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Efficient Estimation of Extreme Quantiles using Adaptive Kriging and Importance Sampling

Nassim Razaaly 1 Daan Crommelin 2 Pietro Marco Congedo 3
1 DeFI - Shape reconstruction and identification
Inria Saclay - Ile de France, CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
3 PLATON - Uncertainty Quantification in Scientific Computing and Engineering
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
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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https://hal.inria.fr/hal-02436230
Contributor : Pietro Marco Congedo <>
Submitted on : Sunday, January 12, 2020 - 7:08:35 PM
Last modification on : Wednesday, March 24, 2021 - 1:58:08 PM

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Nassim Razaaly, Daan Crommelin, Pietro Marco Congedo. Efficient Estimation of Extreme Quantiles using Adaptive Kriging and Importance Sampling. International Journal for Numerical Methods in Engineering, Wiley, 2020, 121 (9), pp.2086-2105. ⟨10.1002/nme.6300⟩. ⟨hal-02436230⟩

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