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Metamodel-based subset simulation adaptable to target computational capacities: the case for high-dimensional and rare event reliability analysis

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Abstract

Metamodel-based approaches to reliability analysis, e.g., adaptive Kriging, are computationally challenged by the complexity of reliability problems, thus limiting the application of these methods to problems that are low-dimensional or not rare. Here, we propose a reliability analysis approach via a deep integration of subset simulation and adaptive kriging (RASA) for an unbiased estimation of failure probabilities of high-dimensional or rare event problems. Concepts of conditional failure probability curves and dynamic learning function are introduced to decompose the original problem to subreliability problems and adaptively identify intermediate failure thresholds of limit state functions corresponding to the subreliability problems. The reliability decomposition and the establishment of target intermediate failure thresholds are guided by the available computational capacity, thus, enabling RASA to control the computational cost associated with the estimation of the intermediate failure thresholds in each subset and consequently to analyze the reliability of medium to high-dimensional problems or rare events. Three numerical examples are investigated as benchmark to explore the performance of the proposed method. Results indicate that the proposed method has high accuracy and has the ability to adjust to available computational resources.

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Abbreviations

:

Function of the computational capacity

C r :

Required computational storage

\( {COV}_{{\hat{P}}_f^{mcs}} \) :

Coefficient of variation of \( {\hat{P}}_f^{mcs} \)

\( {COV}_{{\hat{P}}_f^{ss}} \) :

Coefficient of variation of \( {\hat{P}}_f^{ss} \)

COV thr :

Coefficient of variation of \( {\hat{P}}_f^{mcs} \)

f :

The Kriging basis function

F :

The vector of f

g :

True limit state function

\( \hat{g} \) :

Estimated limit state function

g i :

True sub-limit state function in SS

\( {\hat{g}}_i \) :

Estimated sub-limit state function in SS

I g :

Indicator based on g

\( {I}_{\hat{g}}^{si} \) :

Indicator based on \( \hat{g} \)

m :

Number of samples for Kriging training

n ss :

Number of subsets

N cds :

Number of candidate design samples for any algorithm

N d :

Number of the dimension of random variables

N MCS :

Number of samples for MCS

N ss :

Number of candidate design samples in SS

\( {N}_{ss}^{max} \) :

Maximum number of candidate design samples in Kriging-based SS

:

Parameter sets for Kriging surrogate model

p 0 :

Intermediate probability of failure

\( {p}_0^{\ast } \) :

Optimal intermediate probability of failure

P f :

Probability of failure (ground truth)

\( {\hat{P}}_f^{di} \) :

Probability of failure estimated with deterministic indicator

\( {P}_f^{mcs} \) :

Probability of failure estimated through MCS

\( {\hat{P}}_f^{mcs} \) :

Probability of failure estimated through Kriging-based MCS

\( {\hat{P}}_f^{si} \) :

Probability of failure estimated with stochastic indicator

\( {P}_f^{ss} \) :

Probability of failure estimated with SS

\( {\hat{P}}_f^{ss} \) :

Probability of failure estimated through Kriging-based SS

p k :

Intermediate failure probability for SS

\( {\hat{p}}_k \) :

Intermediate failure probability for Kriging-based SS

r :

Correlation vector

R :

The Kriging correlation function between two points

S :

Pool of generated samples in Ωf

t i :

True intermediate failure thresholds

\( {\hat{t}}_{\boldsymbol{i}} \) :

Estimated intermediate failure thresholds

x :

The vector of random variables

x tr :

Training samples

\( {\boldsymbol{x}}_{tr}^{\ast } \) :

Next best training samples

Y :

Vector of the responses from Kriging

Z :

The Gaussian process

β :

The vector of coefficients for Kriging basis

Γ thr :

Stopping criterion threshold

θ :

The vector of hyperparameters for Kriging

Θ t :

pdf of identified intermediate failure thresholds

\( {\mu}_{\hat{g}} \) :

The mean value of Kriging responses

ρ :

Probability density function (pdf)

σ 2 :

Variance of Gaussian process

\( {\sigma}_{\hat{g}}^2 \) :

The variance of Kriging responses

φ :

pdf of Gaussian distribution

Φ:

cdf of Gaussian distribution

Ω:

Probabilistic domain of x

Ωf :

Failure domain of Ω

Ω i :

The ith subsets in subset simulation

Ωs :

Failure domain of Ω

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Acknowledgements

This research was supported in part by the U.S. National Science Foundation (NSF) through awards CMMI-1762918 and 2000156, as well as ‘Shuimu Tsinghua Scholar’ Plan by Tsinghua University. The first author also thanks Prof. Quanwang Li for the support of postdoctoral research at Tsinghua university. Any opinions, findings, and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of these supports. These supports are greatly appreciated.

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Authors and Affiliations

  1. Department of Civil Engineering, Tsinghua University, Beijing, 100084, China

    Zeyu Wang

  2. The Ohio State University, Columbus, OH, 43210, USA

    Zeyu Wang & Abdollah Shafieezadeh

Authors
  1. Zeyu Wang
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  2. Abdollah Shafieezadeh
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Corresponding author

Correspondence to Abdollah Shafieezadeh.

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The authors declare that they have no conflict of interest.

Replication of results

The algorithms and step-by-step implementation approach for the proposed reliability analysis with subset simulation using adaptive Kriging (RASA) are presented in Algorithms 1 to 5 in the paper. Readers can use MATLAB and UQLab Kriging package to implement the algorithms and generate the results presented in the paper.

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Wang, Z., Shafieezadeh, A. Metamodel-based subset simulation adaptable to target computational capacities: the case for high-dimensional and rare event reliability analysis. Struct Multidisc Optim 64, 649–675 (2021). https://doi.org/10.1007/s00158-021-02864-9

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  • DOI: https://doi.org/10.1007/s00158-021-02864-9

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