First-passage Probability Estimation of Stochastic Dynamic Systems by a Parametric Approach

Abstract

First-passage probability estimation of stochastic dynamic systems is an important but still challenging problem in various science and engineering fields. This paper proposes a novel parametric approach, termed 'fractional moments-based mixture distribution' (FMs-MD), to address this challenge. Such method is based on capturing the extreme value distribution (EVD) of the studied stochastic system response in the first place. The concept of FM is then introduced to characterize the EVD, which is by definition a multi-(high-) dimensional integration. To efficiently evaluate the FM, a parallel adaptive strategy is developed by applying a sequential sampling technique, namely, refined Latinized stratified sampling (RLSS). By taking advantage of RLSS, both variance-reduction and parallel computing are possible in the process of FM computation. From the knowledge of low-order FMs, the EVD is then intended to be reconstructed. One flexible MD model is proposed on the basis of the extended Lognormal and generalized inverse Gaussian distributions. By fitting a set of FMs, the EVD can be reconstructed via this mixture model. The performance of the proposed method is verified by a numerical example consisting of a Duffing oscillator with random parameters under Gaussian white noise.

Full text

First-passage probability estimation of stochastic dynamic systems by a parametric approach
Chen Ding1, Chao Dang∗1, Matteo Broggi1, and Michael Beer1,2,3
1Institute for Risk and Reliability, Leibniz University Hannover, Hannover, Germany
2Institute for Risk and Uncertainty, University of Liverpool, Liverpool, UK
3International Joint Center for Engineering Reliability and Stochastic Mechanics, Tongji University, Shanghai, China
Abstract
First-passage probability estimation of stochastic dynamic systems is an important but still challenging problem in various science
and engineering fields. This paper proposes a novel parametric approach, termed ‘fractional moments-based mixture distribution’
(FMs-MD), to address this challenge. Such method is based on capturing the extreme value distribution (EVD) of the studied stochastic
system response in the first place. The concept of FM is then introduced to characterize the EVD, which is by definition a multi-
(high-) dimensional integration. To efficiently evaluate the FM, a parallel adaptive strategy is developed by applying a sequential
sampling technique, namely, refined Latinized stratified sampling (RLSS). By taking advantage of RLSS, both variance-reduction and
parallel computing are possible in the process of FM computation. From the knowledge of low-order FMs, the EVD is then intended
to be reconstructed. One flexible MD model is proposed on the basis of the extended Lognormal and generalized inverse Gaussian
distributions. By fitting a set of FMs, the EVD can be reconstructed via this mixture model. The performance of the proposed method
is verified by a numerical example consisting of a Duffing oscillator with random parameters under Gaussian white noise.
Keywords: first-passage probability, stochastic dynamic systems, extreme value distribution, mixture distribution, fractional moments
1. Introduction
Stochastic dynamic systems are widely encountered in
many science and engineering fields, such as circuit the-
ory, quantum optics, mechanical engineering and civil en-
gineering. There are various related topics on stochastic
dynamic systems, among which the first-passage probabil-
ity analysis is of great significance. Briefly, first-passage
probability is defined by the probability that the interested
response of a stochastic dynamic system reaches or ex-
ceeds the specified threshold for the first time within a given
time interval. Although the definition is simple, it is hard
to obtain the analytical solution for problems with strong
nonlinear and high-dimensional properties. To tackle with
this analysing difficulty, several methods are developed, in-
cluding semi-analytical approaches for solving the partial
differential equations [1]–[3] and simulation-based meth-
ods, such as Monte Carlo simulation (MCS) [4] and subset
simulation (SS) [5]. Nevertheless, for a general stochas-
tic dynamic system, these methods usually fail to analyze
first-passage probability with sufficient accuracy under an
acceptable computational workload.
In fact, first-passage probability estimation is closely re-
lated to the extreme value distribution (EVD) of interested
system response. Specifically, the first-passage probabili-
ties under different thresholds can be determined once the
corresponding EVD is obtained. Generally, two kinds of
approaches are able to obtain the EVD: one is probability
density evolution method (PDEM) and direct probability in-
tegral method (DPIM), the other is moments-based method.
Both the PDEM [6], [7] and DPIM[8] have the solid theoret-
ical basis according to the probability conservation theorem,
however, these approaches cannot easily be applied to prob-
lems with very high-dimensional random inputs or rare fail-
ure events due to numerical difficulty. The moments-based
method attempts to fit a proper distribution model to the
EVD from the knowledge of its estimated moments. The
integer moments can be used to characterize information
∗E-Mail: chao.dang@irz.uni-hannover.de
of EVD [9]. However, on one hand, low-order integer mo-
ments are not enough to describe tail information of EVD.
On the other hand, although more information of EVD can
be contained by adopting high-order integer moments, they
are much more difficult to estimate with sufficient accuracy.
To overcome such shortage of integer moments, fractional
moments are widely used for EVD characterization. Be-
sides, a single fractional moment embodies information of
multiple integer moments in theory. Due to the attractive
features inherent in fractional moments, various methods
have been developed based on this kind of moments. For
example, the fractional moment-based maximum entropy
method [10], [11] enables to reconstruct the EVD of re-
sponse of a nonlinear structure considering uncertainties in
both structural parameters and dynamic excitations. Never-
theless, the polynomial involved in the MEM density may
cause tail oscillations and hence inaccurate results for first-
passage probabilities. In addition, two methods, i.e., a mix-
ture distribution model with fractional moments [12] and a
mixture distribution with moment generating function [13],
have also been developed to represent the EVD of response
of a nonlinear structure with uncertain parameters under
stochastic ground motions. Nevertheless, the flexibility of
these methods are still limited.
Overall, the fractional moments-based methods are ad-
vantageous that they are able to tackle with both high-
dimensions and strong nonlinearity, as well as small first-
passage probabilities. Although this kind of methods are ex-
tensively studied, the computational efficiency and accuracy
of first-passage probability evaluation can be further im-
proved. As observed, the existing fractional moments-based
methods usually adopt sampling-based moment evaluation
methods with predetermined fixed sample sizes to deal
with different first-passage problems. However, the optimal
sample size for moment evaluation is actually problem-
dependent. With predefined sample size, the moment esti-
mation may be trapped into oversampling or undersampling,
leading to unnecessary large computational efforts or im-
precise moment results. Furthermore, the probability distri-

