A new reliability method combining adaptive Kriging and active variance reduction using multiple importance sampling

Abstract

This article describes a new adaptive Kriging method combined with adaptive importance sampling approximating the optimal auxiliary by iteratively building a Gaussian mixture distribution. The aim is to iteratively reduce both the modeling and sampling errors simultaneously, thus avoiding limitations in cases of very rare failure events. At each iteration, a near optimal auxiliary Gaussian distribution is defined and new samples are drawn from it following the scheme of adaptive multiple importance sampling (MIS). The corresponding estimator is provided as well as its variance. A new learning function is developed as a generalization of the U learning function for MIS populations. A stopping criterion is proposed based on both the modeling error and the variance of the estimator. Results on benchmark problems show that the method exhibits very good performances on both efficiency and accuracy.

Appendices

Appendix 1: Discussion of the p hyperparameter

As mentioned in Sect. 2.2 the choice of the p hyperparameter is at least partly arbitrary since it does not directly relate to an interpretable performance metric. In order to choose an adequate value one can only rely on empirical evidences. A good basis for such evidence is to study the sensitivity of the performances with respect to the parameter. The results of such a study are presented in this Appendix where the method is applied on three analytical examples (examples one and three from Sects. 3.1 and 3.2 and example four from Appendix 2) with varying values of p.

Thirty values of p are tested in the interval $p\in [1, 6]$. For each, the method is applied ten times to observe the variation of performances (the same ten random number regenerator seeds are used for each value of p). The results are illustrated in Figs. 14 and 15 presenting respectively the mean and amplitude of number of calls to the performance function and relative error.

Fig. 14

Average and extrema of the number of calls to the performance function as a function of the p hyperparameter for a example 1 (case 2), b example 3, and c example 4 (see) Appendix 2

Fig. 15

Average and extrema of the failure probability error as a function of the p hyperparameter for a example 1 (case 2), b example 3, and c example 4 (see) Appendix 2

It can be observed from Fig. 14 that, in terms of computation time, there is a tendency for very high values of p to perform poorly, especially for the first two examples (a) and (b). For p between one and four no strong tendency is observable for the last two examples (b) and (c). For the first example (a) there seem to be an optimal value between $p=3$ and $p=4$.

In Fig. 15 it can be observed that for the first example (a) small values of p induce a higher variance of performance. In contrast, on the second example (b), the same tendency is observed for high values of p. No strong tendency can be observed from example three (c). From Fig. 15a the optimal value seem to be between $p=3$ and $p=5$, while for Fig. 15b the optimal value seem to be between $p=1$ and $p=3$

From these results, the choice of $p=3$ as proposed in Sect. 2.2 appears as a reasonable compromise providing satisfying performances and computation time on all tested examples.

Appendix 2: Additional performance comparison

In addition to the four examples presented in Sect. 3 the performances of the proposed approach are compared to two other relevant methods form the literature, i.e., AK-ALIS (Liu et al. 2020) and AKOIS (Zhang et al. 2020). The study is limited to the analytical example function presented in the respective articles and the reference results are reported as published. The performances of the proposed AK-AMIS method are studied as described in Sect. 3 and a hundred application have been performed on each application.

Overall the proposed AK-AMIS method performs very well on these four additional examples and reaches satisfying precision on all of them. AK-AMIS converges around 25% quicker than AK-ALIS while consistently staying below 2.4% error (Table 7). On examples six and seven AK-AMIS successfully converges but requires about 25% more evaluations of the performance function than AK-OIS (Tables 8 and 9). On example 8 (Table 10) AK-AMIS converges around 50% faster than AK-OIS while inducing a slightly higher error of about 2%.

Table 7 Results for application 5

Table 8 Results for application 6

Table 9 Results for application 7

Table 10 Results for application 8A

Case 5: An analytical function from Liu et al. (2020),

Case 6: A bivariate example (Zhang et al. 2020),

Case 7: A polynomial function with multiple MPPs (Zhang et al. 2020),

Case 8: A polynomial function with multiple MPPs (Zhang et al. 2020).