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

TL;DR

This paper introduces a domain-decomposed Bayesian inversion method that uses local Karhunen-Loève expansions to efficiently handle high-dimensional inverse problems governed by PDEs, enabling parallel computation and improved scalability.

Contribution

The paper presents a novel domain decomposition approach combining local KL expansions and a projection method for global reconstruction in Bayesian inverse problems.

Findings

Efficient parallel local Bayesian inversions in subdomains.

Significant reduction in computational complexity.

Effective global field reconstruction from local samples.

Abstract

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.