bution models developed in those state-of-art methods still
lack sufficient flexibility. For EVDs with heavy tails, the ex-
isting probability distribution models can produce deviated
results, which then affects the accuracy of assessed first-
passage probability. Therefore, it is necessary to develop
an adaptive scheme to estimate the fractional moments, and
a distribution model with sufficient flexibility to reconstruct
the EVD for different first-passage problems.
In this study, we develop an adaptive parametric method
to estimate the first-passage probability of stochastic dy-
namic systems. For a specified first-passage problem, the
sample size for estimation of fractional moments is de-
termined adaptively by introducing a parallel sequential
sampling technique. Besides, a novel and flexible mix-
ture distribution model is developed, where the distribution
parameters are specified by fitting the estimated fractional
moments.
2. First-passage Probability Estimation of Stochastic
Dynamic Systems
Without loss of generality, the response of a stochas-
tic dynamic system could be described by one state-space
equation with an initial condition:
¤
Y(𝑡)=Q(Y(𝑡),U, 𝑡),Y(0)=𝑦0(1)
where Y=(𝑌1, 𝑌2, ... , 𝑌𝑑)denotes a 𝑑-dimensional state
vector; Q=(𝑄1, 𝑄2, .. ., 𝑄𝑑)is a deterministic operator
vector; U=(𝑈1, 𝑈2, .. ., 𝑈𝑠)represents a 𝑠-dimensional
parameter vector concerning uncertain system properties
and external loads, which assumes to follow a specified
joint probability density function (PDF) 𝑝U(𝑢); and 𝑡is the
time.
For a well-posed stochastic dynamic system, the unique
solution of Eq.1 exists and depends on the parameter vector
U. If a set of quantities of interested system responses
are considered for reliability analysis, such as the strains or
stresses at certain points of a system, they can be represented
by a relationship equation of the state vector, reading:
𝓨(𝑡)=𝛹Y(𝑡),¤
Y(𝑡)=𝑯(U, 𝑡),𝜕
𝜕𝑡 𝑯(U, 𝑡 )(2)
where 𝓨={𝒴
1, . .., 𝒴
𝑑};𝑯denotes the deterministic op-
erator. For convenience, we only take a component of 𝓨,
say 𝒴, to illustrate the proposed method in the following.
Given time duration 𝑇, the first-passage probability is
described as
𝑃𝑓=Pr {𝒴(𝑡)∉𝛺safe,∃𝑡∈[0, 𝑇] } (3)
where Pr {·}is probability operator; 𝛺safe denotes the safe
domain. Further, considering a symmetric double bound-
ary, the first-passage probability can be written as:
𝑃𝑓=Pr {|𝒴(𝑡)| > 𝑧 𝑏,∃𝑡∈[0, 𝑇 ] } (4)
where 𝑧𝑏is the prescribed threshold of symmetric double
boundary.
In fact, the first-passage probability could be straight-
forwardly and conveniently obtained from the extreme
value distribution (EVD) of interested response. Consider
the maximum absolute extreme value over the prescribed
[0, 𝑇 ]:
Z=max
𝑡∈[0, 𝑇 ]{|𝒴(𝑡)|} (5)
where Zis the maximum absolute extreme value of inter-
ested response, which could also be treated as a function
relating to U, i.e., Z=𝐺(U). Accordingly, Eq.4 could be
𝑃𝑓=Pr {Z> 𝑧𝑏}(6)
Once the probability distribution of Z, also referred to
EVD, is obtained, the first-passage probability can be eval-
uated by a simple integral of EVD:
𝑃𝑓=∫+∞
𝑧𝑏
𝑓Z(𝑧)d𝑧=1−𝐹Z(𝑧𝑏)(7)
where 𝑓Z(𝑧)and 𝐹Z(𝑧)are the probability density func-
tion (PDF) and cumulative distribution function (CDF) of
Z, separately. However, the PDF and CDF are analytically
intractable for a general stochastic dynamic system. There-
fore, one simple and efficient method is proposed in the next
section to estimate the EVD.
3. A Fractional Moments-based Mixture Distribution
Method
This section will develop a novel fractional moments-
based mixture distribution method for first-passage proba-
bility evaluations. Briefly, two contributions are involved:
first, a parallel adaptive scheme is developed for fractional
moment estimation, where the sample extension continues
until the accuracy of moments is satisfied; second, a new
and flexible mixture distribution model is proposed, which
enables to capture the EVD accurately.
3.1. Parallel Adaptive Evaluation of Fractional Moments
By definition, the 𝑟-th fractional moments of Zreads:
𝑀𝑟
Z=𝐸[Z𝑟]=∫+∞
0
𝑧𝑟𝑓Z(𝑧)d𝑧(8)
where 𝑟could be any real number and 𝐸[·]is the expecta-
tion operator.
Actually, if we implement Taylor expansion to Zaround
its mean 𝜇Z, Eq.8 would become
𝐸[Z𝑟]=∞

