Efficient estimation of extreme quantiles using adaptive kriging and importance sampling |
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Authors: | Nassim Razaaly Daan Crommelin Pietro Marco Congedo |
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Affiliation: | 1. DeFI Team, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France, Palaiseau, France;2. CWI Amsterdam, Amsterdam, the Netherlands
Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, the Netherlands |
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Abstract: | 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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