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.

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Keywords Extreme quantile, Importance sampling, Kriging, Multiple failure regions, Quantile, Rare event, Tail probability
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Journal International Journal for Numerical Methods in Engineering
Razaaly, N, Crommelin, D.T, & Congedo, P.M. (2019). Efficient estimation of extreme quantiles using adaptive kriging and importance sampling. International Journal for Numerical Methods in Engineering. doi:10.1002/nme.6300