𝑘=0
𝑟!
𝑘!(𝑟−𝑘)!𝜇𝑟−𝑘
Z𝐸(𝑧−𝜇Z)𝑘(9)
As seen, one 𝑟-th fractional moment enables to contain a
large number of information of integer moments. Besides,
compared with high-order moments, low-order fractional
moments are less insensitive to sampling variability, which
can be easily estimated accurately from a small number of
simulations. Therefore, a set of low-order fractional mo-
ments are sufficient to represent the main body and tail infor-
mation of EVD, which can be then used for reconstructing a
more accurate EVD with acceptable computational efforts.
Various methods could be implemented to evaluate the
fractional moments. Among them, methods based on

variance-reduction sampling technique are widely used
[11], [12], [14]. Since their variance reduction schemes
do not allow for sample size extension, such methods use
a pre-specified fixed sample size to evaluate fractional mo-
ments. To be specified, the estimator of 𝑟-th fractional
moment takes the form:
ˆ
𝑀𝑟
Z=∫Z
𝑧𝑟𝑓Z(𝑧)d𝑧≈
𝑁

𝑘=1
𝜔𝑘·𝐺𝑟(u𝑘)(10)
where 𝜔𝑘represents the 𝑘-th sample weight, 𝑘=1, ..., 𝑁;𝑁
is the number of samples, which is empirically determined;
u𝑘denotes the 𝑘-th sample of Uin the original variable
space.
However, the sample size for moment evaluation is often
problem-dependent. A fixed sample size can lead to over-
sampling or undersampling, resulting in a waste of compu-
tational efforts or inaccurate moment results. To increase
computational efficiency and ensure accuracy, an optimal
sample size should be determined adaptively for a specified
problem. One feasible way to achieve adaptive sample size
extension is to add samples sequentially until a specified
criterion is satisfied. Denote ˆ
Nto be the total number of
samples generated by the adaptive sample size extension,
and denote 𝜔(𝑘)and u(𝑘)to be the 𝑘-th related weight and
sample, where 𝑘=1, ..., ˆ
N. The corresponding 𝑟-order
fractional moments can be written as:
ˆ
𝑀𝑟
Z=
ˆ
N

