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

Domain-decomposed Bayesian inversion based on local Karhunen-Lo\`{e}ve expansions