choice of metamodel in AK-MCS.
.Kriging Structure Kriging options when field
MetaModel = 'Kriging'. If
none is set, then the default Kriging
options are used (refer to UQLAB
User Manual – Kriging (Gaussian
process modelling) ). Note that a
small nugget of 10−10 is added by
default to the correlation options to
improve numerical stability.
.PCK Structure PC-Kriging options when field
MetaModel = 'PCK'. If none is
set, then the default PC-Kriging
options are used (refer to UQLAB
User Manual – PC-Kriging ). Note
that a small nugget of 10−10 is added
by default to the correlation options
to improve numerical stability.
.LearningFunction
String
default: 'U'
Learning function to determine the
best next sample(s) to be added to
the experimental design.
'U' U-function (see Eq. (1.59)).
'EFF' Expected feasibility function (see
Eq. (1.61)).
.Convergence String
default: 'stopU'
Convergence criterion for the
adaptive experimental design
algorithm.
'stopU' Convergence when min U(x)≥2
(see Echard et al. (2011)) on the
candidate set.
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'stopPf' Convergence criterion based on the
convergence of the failure
probability estimate (see Eq. (1.63)).
'stopBeta' Convergence criterion based on the
convergence of the reliability index
estimate (see Eq. (1.66)).
.MaxAddedED Integer
default: 1000
Number of samples added to the
experimental design of the Kriging
metamodel.
.IExpDesign See Table 12 Specification of the initial
experimental design of the
metamodel.
The initial experimental design of AK-MCS can either be given by a number of samples and
a sampling method or by a matrix containing the set of input samples and the corresponding
values of the limit-state function.
Table 12: ROpts.AKMCS.IExpDesign
.N Integer
default: 10
Number of samples in the initial
experimental design.
.Sampling String
default: 'LHS'
Sampling techniques of the initial
experimental design. See UQLAB
User Manual – The INPUT module for
more sampling techniques.
.X N×MDouble Matrix containing the initial
experimental design.
.G N×Nout Double Vector containing the responses of
the limit-state function
corresponding to the initial
experimental design, corrected by
the threshold k(see also Table 4):
(g(x)−T)for Criterion = '<=',
(T−g(x)) for Criterion = '>='.
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Structural Reliability
(Rare Events Estimation)
3.2 Accessing the results
Syntax
myAnalysis = uq_createAnalysis(ROpts)
Output
The information stored in the myAnalysis.Results structure depends on which kind of
analysis is performed. In the sequel, the results for each of the methods are reviewed.
• Monte Carlo - Table 13
• FORM - Table 15
• SORM - Table 17
• Importance sampling - Table 18
• Subset simulation - Table 19
• AK-MCS - Table 21
3.2.1 Monte Carlo
The results are summarized in Table 13.
Table 13: myAnalysis.Results
.Pf Double Estimator of the failure probability, Pf,MCS.
.Beta Double Associated reliability index,
βMCS =−Φ−1(Pf,MCS ).
.CoV Double Coefficient of variation.
.ModelEvaluations Integer Total number of model evaluations performed
during the analysis.
.PfCI 1×2Double Confidence interval of the failure probability.
.BetaCI 1×2Double Confidence interval of the associated reliability
index.
.History See Table 14 If the simulation is carried out using batches of
points, History(i) contains results obtained
after the i-th batch.
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If the simulation has been carried out by using various batches of points, the information on
the convergence in each step is stored in the structure History. Its contents are described in
Table 14.
Table 14: myAnalysis.Results.History
.Pf Double Failure probability estimate after each batch.
.CoV Double Coefficient of variation after each batch.
.Conf Double Confidence interval after each batch.
.X N×MDouble Matrix containing the input vectors in the
original space evaluated by the limit-state
function.
.U N×MDouble Matrix containing the input vectors in the
standard normal space evaluated by the
limit-state function.