𝑘=1
𝜔(𝑘)·𝐺𝑟u(𝑘)(11)
Actually, one recently developed sequential sampling
technique, termed the refined Latinized stratified sampling
(RLSS) [15], is suitable for our purpose of sample size ex-
tension. Besides, RLSS is applicable to any dimensional
problems due to its ability of space filling and effective vari-
ance reduction caused by both main or additive effects of
transformations and variable interactions [15]. Therefore,
RLSS is applied to evaluate fractional moments herein. The
corresponding estimator is given as:
ˆ
𝑀𝑟
Z=
ˆ
N

𝑘=1
𝜔(𝑘)·𝐺𝑟Γˆ
𝝑(𝑘) (12)
where 𝜔(𝑘)and ˆ
𝝑(𝑘)are the 𝑘-th weight and sample of
RLSS; Γ(·)is the isoprobabilistic transformation operator
that transforms the 𝑘-th sample u(𝑘)in the original variable
space to ˆ
𝝑(𝑘)in the [0,1]𝑠space.
Briefly, RLSS proceeds as follows. Denote D𝑠be
a𝑠-dimensional hypercube space [0,1]𝑠. First, de-
fine a Latinized stratified sampling (LSS) design of size
Nwith samples 𝝑(𝑘1)and strata Ω(𝑘1), 𝑘1=1, ..., N,
by connecting each component sample of Lain hyper-
cube sampling (LHS) with a stratified sampling (SS) de-
sign. The mutually exclusive and exhaustive LSS stratum
Ω(𝑘1)are produced by dividing D𝑠into Nstrata evenly,
which coincides with relative LHS one-dimensional stra-
tum Ω𝑖 𝑗 , 𝑖 =1, ..., 𝑠;𝑗=1, ..., Nand is an equal-weighted
hyper-rectangle. By specifying the starting coordinates
near origin 𝜒(𝑘1)=n𝜒(𝑘1)
1, ..., 𝜒 (𝑘1)
𝑛𝑠oand the side length
𝜁(𝑘1)=n𝜁(𝑘1)
1, .. ., 𝜁 (𝑘1)
𝑠o, the strata weights are calculated
as
𝜔(𝑘1)=
𝑠
Ö
𝑖=1
𝜁(𝑘1)
𝑖(13)
where ÍN
𝑘1=1𝜔(𝑘1)=1. The LSS samples 𝝑(𝑘1)are formed
by filling each LHS sample component in corresponding
Ω(𝑘1)without replacement. Then, produce candidate sam-
ples and strata of RLSS based on Hierarchical Latinized
stratified sampling (HLSS) [15]. Briefly, first apply a 𝜉-
level sample expansion to 𝝑(𝑘1)on the basis of hierarchical
Latin hypercube sampling (HLHS) [15], where 𝜉⩾1. Af-
ter that, identify the strata Ω(𝑘1), 𝑘1=1, . .., Nwith largest
strata weights 𝜁⋆=max
𝑖n𝜁(𝑘1)
𝑖o, and refine them into 𝜉
equal substrata along the dimension with largest side length
𝜔⋆=max
𝑘1𝜔(𝑘1). Accordingly, the HLSS strata are
formed, denoted as Ω(𝑘2), 𝑘2=1, ..., N(𝜉+1). Identify
the substrata 𝛩(𝑘2)
𝑖=nΩ𝑖 𝑗 ∈h𝜒(𝑘2)
𝑖, 𝜒(𝑘2)
𝑖+𝜁(𝑘2)
𝑖iowhich
intersect with LHS strata Ω𝑖 𝑗 in every dimension, and then
count the total number of such substrata as 𝜚(𝑘2)
𝑖, 𝑖 =1, ..., 𝑠.
Calculate the minimum value of substrata number in each
𝑖-th dimension by ˜𝜚∗
𝑖=min
𝑘2n𝜚(𝑘2)
𝑖o. Then generate HLSS
samples ˜
𝝑(𝑘2), 𝑘2=1, ..., N(𝜉+1)inside specific 𝛩(𝑘2)
𝑖
with 𝜚(𝑘2)
𝑖=˜𝜚∗
𝑖following the criterion: for each dimension,
if ˜𝜚∗
𝑖=1, sample from Ω𝑖 𝑗 ; if ˜𝜚∗
𝑖>1, sample randomly
from HLHS candidates inside 𝛩(𝑘2)
𝑖. Repeat such sample
generation till all the dimensions are filled. Afterwards, set
the number of parallel computing cores o⩾1according
to user needs. The RLSS strata could be produced by ran-
domly apply ostrata refinement from HLSS design using
parallel calculating scheme. Accordingly, draw the corre-
sponding HLSS samples to form onumbers of RLSS sam-
ples ˆ
𝝑(𝑘), 𝑘 =1, ..., ˆ
N, and then compute the strata weights
by 𝜔(𝑘)=Î𝑠
𝑖=1𝜁(𝑘)
𝑖, 𝑘 =1, ..., ˆ
N. Subsequently, repeat
the RLSS strata refinement until satisfying the stopping
condition or a new candidate sample expansion of HLSS
design is required. It should be mentioned here that the
total number ˆ
Nof RLSS samples depends on the number
of RLSS refinement repetition and the number of required
HLSS sample expansion, which shows the adaptiveness to
different requirements. Denote the repetition number of
ostrata refinement as ℏ0when there is no need to extend
HLSS, and denote the number of HLSS expansion as ℏ. We
have the adaptive sample size as
ˆ
N=N(𝜉+1)ℏ−1+ℏ0×o(14)
where ℏ⩾1,ℏ0⩾1and ℏ0·o⩽N(𝜉+1)ℏ.
How to define one appropriate stopping condition of
RLSS method is of great importance, since it may directly
affect the generated sample size and the accuracy of frac-
tional moments by Eq.12. As known, without sufficient
sample size, the high order fractional moments, i.e., second-

