A phase-field approach for the interface reconstruction in a nonlinear elliptic problem arising from cardiac electrophysiology

TL;DR

This paper presents a phase-field method for reconstructing discontinuous coefficients in a nonlinear elliptic PDE, motivated by cardiac electrophysiology, with theoretical convergence proofs and numerical validation.

Contribution

It introduces a phase-field relaxation for interface reconstruction in nonlinear elliptic problems, with proven Γ-convergence and a new iterative algorithm.

Findings

The phase-field approach effectively reconstructs arbitrarily-shaped inclusions.

The algorithm demonstrates robustness and convergence in numerical tests.

Comparison shows advantages over shape derivative methods.

Abstract

In this work we tackle the reconstruction of discontinuous coefficients in a semilinear elliptic equation from the knowledge of the solution on the boundary of the domain, an inverse problem motivated by biological application in cardiac electrophysiology. We formulate a constraint minimization problem involving a quadratic mismatch functional enhanced with a regularization term which penalizes the perimeter of the inclusion to be identified. We introduce a phase-field relaxation of the problem, replacing the perimeter term with a Ginzburg-Landau-type energy. We prove the $\Gamma$-convergence of the relaxed functional to the original one (which implies the convergence of the minimizers), we compute the optimality conditions of the phase-field problem and define a reconstruction algorithm based on the use of the Fr\`echet derivative of the functional. After introducing a discrete version of the problem we implement an iterative algorithm and prove convergence properties. Several numerical results are reported, assessing the effectiveness and the robustness of the algorihtm in identifying arbitrarily-shaped inclusions. Finally, we compare our approach to a shape derivative based technique, both from a theoretical point of view (computing the sharp interface limit of the optimality conditions) and from a numerical one.