Uncertainty Quantification Seminar Nassim Razaaly (INRIA)

An efficient reliability analysis tool, for the computation of low tail probabilities and extreme quantiles characterized by multiple failure regions

Calculation of tail probabilities and small quantiles is of fundamental importance in several domains, such as for example risk assessment or optimization. One major challenge consists in their computation when characterized by multiple-failure regions and rare event, say an occurrence probability smaller than 1e-7. Here, we focus on cases where the function of interest is the output of an computationally expensive code such as CFD or structural analysis. We propose a novel algorithm permitting to build an accurate Kriging metamodel, and exploit it using Importance Sampling techniques in order to estimate the required statistics (either quantile or tail probability). In fact, it relies on a novel metamodel building strategy, which aims to refine the limit-state region in all the branches ”equally”, even in the case of multiple failure regions. Due to Kriging limitations, the method is suitable for low stochastic dimension (say less than 10). This refinement step is formulated in such a way that the computation of both small probabilities of failure and extreme quantiles is unified (Schobi 2016). Parallel strategies are proposed. Several numerical examples taken from Bect 2017 and Schobi 2016 are carried out (2D, 6D, 8D), where small failure probabilities (10^-6 - 10^-9) are efficiently and accurately evaluated with a low number of the original performance function (less than 100). Corresponding inverse problem consisting in evaluating the associated quantile associated to the true failure probability are evaluated with the proposed framework, with the same order of performance function calls. The main novelty consists in adapting the method proposed by Schobi (2016) for the efficient evaluation of extreme quantiles.