First-passage probability estimation of high-dimensional nonlinear stochastic dynamic systems by a fractional moments-based mixture distribution approach_2

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This manuscript has been published in Mechanical Systems and Signal Processing. The DOI of this manuscript
is: https://doi.org/10.1016/j.ymssp.2022.109775. Please cite as: C. Ding, C. Dang, M. A. Valdebenito, M. G. Faes,
M. Broggi, M. Beer, First-passage probability estimation of high-dimensional nonlinear stochastic dynamic systems
by a fractional moments-based mixture distribution approach, Mechanical Systems and Signal Processing 185 (2023)
109775.
First-passage probability estimation of high-dimensional nonlinear stochastic
dynamic systems by a fractional moments-based mixture distribution approach
Chen Dinga, Chao Danga,∗, Marcos A. Valdebenitob, Matthias G.R. Faesc, Matteo Broggia, Michael Beera,d,e
aInstitute for Risk and Reliability, Leibniz University Hannover, Callinstr. 34, Hannover 30167, Germany
bFaculty of Engineering and Sciences, Universidad Adolfo Ib´
a˜
nez, Av. Padre Hurtado 750, 2562340 Vi˜
na del Mar, Chile
cChair for Reliability Engineering, TU Dortmund University, Leonhard-Euler-Str. 5, Dortmund 44227, Germany
dInstitute for Risk and Uncertainty, University of Liverpool, Peach Street, Liverpool L69 7ZF, United Kingdom
e
International Joint Research Center for Resilient Infrastructure & International Joint Research Center for Engineering Reliability and Stochastic
Mechanics, Tongji University, Shanghai 200092, PR China
Abstract
First-passage probability estimation of high-dimensional nonlinear stochastic dynamic systems is a significant task to
be solved in many science and engineering fields, but remains still an open challenge. The present paper develops a
novel approach, termed ‘fractional moments-based mixture distribution’, to address such challenge. This approach is
implemented by capturing the extreme value distribution (EVD) of the system response with the concepts of fractional
moment and mixture distribution. In our context, the fractional moment itself is by definition a high-dimensional
integral with a complicated integrand. To efficiently compute the fractional moments, a parallel adaptive sampling
scheme that allows for sample size extension is developed using the refined Latinized stratified sampling (RLSS).
In this manner, both variance reduction and parallel computing are possible for evaluating the fractional moments.
From the knowledge of low-order fractional moments, the EVD of interest is then expected to be reconstructed. Based
on introducing an extended inverse Gaussian distribution and a log extended skew-normal distribution, one flexible
mixture distribution model is proposed, where its fractional moments are derived in analytic form. By fitting a set
of fractional moments, the EVD can be recovered via the proposed mixture model. Accordingly, the first-passage
probabilities under different thresholds can be obtained from the recovered EVD straightforwardly. The performance
of the proposed method is verified by three examples consisting of two test examples and one engineering problem.
Keywords:
First-passage probability, Stochastic dynamic system, Extreme value distribution, Fractional moment, Mixture
distribution
∗Corresponding author
Email address: chao.dang@irz.uni-hannover.de (Chao Dang)
Preprint submitted to Elsevier October 11, 2022
arXiv:2210.04715v1 [stat.ME] 10 Oct 2022
1. Introduction
Stochastic dynamic systems which involve the randomness in internal system properties and/or external dynamic
loads are widespread in various science and engineering fields, such as meteorology, quantum optics, circuit theory and
structural engineering [
1
]. To assess the effects of input randomness on the system performance, dynamic reliability
analysis has drawn increasing attention. Generally, dynamic reliability analysis for stochastic dynamic systems can be
classified as the first-passage probability evaluation and the fatigue failure probability estimation [
2
]. In the literature,
the first-passage probability evaluation has been extensively studied over the past several decades. However, finding
efficient and accurate solutions to the first-passage problem still remains challenging. The reason is twofold: (1) the
high-dimensional input randomness and strongly nonlinear behavior of stochastic dynamic systems may be encountered
simultaneously; (2) the first-passage probabilities of such systems under certain thresholds may be relatively small.
