A recursive dimension-reduction method for high-dimensional reliability analysis with rare failure event

https://doi.org/10.1016/j.ress.2021.107710Get rights and content

Highlights

  • A recursive reduction procedure is introduced for reliability problem with rare event.

  • Active subspace method is employed to solve high-dimensional problem.

  • Kriging method is used to fit the data into a robust low-dimensional surrogate model.

Abstract

A new dimension-reduction method (DRM), called ’subset active subspace method (SASM)’, is proposed to compute small failure probabilities encountered in high-dimensional reliability analysis of engineering systems. The basic idea is to introduce a recursive procedure to improve the efficiency, accuracy and applicability of the conventional active subspace method (ASM). For the reliability problems with a rare event, SASM firstly transfers the original high-dimensional reliability problem into a low-dimensional reliability problem in a proper failure domain. Then, a simplified low-dimensional surrogate model is built in order to improve the result of reliability analysis by increasing significantly the samples with a minimum additional computational effort. The proposed method is verified by three nonlinear numerical examples, including theoretical and industrial, explicit and implicit performance functions. Besides, some other existing methods are also investigated and compared to the proposed method. It is found that the proposed method can keep the trade-off between accuracy and efficiency.

Introduction

With the increasing attention for nonlinear stochastic analysis, the existence of randomness in both external excitation and system properties is now widely recognised by scientists and engineers. In many practical uncertainty quantification problems [1], [2], [3], [4], the difficulties of reliability analysis are simultaneously related to high-dimensionality and small failure probability. Plenty of methods have been proposed to resolve the reliability problem such as Monte Carlo Simulation (MCS) [5], the random perturbation method [6], surrogate model methods [7], subset simulation (SS) [8], and Probability Density Evolution Method (PDEM), etc. The MCS is one of the most commonly used methods in reliability analysis. However, despite the universality of MCS, its numerical efficiency, especially the sampling efficiency declines rapidly with the growth of the problem’s dimension [9]. The orthogonal polynomials expansion method has no secular terms problem arising and therefore is particularly of value in dynamic response analysis, achieves a somewhat relative balance between accuracy and efficiency [10]. By using the probability conservation principle with a new strategy of point selection, PDEM represents a new method to reach the balance between computational efficiency and accuracy. However, PDEM encounters substantial difficulties when the failure probability is small [11]. The surrogate model methods have been widely investigated in recent years, many surrogate model methods (e.g., Neural Networks [12], Support Vector Machines [13], Kriging [10], [14], [15], [16]) have been introduced to improve the efficiency of the reliability analysis. However, most surrogate model methods are haunted by the ”curse of dimensionality” problem, which means that the computational consumption of building a reliable model and the predicting error rise sharply with the growth of the problem’s dimension [17]. In recent years, the combination of active learning Gaussian process model combined with the advanced Monte Carlo simulation has witnessed rapid development [18], [19], and they are proven to be more and more powerful for fixing the challenge of high-dimensional and small failure probability problems.

Employing dimension reduction methods (DRMs) is an attractive option to resolve the high-dimensional problem. A lot of DRM methods have been proposed to build the low-dimensional approximation of the function of quantity of interest (QOI), such as sensitivity analysis [20], high dimensional model representation (HDMR) [21], analysis of variance (ANOVA) [22], separation of variables and polynomial chaos expansions (PCE) [23], [24], etc. However, all the above model-based DRMs require strong hypotheses which can hardly be satisfied in practical engineering problems. For example, sparse grids interpolation assume that the high-dimensional performance function can be approximated by the summation of univariate functions as follows: h(x)h1(x)++hmxmwhere m is the number of inputs, x is an m-vector containing the input parameters of the function, h(x) is the scalar QOI that depends on the inputs. The separation of variables method assume that a low-rank structure is exists as h(x)i=1rhi,1(x)hi,mxmwhere r is much smaller than m. When h can be decomposed by some basis functions, then the approximation h(x)i=1naiϕi(x)performs well if many ai are close to zero.

c called active subspace method (ASM) [25] has gained increasing attention due to its data-based property. By exploiting the existing data, ASM can project the original high-dimensional samples into a lower input space. Then, the accuracy and efficiency can be improved for either optimisation, integration, or regression problems. By far ASM has already been widely used in many fields such as geoscience [26], aerospace engineering [27], civil engineering [28], and electronic engineering [29]. However, one major problem of applying ASM is that when the performance function does not accept a latent low-dimensional active subspace, the errors caused by the dimension reduction procedure are nonnegligible [30]. In this regard, improvements to the applicability and accuracy of the original ASM could be of great necessity.