.G N×Nout
Double
Values of the limit-state function along the
FORM iterations i.e.
g(xk)−Tfor Criterion = '<=',
T−g(xk)for Criterion = '>='.
3.2.2 FORM and SORM
FORM and SORM methods are very close in terms of calculations. Indeed, SORM can be
understood as a correction of the FORM estimation of the probability. Therefore, the results
structures are very similar. The results of FORM are shown in Table 15. When executing
SORM, some fields will be added to the FORM Results structure, shown in Table 17.
Table 15: myAnalysis.Results
.BetaHL Double Hasofer-Lind reliability index, βHL.
.Pf Double Estimator of the failure probability,
Pf,FORM = Φ(−βHL).
.ModelEvaluations Integer Total number of model evaluations performed
to solve the analysis.
.Ustar 1×MDouble Design point U∗in the standard normal space.
.Xstar 1×MDouble Design point X∗the original space.
.Importance 1×MDouble Importance factors Si.
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.Iterations Integer Number of iterations carried out by the
optimization algorithm.
.Method String Method used to solve the analysis.
.History See Table 16 Structure with information about the algorithm
steps and runtime information.
The field History of the results contains more detailed information extracted from the algo-
rithm steps.
Table 16: myAnalysis.Results.History
.ExitFlag String Reason why the algorithm stopped.
.BetaHL Double Values of the reliability index along the FORM
iterations.
.OriginValue Double Value of the limit-state function G(u=0).
.G N×Nout
Double
Values of the limit-state function along the
FORM iterations i.e.
g(xk)−Tfor Criterion = '<=',
T−g(xk)for Criterion = '>='.
.Gradient Double Values of the gradient of the limit-state
function along the FORM iterations.
.StopU Double Values of the stopping criterion on the design
point convergence, |Uk+1 −Uk|, along the
FORM iterations.
.StopG Double Values of the stopping criteria on the limit-state
function, |G(Uk)
G(U0)|, along the FORM iterations.
.U N×MDouble Coordinates of the points Ukin the standard
normal space.
.X N×MDouble Coordinates of the points Xkin the original
space.
If SORM is also performed, two fields are added to the Results, and two fields are added to
Results.History, as shown in Table 17.
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Table 17: myAnalysis.Results
.PfSORM Double SORM estimator of the failure probability,
Pf,SORM, using Hohenbichler’s formula.
.PfSORMBreitung Double SORM estimator of the failure probability,
Pf,SORM, using Breitung’s formula.
.BetaSORM Double Associated reliability index
βSORM =−Φ−1(Pf,SORM), using Hohenbichler’s
formula.
.BetaSORMBreitung Double Associated reliability index
βSORM =−Φ−1(Pf,SORM), using Breitung’s
formula.
.History.FORMEvals Integer Number of model evaluations carried out to
perform the FORM analysis.
.History.Hessian M×MDouble Hessian matrix of the limit-state function at the
design point, U∗.
3.2.3 Importance sampling
Since importance sampling is a simulation method, the structure of the results is similar to
the one of Monte Carlo simulation. The results are listed in Table 18.
Table 18: myAnalysis.Results
.Pf Double Estimator of the failure probability, Pf,IS.
.Beta Double Associated reliability index, βIS =−Φ−1(Pf,IS).
.CoV Double Coefficient of variation.
.ModelEvaluations Integer Total number of model evaluations performed
during the analysis.
.PfCI 1×2Double Confidence interval of the failure probability.
.BetaCI 1×2Double Confidence interval of the associated reliability
index.
.FORM See Table 15 Results of the FORM analysis used to find the
design point.
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.History See Table 14 If importance sampling is carried out using
batches of points, History(i) contains
results obtained after the i-th batch.
3.2.4 Subset simulation
Since subset simulation is a simulation method, the structure of the results is similar to the
one of Monte Carlo simulation. The results are listed in Table 19 and Table 20.
Table 19: myAnalysis.Results
.Pf Double Estimator of the failure probability, Pf,SS.
