Calculating Bayesian model evidence for porous-media flow using a multilevel estimator

Abstract

We consider calculation of the Bayesian model evidence, which is an essential component in realistic uncertainty quantification. The main motivation is large-scale porous-media-flow problems, where plain Monte-Carlo (MC) integration is computationally infeasible. We propose a simplistic multilevel (ML) estimator and argue why this estimator is expected to perform well within a certain problem class, and point out that many important problems, including many porous-media-flow problems, belong to this class. Four test cases are utilized to assess the performance of the proposed ML estimator. Test cases I and II contain two strongly linked toy models, but only Test case II belongs to the problem class where the proposed ML estimator is expected to perform well. Test cases III and IV are concerned with single-phase and two-phase porous-media flows, respectively. The results show that the ML estimator clearly and consistently outperforms MC integration in Test cases II, III, and IV, while this is not the case for Test case I.