A Bayesian approach to multiscale inverse problems using the sequential Monte Carlo method

A new Bayesian computational approach is developed to estimate spatially varying parameters. The sparse grid collocation method is adopted to parameterize the spatial field. Based on a hierarchically structured sparse grid, a multiscale representation of the spatial field is constructed. An adaptive refinement strategy is then used for computing the spatially varying parameter. A sequential Monte Carlo (SMC) sampler is used to explore the posterior distributions defined on multiple scales. The SMC sampling is directly parallelizable and is superior to conventional Markov chain Monte Carlo methods for multi-modal target distributions. The samples obtained at coarser levels of resolution are used to provide prior information for the estimation at finer levels. This Bayesian computational approach is rather general and applicable to various spatially varying parameter estimation problems. The method is demonstrated with the estimation of permeability in flows through porous media.