Multilevel approximate Bayesian approaches for flows in highly heterogeneous porous media and their applications

Estimation of quantities related to high-contrast flow problems such as permeability field plays an important role in porous media characterization. A Generalized Multiscale Finite Element Method (GMsFEM) can be used for solving parameter-dependent (or stochastic) flow problems with multiscale nature. A hierarchy of approximations of different resolution can be provided by GMsFEM. Hence, it can be coupled with Multilevel Markov Chain Monte Carlo (MLMCMC) to generate samples in different levels and form the multilevel estimator. Karhunen-Lo'eve Expansion (KLE) is used to parameterize the underlying random field by a function of Gaussian random field. Instead of MCMC, an Approximate Bayesian Computation (ABC) method can be used within the Multilevel Monte Carlo framework. ABC can be incorporated in different levels to reduce the computational cost and to produce an approximate solution by ensembling different levels.