order moment of Z,ˆ
𝑀2
Z, cannot be obtained robustly. To
ensure the accuracy of estimated fractional moments, we
therefore recommend one criterion by judging the robust-
ness of ˆ
𝑀2
Zevaluated by RLSS. To specify, the RLSS stops
when the coefficient of variation (COV) of ˆ
𝑀2
Zis less than
one prescribed small value 𝜖,
COV ˆ
𝑀2
Z< 𝜖 (15)
For RLSS, there is no analytical solution of COV nˆ
𝑀2
Zo
available. Alternatively, the bootstrap scheme [16] can be
adopted to obtain the approximated result of it in a conve-
nient way.
3.2. Proposed Mixture Distribution Model
Once the fractional moments are evaluated, the EVD of
Zcan then be represented by a proper distribution model
from the knowledge of its estimated fractional moments.
In order to obtain a precise first-passage probability, the
keypoint of EVD reconstruction lies on capturing not only
its main body, but also its tail information. Therefore, a
sufficiently flexible distribution model is needed to ensure
its applicability to various-shaped EVDs. Here, we pro-
pose a novel mixture distribution model, which mixes two
extended basic distributions with extra shape parameters.
Specifically, the Lognormal distribution and generalized
inverse Gaussian distribution are extended, both of which
are positively skewed distribution taking only positive real
values.
First, suppose that Zfollows the Lognormal distribution
with corresponding PDF 𝑓LD (𝑧). Perform 𝑋=Z1/𝜃to ex-
tend the Lognormal distribution by introducing one shape
parameter 𝜃 > 0. According to the principle of probability
conservation, the PDF of 𝑋can be derived via 𝑓ELD (𝑥)=
𝑓LD (𝑧)𝑑𝑧
𝑑𝑥 =𝑓LD 𝑥𝜃·𝜃·𝑥𝜃−1, while the fractional mo-
ments of 𝑋can be obtained by 𝐸[Z𝑟]=𝐸𝑋𝑟/𝜃. Thus,
the PDF of extended Lognormal distribution (ELD) with
positive 𝑋reads:
𝑓ELD (𝑥;𝛼, 𝛽, 𝜃)=𝜃
𝑥𝛽√2𝜋exp −(𝜃log (𝑥)−𝛼)2
2𝛽2(16)
where the location parameter 𝛼can be any real number and
scale parameter 𝛽 > 0. Accordingly, the 𝑟-th fractional
moments of 𝑋is:
𝐸𝑋𝑟
ELD=exp 𝑟 𝛼
𝜃+𝑟2𝛽2
2𝜃2(17)
Similarly, if Zfollows the generalized inverse Gaussian
distribution with corresponding PDF 𝑓GIGD (𝑧)and let 𝑋=
Z1/𝜂with shape parameter 𝜂 > 0, the extended generalized
inverse Gaussian distribution (EGIGD) will be produced
with PDF of 𝑋[17]:
𝑓EGIGD (𝑥;𝑐, 𝑑 , 𝜀, 𝜂)=(𝜀/𝑑)𝑐𝜂𝑥𝑐 𝜂−1
2𝐾𝑐(𝑑𝜀)exp −𝑑2𝑥−𝜂+𝜀2𝑥𝜂
2
(18)
and fractional moments of 𝑋:
𝐸𝑋𝑟
EGIGD=𝐾𝑐+𝑟/𝜂(𝑑𝜀)
𝐾𝑐(𝑑𝜀)𝑑
𝜀𝑟/𝜂
(19)
where 𝐾𝑐(·)denotes the modified Bessel function of second
kind with parameter 𝑐∈(−∞,+∞),𝑑 > 0and 𝜀 > 0.
Then, ELD and EGIGD are mixed together via intro-
ducing a weight parameter 𝑤∈[0,1]to form a mixture
distribution of ELD and EGIGD (M-ELD-EGIGD) with
eight parameters 𝛬={𝑤, 𝛼, 𝛽, 𝜃, 𝑐, 𝑑, 𝜀, 𝜂}such that
𝑓M−ELD−EGIGD (𝑥;𝛬)
=𝑤 𝑓ELD (𝑥;𝛼, 𝛽, 𝜃)+(1−𝑤)𝑓EGIGD (𝑥;𝑐, 𝑑, 𝜀, 𝜂)
=𝑤Φ𝜃log(𝑥)−𝛼
𝛽
+(1−𝑤)(𝜀/𝑑)𝑐𝜂𝑥𝑐 𝜂−1
2𝐾𝑐(𝑑 𝜀)exp −𝑑2𝑥−𝜂+𝜀2𝑥𝜂
2,with 𝑥 > 0
(20)
The corresponding fractional moments of M-ELD-EGIGD
is:
𝐸𝑋𝑟
M−ELD−EGIGD;𝛬=𝑤 𝐸 𝑋𝑟
ELD+(1−𝑤)𝐸𝑋𝑟
EGIGD
=𝑤exp 𝑟 𝛼
𝜃+𝑟2𝛽2
2𝜃2+(1−𝑤)𝐾𝑐+𝑟/𝜂(𝑑 𝜀)
𝐾𝑐(𝑑 𝜀)𝑑
𝜀𝑟/𝜂
(21)
Generally, fractional moment matching scheme could be
utilized to determine the free parameters 𝛬of M-ELD-
EGIGD. Using a nonlinear least squares solver lsqnonlin
in Matlab, the unknown parameters are obtained by solv-
ing a system of eight nonlinear equations involving eight
fractional moments:
ˆ
𝑀ˆ𝒓
Z=𝐸𝑋ˆ𝒓
M−ELD−EIGD;𝛬(22)
where ˆ𝒓=1
8×[1,2, ..., 8]. Usually, the initial solu-
tion of Eq.22 could be derived from first- and second-
order central moments of Z(𝜇Z=ˆ
𝑀1
Zand 𝜎Z=
ˆ
𝑀2
Z−ˆ
𝑀1
Z2
) such that 𝛼0=log 𝜇2
Z/𝜎2
Z+𝜇2
Z,
𝛽0=log 𝜎2
Z/𝜇2
Z+1,𝜃0=1,𝑐0=−0.5,𝑑0=
𝜇3
Z/𝜎2
Z,𝜀0=𝜇Z/𝜎2
Z,𝜂0=1and 𝑤0=0.5. However,
if such initial values are directly used for solving Eq.22, we
may obtain a locally optimized solution that results in in-
accurate EVD. To avoid such problem, an improved initial
solution ˆ
𝛬0=𝑤0,ˆ𝑎0,ˆ
𝑏0,ˆ
𝜃0,ˆ𝑐0,ˆ
𝑑0,ˆ𝜀0,ˆ𝜂0is proposed by
using another low-order fractional moment matching algo-
rithm such that










