  This work presents a multiscale model reduction approach to discontinuous
fields identification problems in the framework of Bayesian inference. An
ensemble-based variable separation (VS) method is proposed to approximate
multiscale basis functions used to build a coarse model. The
variable-separation expression is constructed for stochastic multiscale basis
functions based on the random field, which is treated Gauss process as prior
information. To this end, multiple local inhomogeneous Dirichlet boundary
condition problems are required to be solved, and the ensemble-based method is
used to obtain variable separation forms for the corresponding local functions.
The local functions share the same interpolate rule for different physical
basis functions in each coarse block. This approach significantly improves the
efficiency of computation. We obtain the variable separation expression of
multiscale basis functions, which can be used to the models with different
boundary conditions and source terms, once the expression constructed. The
proposed method is applied to discontinuous field identification problems where
the hybrid of total variation and Gaussian (TG) densities are imposed as the
penalty. We give a convergence analysis of the approximate posterior to the
reference one with respect to the Kullback-Leibler (KL) divergence under the
hybrid prior. The proposed method is applied to identify discontinuous
structures in permeability fields. Two patterns of discontinuous structures are
considered in numerical examples: separated blocks and nested blocks.
