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In this paper, we study porous media flows in heterogeneous stochastic media.We propose an efficient forward simulation technique that is tailored forvariational Bayesian inversion. As a starting point, the proposed forwardsimulation technique decomposes the solution into the sum of separablefunctions (with respect to randomness and the space), where each term iscalculated based on a variational approach. This is similar to ProperGeneralized Decomposition (PGD). Next, we apply a multiscale technique to solvefor each term and, further, decompose the random function into 1D fields. As aresult, our proposed method provides an approximation hierarchy for thesolution as we increase the number of terms in the expansion and, also,increase the spatial resolution of each term. We use the hierarchical solutiondistributions in a variational Bayesian approximation to perform uncertaintyquantification in the inverse problem. We conduct a detailed numerical study toexplore the performance of the proposed uncertainty quantification techniqueand show the theoretical posterior concentration.

Bayesian and Variational Bayesian approaches for flows in heterogenous random media