  In many Bayesian inverse problems the goal is to recover a spatially varying
random field. Such problems are often computationally challenging especially
when the forward model is governed by complex partial differential equations
(PDEs). The challenge is particularly severe when the spatial domain is large
and the unknown random field needs to be represented by a high-dimensional
parameter. In this paper, we present a domain-decomposed method to attack the
dimensionality issue and the method decomposes the spatial domain and the
parameter domain simultaneously. On each subdomain, a local Karhunen-Lo`eve
(KL) expansion is constructed, and a local inversion problem is solved
independently in a parallel manner, and more importantly, in a
lower-dimensional space. After local posterior samples are generated through
conducting Markov chain Monte Carlo (MCMC) simulations on subdomains, a novel
projection procedure is developed to effectively reconstruct the global field.
In addition, the domain decomposition interface conditions are dealt with an
adaptive Gaussian process-based fitting strategy. Numerical examples are
provided to demonstrate the performance of the proposed method.
