A recursive dimension-reduction method for high-dimensional reliability analysis with rare failure event
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: where is the number of inputs, is an -vector containing the input parameters of the function, is the scalar QOI that depends on the inputs. The separation of variables method assume that a low-rank structure is exists as where is much smaller than . When can be decomposed by some basis functions, then the approximation performs well if many 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 continuous inputs, with a bounded and continuous probability density function: Assume is a continuous and differentiable function, so that the gradient of exists as, Then, a matrix can be subsequently deduced as which 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 , where 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.
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