The existing approaches for first-passage probability estimation can be broadly divided into four kinds: the out-
crossing rate approaches, the diffusion process approaches, the stochastic simulation approaches and the extreme
value distribution (EVD) estimation approaches. For the out-crossing rate approaches, the first-passage probability
is evaluated considering the time of out-crossing within a time duration on the basis of Rice’s formula [
3
–
6
]. Such
approaches are based on the Poisson assumption that level-crossing events are mutually independent and each happens
at most once, or the Markovian assumption that the next crossing event only relates to the present event [
7
]. Although
these solutions can be accurate in some special cases, they may be not applicable for general cases. Besides, it is
hard to derive the joint probability density function (PDF) and its derivatives of the system response of interest when
complicated nonlinear stochastic dynamic systems are encountered. The diffusion process approaches evaluate the
first-passage probability by solving a partial differential equation, such as the Kolmogorov backward equation [
8
] or
the Fokker Planck equation [
9
]. Solutions to such equations could be derived via the path integration method [
10
–
12
],
stochastic average technique [
13
,
14
], ensemble-evolving-based generalized density evolution equation [
2
,
15
], etc.
Nevertheless, this kind of approach is mostly applicable for nonlinear stochastic dynamic systems enforced by white
noise. For the stochastic simulation approach, the extensively used Monte Carlo simulation (MCS) [
16
] is able to
address problems regardless of their dimensions and nonlinearities. However, MCS is inefficient and even infeasible
to assess a small probability for an expensive-to-evaluate model since a considerably large number of simulations
are required. Although some variants of MCS have been developed, such as important sampling [
17
–
20
] and subset
simulation [21–23], they still suffer their respective limitations concerning efficiency, accuracy and applicability, etc.
Recently, the EVD estimation approaches have attracted lots of attention. This is because once the EVD of system
response of interest is obtained, the first-passage probability can be straightforwardly and conveniently evaluated [
24
].
Nevertheless, the analytical solution to the EVD is difficult and even impossible to be obtained for a general nonlinear
stochastic dynamic system. Therefore, various approximation methods have been developed to estimate the EVD, which
can be roughly classified as probability conservation-based methods and moments-based methods. According to the
principle of probability conservation, the probability density evolution method (PDEM) [7,24] and direct probability
integral method (DPIM) [
25
] are derived, which can be used for the purpose of EVD estimation. However, since such
methods are typically dependent on the partition of random variable space, their application for high-dimensional
problems may be challenging. Moment-based methods, on the other hand, estimate the first-passage probability by
fitting an appropriate parametric distribution model to the EVD, and the free parameters of the distribution model are
obtained from the estimated moments of the EVD. The integer moments-based methods can be adopted to recover the
2
EVD [
26
,
27
], where high-order integer moments, i.e., skewness and kurtosis, need to be considered. Yet it is difficult
to evaluate such high-order integer moments using a small sample size, due to their large variability [
28
]. To alleviate
such difficulty, a series of methods based on non-integer moments, such as fractional moments and linear moments,
have been developed. The fractional moments-based maximum entropy methods [
29
–
32
] can estimate the first-passage
probabilities of nonlinear stochastic dynamic systems from low to high dimensions. However, it is difficult to solve the
non-convex optimization problem that is typically encountered, and the obtained results can be easily trapped into local
optimum. Besides, due to the polynomials involved in the maximum entropy density, the recovered EVD can have
unexpected oscillating distribution tail, which then leads to an inaccurate evaluation of the first-passage probability. Two
mixture parametric distribution methods in conjunction with fractional moments [33] or moment-generating function
[
34
] are developed. These methods enable to evaluate first-passage probabilities of high-dimensional and strongly
nonlinear stochastic dynamic systems from a small number of simulations. Furthermore, a fractional moments-based
shifted generalized lognormal distribution method [
35
] is utilized to assess seismic reliability of a practical bridge
subjected to spatial variate ground motions. Besides, the linear moments-based polynomial normal transformation
distribution method [
36
] is developed to analyze high-dimensional dynamic systems with deterministic structural
parameters subjected to stochastic excitations.
Overall, the fractional moments-based methods offer the possibility to deal with both high-dimensional and strongly
nonlinear stochastic dynamic systems from a reduced number of simulations, even with small first-passage probabilities.
