  Estimation of spatially-varying parameters for computationally expensive
forward models governed by partial differential equations is addressed. A novel
multiscale Bayesian inference approach is introduced based on deep
probabilistic generative models. Such generative models provide a flexible
representation by inferring on each scale a low-dimensional latent encoding
while allowing 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 efficiently obtain posterior
parameter samples at various scales. The estimation of coarse-scale parameters
using a low-dimensional latent embedding captures global and notable parameter
features using an inexpensive but inaccurate solver. MCMC sampling 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 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.
