Bayesian inversion using adaptive Polynomial Chaos Kriging within Subset Simulation

Abstract

In this paper, we propose a Bayesian inversion approach combining adaptive Polynomial Chaos Kriging (PCK) surrogate models and a rare event estimation method called Subset Simulation (SuS). It is based on the recently introduced Bayesian Updating with Structural reliability (BUS) framework that enables to reformulate the classical Bayesian inference into a rare event estimation problem. In this context, the SuS method aims at drawing samples from the posterior distribution as well as estimating the model evidence, which is usually computationally intractable when considering classical MCMC approaches. The proposed approach involves the construction of a PCK surrogate model which provides both global and local approximations of the likelihood function, through the combination of Polynomial Chaos and Kriging surrogates. Furthermore, we propose an adaptive scheme for enriching the PCK surrogate throughout the SuS sampling procedure, in order to improve its accuracy near informative regions. The applicability and the efficiency of the proposed approach are assessed through several cases studies with increasing complexity. Results highlight that the proposed approach enables to accurately approximating posteriors with a limited amount of full model calls, even in the case of multi-modal posteriors, which are usually difficult to sample when using classical MCMC algorithms.