Diffuse Interface Methods for Inverse Problems: Case Study for an Elliptic Cauchy Problem

TL;DR

This paper introduces a diffuse domain method for inverse problems with complex geometries, demonstrating its application to ECG inversion and establishing convergence rates despite topology-dependent perturbations.

Contribution

It systematically applies diffuse interface methods to inverse problems, particularly for ECG inversion, and develops a novel saddle-point approach to handle topology-dependent perturbations.

Findings

Proved convergence rates for the diffuse domain inverse problem.

Developed a saddle-point framework for topology-dependent perturbations.

Applied method successfully to ECG inversion with complex geometries.

Abstract

Many inverse problems have to deal with complex, evolving and often not exactly known geometries, e.g. as domains of forward problems modeled by partial differential equations. This makes it desirable to use methods which are robust with respect to perturbed or not well resolved domains, and which allow for efficient discretizations not resolving any fine detail of those geometries. For forward problems in partial differential equations methods based on diffuse interface representations gained strong attention in the last years, but so far they have not been considered systematically for inverse problems. In this work we introduce a diffuse domain method as a tool for the solution of variational inverse problems. As a particular example we study ECG inversion in further detail. ECG inversion is a linear inverse source problem with boundary measurements governed by an anisotropic diffusion equation, which naturally cries for solutions under changing geometries, namely the beating heart. We formulate a regularization strategy using Tikhonov regularization and, using standard source conditions, we prove convergence rates. A special property of our approach is that not only operator perturbations are introduced by the diffuse domain method, but more important we have to deal with topologies which depend on a parameter $\eps$ in the diffuse domain method, i.e. we have to deal with $\eps$-dependent forward operators and $\eps$-dependent norms. In particular the appropriate function spaces for the unknown and the data depend on $\eps$. This prevents to apply some standard convergence techniques for inverse problems, in particular interpreting the perturbations as data errors in the original problem does not yield suitable results. We consequently develop a novel approach based on saddle-point problems.