On the Reconstruction of Cavities in a Nonlinear Model Arising from Cardiac Electrophysiology

Abstract

In this paper we deal with the problem of determining perfectly insulating regions (cavities) from one boundary measurement in a nonlinear elliptic equation arising from cardiac electrophysiology. Based on the results obtained in [9] we propose a new reconstruction algorithm based on Γ-convergence. The relevance and applicability of this approach is then shown through several numerical experiments.

A phase-field approach for detecting cavities via a Kohn–Vogelius type functional

“…In the inverse problem context, applications of a phase-field approach have been proposed in [33,70,71] for a linear elliptic equation, in [14,15] for a semilinear elliptic equation, and very recently in [8] for the Lamé system and in [58] for a quasilinear Maxwell system.…”

“…Recently, a phase-field approach has been proposed in [33] and then applied also in [15] for the identification of inclusions in the framework of a linear and a semilinear elliptic equation, respectively. The same approach has been also extended to the detection of cavities in the case of a semilinear elliptic equation in [14] and of linear elasticity in [8]. All these papers propose an algorithm rephrasing the inverse problem as an optimization procedure, where the goal is to minimize a suitable misfit functional, defined on the boundary of Ω, with the addition of a regularization term which involves a relaxation of the perimeter of the domain to be reconstructed.…”