ˆ
𝑀1/2
Z=𝐸h𝑋1/2
ELD; ˆ𝑎0,ˆ
𝑏0,ˆ
𝜃0i
ˆ
𝑀1
Z=𝐸𝑋1
ELD; ˆ𝑎0,ˆ
𝑏0,ˆ
𝜃0
ˆ
𝑀3/2
Z=𝐸h𝑋3/2
ELD; ˆ𝑎0,ˆ
𝑏0,ˆ
𝜃0i(23)














ˆ
𝑀1/2
Z=𝐸h𝑋1/2
EGIGD; ˆ𝑐0,ˆ
𝑑0,ˆ𝜀0,ˆ𝜂0i
ˆ
𝑀1
Z=𝐸𝑋1
EGIGD; ˆ𝑐0,ˆ
𝑑0,ˆ𝜀0,ˆ𝜂0
ˆ
𝑀3/2
Z=𝐸h𝑋3/2
EGIGD; ˆ𝑐0,ˆ
𝑑0,ˆ𝜀0,ˆ𝜂0i
ˆ
𝑀2
Z=𝐸𝑋2
EGIGD; ˆ𝑐0,ˆ
𝑑0,ˆ𝜀0,ˆ𝜂0
(24)

where 𝛬0=[𝑤0, 𝑎0, 𝑏0, 𝜃 0, 𝑐0, 𝑑0, 𝜀0, 𝜂0]is adopted as the
original initial value to solve nonlinear equations in Eq.23
and 24. In this case, such improved initial value can help
the nonlinear equation solution to be closer to the global
optimization value, so that the EVD of Zcould be recon-
structed in a more accurate way.
After obtaining the recovered EVD, the first-passage
probability could be assessed by Eq.7 via defining the
threshold 𝑧𝑏.
3.3. Implementation Procedure
In order to facilitate the numerical implementation of the
proposed method, a brief step-by-step procedure is listed as
follows:
Step 1: Set the value of 𝜉,oand 𝜖. Generate a LSS
design with Ninitial samples and strata. Compute the
initial strata weights by Eq.13.
Step 2: According to RLSS technique, generate candi-
date samples and strata of RLSS by HLSS scheme and then
sequentially produce oRLSS samples and corresponding
strata based on HLSS and LSS initial design with paral-
lel computing scheme. Calculate the strata weights 𝜔(𝑘)
of RLSS samples ˆ
𝝑(𝑘), 𝑘 =1, ..., ˆ
N. Then transform the
RLSS samples to original variable space to form extreme
value Zand compute ˆ
𝑀2
Z.
Step 3: Check whether the stopping condition (Eq.15)
is achieved or not. If satisfied, the RLSS process stops.
Otherwise, repeat generating new oRLSS samples to meet
the stopping condition or until a new round of candidate
sample extension of RLSS is required.
Step 4: Evaluate a set of r-order fractional mo-
ments of Zwith RLSS samples that satisfy the con-
vergence condition according to Eq.12, where r=
{1/8,1/4,3/8,1/2,5/8,3/4,7/8,1,3/2,2}.
Step 5: Calculate the initial values of Eq.22 (𝛬0=
[𝛼0, 𝛽0, 𝜃0, 𝑐0, 𝑑0, 𝜖0, 𝜂0]) based on sample moments of
Z(i.e., 𝜇Z=ˆ
𝑀1
Zand 𝜎Z=ˆ
𝑀2
Z−ˆ
𝑀1
Z2
).
Then estimate the improved initial values ˆ
𝛬0=
𝑤0,ˆ𝑎0,ˆ
𝑏0,ˆ
𝜃0,ˆ𝑐0,ˆ
𝑑0,ˆ𝜀0,ˆ𝜂0by Eqs.23 and 24, where
𝑤0=0.5.
Step 6: Reconstruct the EVD of Zby proposed M-
ELD-EGIGD, where the involved eight free parameters are
estimated by low-order fractional moment matching scheme
(Eq.22) with proposed improved initial value ˆ
𝛬0.
Step 7: Give a prescribed threshold 𝑧𝑏and then assess
the first-passage probability of Zbased on Eq.7.
4. Numerical Example
To demonstrate the efficacy of the proposed method, one
numerical example considering a Duffing oscillator with
random system parameters subjected to Gaussian white
noise is given:
¥
𝑌(𝑡)+𝛾¤
𝑌(𝑡)+𝑌(𝑡)+𝜍𝑌 3(𝑡)=𝒢(𝑡)(25)
where ¥
𝑌,¤
𝑌,𝑌are acceleration, velocity and displacement
of the Duffing oscillator at time 𝑡, respectively; 𝛾is the pa-
