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This work presents a multiscale model reduction approach to discontinuousfields identification problems in the framework of Bayesian inference. Anensemble-based variable separation (VS) method is proposed to approximatemultiscale basis functions used to build a coarse model. Thevariable-separation expression is constructed for stochastic multiscale basisfunctions based on the random field, which is treated Gauss process as priorinformation. To this end, multiple local inhomogeneous Dirichlet boundarycondition problems are required to be solved, and the ensemble-based method isused to obtain variable separation forms for the corresponding local functions.The local functions share the same interpolate rule for different physicalbasis functions in each coarse block. This approach significantly improves theefficiency of computation. We obtain the variable separation expression ofmultiscale basis functions, which can be used to the models with differentboundary conditions and source terms, once the expression constructed. Theproposed method is applied to discontinuous field identification problems wherethe hybrid of total variation and Gaussian (TG) densities are imposed as thepenalty. We give a convergence analysis of the approximate posterior to thereference one with respect to the Kullback-Leibler (KL) divergence under thehybrid prior. The proposed method is applied to identify discontinuousstructures in permeability fields. Two patterns of discontinuous structures areconsidered in numerical examples: separated blocks and nested blocks.

Bayesian identification of discontinuous fields with an ensemble-based variable separation multiscale method