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Title: Bayesian inverse problems for recovering coefficients of two scale elliptic equations
Authors: Hoang, Viet Ha
Quek, Jia Hao
Keywords: Science::Mathematics
Issue Date: 2019
Source: Hoang, V. H. & Quek, J. H. (2019). Bayesian inverse problems for recovering coefficients of two scale elliptic equations. Inverse Problems, 35(4), 045005-.
Project: RG30/16
Journal: Inverse Problems
Abstract: We consider the Bayesian inverse homogenization problem of recovering the locally periodic two scale coefficient of a two scale elliptic equation, given limited noisy information on the solution. We consider both the uniform and the Gaussian prior probability measures. We use the two scale homogenized equation whose solution contains the solution of the homogenized equation which describes the macroscopic behaviour, and the corrector which encodes the microscopic behaviour. We approximate the posterior probability by a probability measure determined by the solution of the two scale homogenized equation. We show that the Hellinger distance of these measures converges to zero when the microscale converges to zero, and establish an explicit convergence rate when the solution of the two scale homogenized equation is sufficiently regular. Sampling the posterior measure by Markov Chain Monte Carlo (MCMC) method, instead of solving the two scale equation using fine mesh for each proposal with extremely high cost, we can solve the macroscopic two scale homogenized equation. Although this equation is posed in a high dimensional tensorized domain, it can be solved with essentially optimal complexity by the sparse tensor product finite element method, which reduces the computational complexity of the MCMC sampling method substantially. We show numerically that observations on the macrosopic behaviour alone are not sufficient to infer the microstructure. We need also observations on the corrector. Solving the two scale homogenized equation, we get both the solution to the homogenized equation and the corrector. Thus our method is particularly suitable for sampling the posterior measure of two scale coefficients.
ISSN: 0266-5611
DOI: 10.1088/1361-6420/aafcd6
Rights: © 2019 IOP Publishing Ltd. All rights reserved.
Fulltext Permission: none
Fulltext Availability: No Fulltext
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