Recent Developments in Inverse Problems - Abstract

Schlottbom, Matthias

Diffuse interface methods for inverse problems: Case study for an elliptic Cauchy problem

joint work with Martin Burger (WWU Münster) and Ole L. Elvetun (Norwegian University of Life Sciences) medskip Several inverse problems have to deal with not exactly known geometries. This makes it desirable to use methods which are robust with respect to perturbed domains. In this work we discuss a diffuse interface method as a tool for the solution of variational inverse problems. As a particular example we study ECG inversion in some detail. ECG inversion is a linear inverse source problem with boundary measurements governed by an anisotropic diffusion equation. We formulate a regularization strategy using Tikhonov regularization and, using standard source conditions, we prove convergence rates. A special difficulty in the analysis of the regularization method is that not only operator perturbations are introduced by the diffuse interface method, but more important, we have to deal with topologies which depend on a parameter $varepsilon$ in the diffuse interface method, i.e. we have to deal with $varepsilon$-dependent forward operators and $varepsilon$-dependent norms. Our results are supported by numerical examples.