In view of this, the present paper mainly focuses on such methods. Despite those attractive features, the fractional
moments-based methods still have two main problems to be solved. On one hand, the sample size for evaluating
fractional moments is usually empirically fixed. This is primarily because the sampling-based schemes adopted
by the existing methods do not allow for the sample size extension. However, the optimal sample size should be
problem-dependent. With a predetermined sample size, the adopted sampling methods may encounter over-sampling or
under-sampling, leading to a waste of over-all computational efforts or unsatisfactory accuracy of estimated fractional
moments. On the other hand, the success of fractional moments-based methods for first-passage probability evaluation
also depends on the selection of an appropriate distribution model. Although the existing distribution models are
capable of representing EVDs for some problems, their flexibility and applicability are limited. Hence, for a wide
range of problems, they may still lack the ability to accurately recover the EVDs over the entire distribution domain,
especially for the tails.
In this paper, we propose a fractional moments-based mixture distribution approach to estimate the first-passage
probabilities of high-dimensional and strongly nonlinear stochastic dynamic systems. It is worth mentioning that the
randomness from both internal system properties and external excitations is taken into account. The main contributions
of this study are summarized as follows. First, a parallel adaptive sampling scheme is proposed for estimating the
fractional moments, as opposed to the traditional fixed sample size scheme. Such a new scheme enables to extend
the sample size sequentially, i.e., one at a time or several at a time. The optimal sample size for fractional moment
estimation is determined by introducing a convergence criterion. In fact, a sequential sampling method with the ability
to effectively reduce variance in high-dimensional problems, named Refined Latinized stratified sampling (RLSS) [
37
],
is suitable for achieving our purposes and is employed within the proposed scheme. Second, one novel and versatile
mixture distribution model is proposed to reconstruct the EVD with the knowledge of its estimated fractional moments.
This model is based on the extension of the conventional inverse Gaussian distribution and the log transformation of the
extended skew-normal distribution. The analytical expression of the fractional moments for such mixture distribution is
3
derived, and a fractional moments-based parameter estimation technique is developed.
The remainder of this paper is organized as follows. Section 2outlines the first-passage probability estimation of a
stochastic dynamic system from the perspective of EVD. In section 3, the proposed fractional moments-based mixture
distribution approach is described in detail, including a parallel adaptive scheme for fractional moments evaluation and
a flexible mixture distribution model for EVD reconstruction. Three examples are given in section 4to demonstrate the
performance of the proposed method. The paper is closed with some concluding remarks in section 5.
2. First-passage probability estimation of stochastic dynamic systems
2.1. Stochastic dynamic systems
Consider a stochastic dynamic system that is governed by the following state-space equation:
˙
Y(t) = Q(Y(t),U, t),(1)
with an initial condition
Y(0) = y0,(2)
where
Y= (Y1, Y2, ..., Ynd)
is a
nd
-dimensional state vector;
Q= (Q1, Q2, ..., Qnd)
is a dynamics operator vector;
U= (U1, U2, ..., Uns)
is a
ns
-dimensional random parameter vector with a known joint probability density function
(PDF)
pU(u)
;
u= (u1, u2, ..., uns)
denotes a realization of
U
;
t
denotes the time. Note that Eq. (1) can be strongly
nonlinear, which may be caused by material, geometrical, or contact nonlinearities inherent in the stochastic dynamic
system. In addition, hundreds or thousands of random variables can be included in the vector
U
due to the randomness
from system properties and external excitations.
For a well-posed stochastic dynamic system, the solution to Eq. (1) is unique and depends on the vector
U
, which
can be assumed to be:
hY(t),˙
Y(t)i=HY(U, t),∂HY(U, t)
∂t ,(3)
where HYand ∂HY
∂t are the deterministic operators.
If we consider the system responses of interest for reliability analysis, say
W(t) = {W1(t),W2(t), ..., Wnd(t)}
,
they can be evaluated from their relations to the state vectors:
W(t) = ΨhY(t),˙
Y(t)i=H(U, t),(4)
where
Ψ
is the transfer operator; and
H
denotes the mapping relation from
U
and
t
to
W(t)
. Accordingly, the
q
-th
component of
W(t)
is denoted by
Wq(t) = Hq(U, t), q = 1, ..., nd
. For notational simplicity, the subscript
q
is
omitted hereafter, and only a component W(t)is considered in the following.
