Implicit sampling for hierarchical Bayesian inversion and applications in fractional multiscale diffusion models

Abstract

This paper presents an implicit sampling method for hierarchical Bayesian inverse problems. A widely used approach for sampling posterior distribution is Markov chain Monte Carlo (MCMC). However, the samples generated by MCMC are usually strongly correlated, which may lead to a small size of effective samples. The implicit sampling method proposed in Chorin and Tu (2009) can generate independent samples and capture inherent non-Gaussian features of the posterior. In implicit sampling method, the posterior samples are generated by a map and distribution around the maximum a posterior (MAP) point. For practical applications, some parameters in prior density are often unknown and the hierarchical Bayesian formulation is used to estimate the MAP point effectively. It is applied to the Bayesian inverse problems of multi-term time fractional diffusion models in heterogeneous media. To effectively capture the heterogeneity effect, we present a mixed generalized multiscale finite element method (mixed GMsFEM) to substantially speed up the Bayesian inversion. Finally, we carry out a few numerical examples to effectively identify various unknown inputs.