In this paper, a new DRM, which can keep the trade-off between accuracy and applicability, is proposed by employing a recursive procedure to locate a support domain. Hence, the original irreducible high-dimensional performance function becomes reducible for reliability analysis in this support domain. It is found that the proposed method is more accurate and efficient than the conventional DRM. Besides, for those problems which the conventional ASM cannot work properly, the proposed method can deal with them more accurately and robustly. Once the low-dimensional subspace is acquired, a versatile surrogate model such as a kriging model [31] (as shown in Appendix C), whose estimation is unbiased and honours the actually observed value, is fitted to help the evaluation of reliability with a rare event. The rest of the paper is organised as follows. In Section 2, the concepts and numerical implementations of the conventional ASM are generically described. Section 3 presents the details of the proposed method. In Section 4, various numerical examples are investigated to verify the proposed method for high-dimensional and small-failure-probability reliability problems. Concluding remarks and the problems needed to be further studied are summarised in Section 5.

Section snippets

Active subspace method

Considering a multivariable performance function with dh continuous inputs, h=hx,xχ=1,1dhRdhwith a bounded and continuous probability density function: ρ(x)>0,xχ and ρ(x)=0,xχAssume h is a continuous and differentiable function, so that the gradient of h exists as, xhx=hx1,,hxdhTThen, a dh×dh matrix G can be subsequently deduced as G=xhxhTfdxwhich represents the uncentred covariance of the gradient vector.

The numerical evaluation of high-dimensional integrals in Eq. (7) can be

Subset simulation

The subset simulation method [42], [43], [44], as a recursive simulation method, is of great significance for the reliability analysis of rare failure events [8]. By introducing the concept of an intermediate failure event (IFE), subset simulation separates the original probability space into a sequence of subsets, and then the small failure probability can be expressed as a product of larger conditional failure probabilities.

Consider a rare failure event F=x :gx>b0, where gx is the limit state

Numerical examples

In this section, the performance of the SASM is studied by three different examples. The comparison of the results (PDF and CDF, reliability probability, and number of samples) of various methods (original active subspace method, sensitivity method, surrogate methods, Monte Carlo Simulation and some hybrid methods) is studied in the examples.

Concluding remarks

This paper presents a new supervised dimension-reduction method (SASM) for estimating the high dimensional reliability problem with a rare failure event. In this method, the procedure of subset simulation is first proposed to find a local domain in which the active subspace method can be applied effectively. Then, the kriging method is utilised to fit the dimension-reduced data into a robust low-dimensional nonlinear model, which is highly efficient to estimate the failure probability with a

CRediT authorship contribution statement

Zhong-ming Jiang: Conceptualization, Methodology, Software, Data curation, Writing - original draft, Writing - review & editing. De-Cheng Feng: Validation. Hao Zhou: Visualization. Wei-Feng Tao: Investigation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The research reported in this paper was supported by the National Natural Science Foundation of China (51908468), the Fundamental Research Funds for the Central Universities (5330500367) and the Natural Science Foundation of Chongqing (4312000227). The support is gratefully appreciated.

References (55)

  • SuG. et al.

    A Gaussian process-based dynamic surrogate model for complex engineering structural reliability analysis

    Struct Saf

    (2017)
  • SchueremansL. et al.

    Benefit of splines and neural networks in simulation based structural reliability analysis

    Struct Saf

    (2005)
  • PapadopoulosV. et al.

    Accelerated subset simulation with neural networks for reliability analysis

    Comput Methods Appl Mech Engrg

    (2012)
  • YuanZ. et al.

    Parameter selection for model updating with global sensitivity analysis

    Mech Syst Signal Process

    (2019)
  • AuS.K. et al.

    Estimation of small failure probabilities in high dimensions by subset simulation

    Probab Eng Mech

    (2001)
  • SchuëllerG.I. et al.

    A critical appraisal of reliability estimation procedures for high dimensions

    Probab Eng Mech

    (2004)
  • KamińskiM.

    The stochastic perturbation method for computational mechanics

    (2013)
  • DubourgV. et al.

    Metamodel-based importance sampling for structural reliability analysis

    Probab Eng Mech

    (2013)
  • ZuevK.

    Subset simulation method for rare event estimation: An introduction

    (2015)
  • WalterJ.C. et al.

    An introduction to Monte Carlo methods

    Physica A

    (2015)
  • LiuZ. et al.

    Surrogate modeling based on resampled polynomial chaos expansions

    Reliab Eng Syst Saf

    (2020)
  • XuJ. et al.

    Extreme value distribution and small failure probabilities estimation of structures subjected to non-stationary stochastic seismic excitations

    Struct Saf

    (2018)
  • PapadrakakisM. et al.

    Structural reliability analyis of elastic-plastic structures using neural networks and Monte Carlo simulation

    Comput Methods Appl Mech Engrg

    (1996)
  • EchardB. et al.

    AK-MCS: An active learning reliability method combining Kriging and Monte Carlo simulation

    Struct Saf

    (2011)
  • WangJ.T. et al.

    Frequency response function-based model updating using Kriging model

    Mech Syst Signal Process

    (2017)
  • KersaudyP. et al.

    A new surrogate modeling technique combining Kriging and polynomial chaos expansions - Application to uncertainty analysis in computational dosimetry

    J Comput Phys

    (2015)
  • GunstR.F. et al.

    Response surface methodology: Process and product optimization using designed experiments

    Technometrics

    (1996)
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