Bayesian inverse problems for partial differential equations with discontinuous coefficients: an adaptive finite element approach

Abstract

This thesis addresses a challenging class of inverse problems in which one seeks to identify an inclusion within a physical domain by observing noisy, indirect measurements of the underlying system. We adopt a Bayesian framework, treating the unknown parameters as random variables and combining prior information with observational data to form a posterior distribution. This approach naturally quantifies the uncertainty in the inferred inclusion and enables rigorous statistical interpretation of the results.

A key component of the study is the development and comparison of two finite element solvers for the forward problem: a standard solver on uniform meshes and an Adaptive Finite Element Method (AFEM). The discontinuous coefficient arising from the inclusion presents substantial difficulties for uniform refinement, leading to slow convergence and excessive computational cost. By contrast, AFEM adaptively refines the mesh in regions of high error or sharp gradients, delivering significantly improved accuracy per degree of freedom.

We evaluate both methods in terms of their convergence rates, focusing in particular on the Hellinger distance between posterior distributions. Numerical experiments show that the AFEM-based solver converges more rapidly and accurately captures the inclusion parameters, making it a superior tool for Bayesian inversion with localized discontinuities. The findings underscore the benefits of adaptive refinement strategies for ill-posed inverse problems with limited or noisy data.