Bayesian identification of discontinuous fields with an ensemble-based variable separation multiscale method
For practitioners in subsurface flow or inverse problems, this method improves computational efficiency in Bayesian inference for discontinuous fields, though it is an incremental improvement over existing multiscale methods.
This work proposes an ensemble-based variable separation multiscale method for Bayesian identification of discontinuous fields, achieving efficient computation by constructing variable-separation expressions for multiscale basis functions. The method is applied to identify discontinuous permeability structures, with convergence analysis in KL divergence under a hybrid TG prior.
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.