Extension of AK-MCS for the efficient computation of very small failure probabilities

We consider the problem of estimating a probability of failure p_f, defined as the volume of the excursion set of a complex (e.g. output of an expensive-to-run finite element model) scalar performance function J below a given threshold, under a probability measure that can be recast as a multivariate standard gaussian law using an isoprobabilistic transformation. We propose a method able to deal with cases characterized by multiple failure regions, possibly very small failure probability p_f (say ∼10⁻⁶−10⁻⁹), and when the number of evaluations of J is limited. The present work is an extension of the popular Kriging-based active learning algorithm known as AK-MCS, as presented in [1], permitting to deal with very low failure probabilities. The key idea merely consists in replacing the Monte-Carlo sampling, used in the original formulation to propose candidates and evaluate the failure probability, by a centered isotropic Gaussian sampling in the standard space, whose standard deviation is iteratively tuned. This extreme AK-MCS (eAK-MCS) inherits its former multi-point enrichment algorithm allowing to add several points at each iteration step, and provide an estimated failure probability based on the Gaussian nature of the Kriging surrogate. Both the efficiency and the accuracy of the proposed method are showcased through its application to two to eight dimensional analytic examples, characterized by very low failure probabilities: p_f∼10⁻⁶−10⁻⁹. Numerical experiments conducted with unfavorable initial Design of Experiment suggests the ability of the proposed method to detect failure domains.