Efficient estimation of extreme quantiles using adaptive kriging and importance sampling

Abstract

SUMMARY This study considers an efficient method for the estimation of quantiles associated to very small levels of probability (up to O(10−9)), where the scalar performance function J is complex (eg, output of an expensive‐to‐run finite element model), under a probability measure that can be recast as a multivariate standard Gaussian law using an isoprobabilistic transformation. A surrogate‐based approach (Gaussian Processes) combined with adaptive experimental designs allows to iteratively increase the accuracy of the surrogate while keeping the overall number of J evaluations low. Direct use of Monte‐Carlo simulation even on the surrogate model being too expensive, the key idea consists in using an importance sampling method based on an isotropic‐centered Gaussian with large standard deviation permitting a cheap estimation of small quantiles based on the surrogate model. Similar to AK‐MCS as presented in the work of Schöbi et al., (2016), the surrogate is adaptively refined using a parallel infill criterion of an algorithm suitable for very small failure probability estimation. Additionally, a multi‐quantile selection approach is developed, allowing to further exploit high‐performance computing architectures. We illustrate the performances of the proposed method on several two to eight‐dimensional cases. Accurate results are obtained with less than 100 evaluations of J on the considered benchmark cases.

Direct Quantile Function Estimation Using Information Principles and Its Applications in Reliability Analysis

“…Schöbi and Sudret [15] proposed a PC-Kriging-based meta-modeling method to estimate quantiles. Adaptive kriging and importance sampling were recently used in an efficient estimation of extreme quantiles [16].…”

Active learning method combining Kriging model and multimodal‐optimization‐based importance sampling for the estimation of small failure probability

“…25,26 However, the formerly mentioned strategies are not applicable to rare-event estimation where the failure probability is very small, like 10 −6 to 10 −9. [27][28][29] The reason is that a very large number of simulated samples are required by MCS for rare-event estimation. To obtain an optimal training point, predictions at all the simulated samples should be provided by the Kriging model, and hence, the whole learning process becomes very time consuming.…”

An efficient particle swarm optimization-joint probability distribution optimization method for structural reliability analysis

“…In recent years, extensive research has been carried out to develop the sampling process in the reliability analysis. [28][29][30][31][32][33] The purpose of this paper is to provide a new method for reliability analysis of engineering structures and mechanisms with the following characteristics:…”

“…In recent years, extensive research has been carried out to develop the sampling process in the reliability analysis. 28–33 The purpose of this paper is to provide a new method for reliability analysis of engineering structures and mechanisms with the following characteristics: Ability to solve highly nonlinear problems with implicit behavior. Accurate approximation of MPP. Simplicity and reduced complexity of the problem. Lack of need for transformation of random variables from design space to standard normal space. Lack of sensitivity to distribution type of random variables (probabilistic or uniform). …”