Detection of conductivity inclusions in a semilinear elliptic problem arising from cardiac electrophysiology

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 planar bounded domain. The problem is motivated by an 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, employing a Ginzburg-Landau-type energy and assessing the Γ-convergence of the relaxed functional to the original one. After computing the optimality conditions of the phase-field optimization problem and introducing a discrete finite element formulation, we propose an iterative algorithm and prove convergence properties. Several numerical results are reported, assessing the effectiveness and the robustness of the algorithm 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.