rameter controlling damping; 𝜍is the non-linearity control
parameter of restoring force; 𝒢is the force causing by a ran-
dom Gaussian white noise at time 𝑡. In this example, both 𝛾
and 𝜍are considered as random system parameters follow-
ing the Lognormal distribution, where their mean values
take 0.5 and 0.3, and standard deviation values take 0.2 and
0.1, respectively.
The driven force 𝒢(𝑡)could be described as:
𝒢(𝑡𝑛)=𝜓(𝑡𝑛)2𝜋𝑆/𝛥𝑡 (26)
where 𝛥𝑡 denotes the time interval within the time period
𝑇=30𝑠; the discrete time is 𝑡𝑛=𝑛 𝛥𝑡, 𝑛 =1, ..., 𝑛𝑡, and
𝑛𝑡=𝑇/𝛥𝑡 +1=1501; a set of 𝜓(𝑡𝑛)are the stochastic vari-
ables considered in the driven force, following the normal
distributions with zero means and standard deviation values
as 1; and the spectral intensity of the stochastic Gaussian
white noise takes 𝑆=1. In this regard, a total number of
1503 random variables are involved in this example.
Here, we are interested in the maximum absolute ex-
treme value of displacement over the whole time period,
i.e., Z=max𝑡{|𝑌(𝑡)| , 𝑡 ∈(0, 𝑇 ]}. First, set 𝜉=1,o=4
and 𝜖=0.015. Based on RLSS technique, 576 RLSS sam-
ples and relative weights are generated, which satisfy the
stopping condition in Eq.15. Then the r-order fractional
moments of Zare calculated by Eq.12. To illustrate the ac-
curacy RLSS can provide, the first-two central moments by
proposed method are compared with the benchmark results
by Monte Carlo simulation (MCS) (106runs) in Tab. 1. As
Table 1. First-two central moments by proposed method and MCS
Method( ˆ
N)𝜇Z𝜎Z
Proposed(576) 3.6625 0.6629
MCS(106) 3.6565 0.6627
R.E. 0.1641% 0.0302%
Note: R.E. = Relative error of results to be compared.
seen, both the 𝜇Zand 𝜎Zby proposed methods are close
to the MCS results, where the largest relative error of these
compared results is only 0.1641%.
Once we obtain accurate fractional moments, the EVD
of Zcan be recovered by proposed M-ELD-EGIGD model.
The curves of PDF, probability of exceedance (POE) and
CDF of Zare depicted in Fig. 1-3, together with those
by MCS. As observed, PDF and CDF curves by proposed
method accord pretty well with those from MCS. It is ob-
served that since MCS is only performed with 106simula-
tion times, some fluctuations exist at the tail end of POE
curve by MCS. However, apart from the fluctuation part of
MCS result, the tail distribution of POE by the proposed
method is consistent with that by MCS.
Further, giving the threshold of Zas 𝑧𝑏=7.6, the first-
passage probability of Zassessed from Eq.7 is provided
in Tab.2. As observed, result by proposed method is quite
close the MCS result. Therefore, the proposed method is
testified to be both accurate and efficient in this example.
5. Conclusion
In many fields, the first-passage problem of stochastic
dynamical systems is a common but not easy to solve prob-