2.2. First-passage probability estimation by EVD
For a stochastic dynamic system, the first-passage probability is the probability that the system response of interest
exceeds a certain safe domain for the first time within a given time range. Accordingly, assuming
T
is the time duration,
we have
Pf= Pr {W (t)/∈Ωsafe,∃t∈[0, T ]},(5)
4
where
Pf
is first-passage probability;
Pr
is probability operator;
Ωsafe
denotes the safe domain. According to different
application backgrounds, the boundary of
Ωsafe
can be different, such as one boundary, double boundary, and circle
boundary [
7
]. In the case of symmetric double boundary problem, the first-passage probability can be further written
as:
Pf= Pr {|W (t)|> blim,∃t∈[0, T ]},(6)
where
blim
is the given threshold that limits the symmetric bounds of
Ωsafe
, and
|·|
is the absolute value operator. In the
present study, the first-passage probability defined by Eq. (6) is of concern.
Note that if the system response in the time period
[0, T ]
remains below the boundary of
Ωsafe
, the first-passage
probability will be equal to zero. From this perspective, once the extreme value of system response exceeds the
boundary, the system fails. Accordingly, Eq. (6) can be rewritten as
Pf= Pr {max {|W (t)|} > blim,∀t∈[0, T ]}= Pr {Z > blim},(7)
where
Z= max
t∈[0, T ]{|W (t)|}
. Note that
Z
is always positive, and depends on the random parameter vector
U
. If we de-
note the functional relationship between
Z
and
U
as
G
, then we have
Z=G(U)
and
Pf= Pr {Z =G(U)> blim}
.
According to classical probability theory, once the probability distribution of
Z
, which is also referred to as extreme
value distribution (EVD), is obtained, Eq. (7) can be straightforwardly calculated from the EVD. Let
fZ(z)
and
FZ(z)
be the PDF and cumulative distribution function (CDF) of Z. Then the first-passage probability reads
Pf=Z+∞
blim
fZ(z)dz= 1 −FZ(blim).(8)
It should be pointed out that the first-passage probability is easy to be obtained from Eq. (8) once the PDF or CDF
of
Z
is known. However, how to estimate the EVD of
Z
is quite challenging. This is because deriving an analytical
expression for the EVD is intractable even for some simple stochastic responses, not to mention the stochastic responses
of high-dimensional and strong-nonlinear stochastic dynamic systems. Therefore, to tackle such challenge, an EVD
estimation method is proposed in the following section.
Remark 1.
For system failure probability evaluation, the above-mentioned EVD estimation method can also be applied
by using the theory of equivalent extreme-value events [
38
]. Briefly speaking, the system failure can be regarded as a
compound event of multiple random events, where a single random event can be described by an inequality associated
with a single response and its threshold. Based on the inequality relationship between the involved random events,
the compound event can be equated to an equivalent extreme-value event whose threshold can be obtained by a linear
combination of the thresholds of the involved random events. In this manner, the system failure probability can be
assessed by the Eq. (8), where
Z
is the equivalent extreme-value event. The interested readers can refer to Ref. [
38
] for
more details.
3. A fractional moments-based mixture distribution approach
In this section, we propose a novel fractional moments-based mixture distribution approach to approximate the
EVD in an efficient and accurate way. The proposed method consists of two main parts. First, a parallel adaptive
scheme is proposed for fractional moments estimation, which allows sequential sample size extension until a prescribed
convergence criterion is satisfied. Second, from the knowledge of estimated fractional moments, an eight-parameter
5
摘要：

ThismanuscripthasbeenpublishedinMechanicalSystemsandSignalProcessing.TheDOIofthismanuscriptis:https://doi.org/10.1016/j.ymssp.2022.109775.Pleaseciteas:C.Ding,C.Dang,M.A.Valdebenito,M.G.Faes,M.Broggi,M.Beer,First-passageprobabilityestimationofhigh-dimensionalnonlinearstochasticdynamicsystemsbyafracti...

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