.Beta Double Associated reliability index, βSS =−Φ−1(Pf,SS).
.CoV Double Coefficient of variation.
.ModelEvaluations Integer Number of model evaluations during the
analysis.
.PfCI 1×2Double Confidence interval of the failure probability.
.BetaCI 1×2Double Confidence interval of the associated reliability
index.
.NumberSubsets Integer Number of auxiliary subsets during the
analysis.
.History See Table 20 Data related to each subset.
The field History of the results contains more detailed information extracted from the algo-
rithm steps.
Table 20: myAnalysis.Results.History
.delta2 Double δ2
jfor estimating the coefficient of variation
(see Eq. (1.55)).
.q Double Intermediate limit-state thresholds.
.X Cell Samples in the input space of each subset.
.U Cell Samples in the standard normal space of each
subset.
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.G Cell Values of the limit-state function of each
sample in each subset corrected by the
threshold T(see also Table 4):
(g(x)−T)for Criterion = '<=',
(T−g(x)) for Criterion = '>='.
.Pfcond Double Conditional failure probability estimates
P(Di+1|Di).
.gamma Double γjfor computing the coefficient of variation
(see Eq. (1.56)).
3.2.5 AK-MCS
Since AK-MCS relies upon Monte Carlo simulation, the structure of the results is similar to
the one of Monte Carlo simulation. The results are listed in Table 21 and Table 22.
Table 21: myAnalysis.Results
.Pf Double Estimator of the failure probability, Pf,AK-MCS.
.Beta Double Associated reliability index,
βAK-MCS =−Φ−1(Pf,AK-MCS).
.CoV Double Coefficient of variation.
.ModelEvaluations Integer Total number of model evaluations performed
during the analysis.
.PfCI 1×2Double Confidence interval of the failure probability.
.BetaCI 1×2Double Confidence interval of the associated reliability
index.
.Kriging Struct Final Kriging metamodel.
.PCK Struct Final PC-Kriging metamodel.
.History See Table 22 Contains intermediate results along the
AK-MCS iterations.
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Table 22: myAnalysis.Results.History
.Pf Double History of the estimate failure probability.
.PfLower Double History of the estimated lower bound of the
failure probability P−
f.
.PfUpper Double History of the estimated upper bound of the
failure probability P+
f.
.NSamples Integer Number of samples added to the experimental
design at each iteration.
.NInit Integer Number of samples in the initial experimental
design.
.X Cell Samples in the input space of each subset.
.G Cell Values of the limit-state function of each
sample in each subset corrected by the
threshold T(see also Table 4):
(g(x)−T)for Criterion = '<=',
(T−g(x)) for Criterion = '>='.
3.3 Printing/Visualizing of the results
UQLAB offers two commands to conveniently print reports containing contextually relevant
information for a given result object:
3.3.1 Printing the results: uq print
Syntax
uq_print(myAnalysis);
uq_print(myAnalysis, outidx);
Description
uq_print(myAnalysis) prints a report on the results of the reliability analysis stored in
the object myAnalysis. If the model has multiple outputs, only the results for the first
output variable are printed.
uq_print(myAnalysis, outidx) prints a report on the results of the reliability analysis
stored in the object myAnalysis for the output variables specified in the array outidx.
Examples:
uq_print(myAnalysis, [1 3]) prints the reliability analysis results for output variables 1
and 3.
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3.3.2 Graphically display the results: uq display
Syntax
uq_display(myAnalysis);
uq_display(myAnalysis, outidx);
Description
uq_display(myAnalysis) creates a visualization of the results of the reliability analysis
stored in the object myAnalysis, if possible. If the model has multiple outputs, only
the results for the first output variable are visualized.
uq_display(myAnalysis, outidx) creates a visualization of the results of the reliability
analysis stored in the object myAnalysis for the output variables specified in the array
outidx.
Examples:
uq_display(myAnalysis, [1 3]) will display the reliability analysis results for output
variables 1 and 3.
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References
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