0 2 4 6 8 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
MCS
Proposed method
Figure 1. PDF of Z
0 2 4 6 8 10
10-8
10-6
10-4
10-2
100
MCS
Proposed method
Figure 2. POE of Z
0 2 4 6 8 10
10-8
10-6
10-4
10-2
100
MCS
Proposed method
Figure 3. CDF of Z
Table 2. First-passage probabilities by proposed method and MCS
Method MCS Proposed method
ˆ
N106576
𝑃𝑓1.0000 ×10−51.0291 ×10−5
lem. The first-passage probability can be evaluated based on
the extreme value distribution (EVD) of interested system
response, where moments-based EVD reconstruction meth-
ods are drawing increasing attention. However, how to de-
termine an optimal sample size to obtain accurate fractional
moments and how to reconstruct the EVD with sufficient
accuracy based on obtained moments are still challenging.
Therefore, in this contribution, we propose a parallel adap-
tive sampling scheme to estimate fractional moments, and a
flexible mixture distribution model to reconstruct the EVD
from estimated moments. To be specified, a sequential sam-
pling technique is proposed to generate samples and corre-
sponding weights adaptively and parallelly until the accu-
racy of estimated fractional moment is satisfied. After that,
a flexible mixture distribution model is developed, which
mixes two proposed extended Lognormal distribution and
generalized inverse Gaussian distribution. The eight free
parameters involved in the proposed distribution model can
be evaluated by two low-order fractional moment match-
ing schemes. Finally, the first-passage probability can be
easily assessed from the estimated EVD given a prescribed
threshold. One numerical example considering a Duffing
oscillator with random system parameters under stochastic
Gaussian white noise is investigated to illustrate the efficacy
of the proposed method. The results show that the proposed
method is applicable to assess small first-passage proba-
bility of a high-dimensional and high nonlinear stochastic
dynamic system.
Acknowledgement
This paper is partially funded by European Union’s Hori-
zon 2020 research and innovation programme under Marie
Sklodowska-Curie project GREYDIENT – Grant Agree-
ment n°955393 and China Scholarship Council (CSC). The
authors would like to gratefully acknowledge the support.
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