Bayesian multiscale deep generative model for the solution of high-dimensional inverse problems

Estimation of spatially-varying parameters for computationally expensive forward models governed by partial differential equations (PDEs) is addressed. A novel multiscale Bayesian inference approach is introduced based on a multiscale deep generative model (MDGM). Such generative models provide a flexible representation and allow hierarchical parameter generation from coarse- to fine-scales. Combining the multiscale generative model with Markov Chain Monte Carlo (MCMC), inference across scales is achieved enabling us to obtain efficiently posterior parameter samples at various scales. To estimate the most salient features of parameters is essential in the proposed method. Inference of the fine-scale parameters is enabled by utilizing the posterior information in the immediate coarser scale. In this way, the global features are identified in the coarse-scale with the inference of low-dimensional variables and inexpensive forward computation, and the local features are refined and corrected in the fine-scale. The developed method is demonstrated with two types of permeability estimation for flow in heterogeneous media. One is a Gaussian random field (GRF) with uncertain length scales, and the other is channelized permeability with the two regions defined by different GRFs. The obtained results indicate that the method allows high-dimensional parameter estimation while exhibiting stability, efficiency, and accuracy. (C) 2022 Elsevier Inc. All rights reserved.