Communications in Mathematical Sciences

Volume 16 (2018)

Number 7

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

Pages: 1975 – 2002



Elena Beretta (Dipartimento di Matematica, Politecnico di Milano, Italy)

Luca Ratti (Dipartimento di Matematica, Politecnico di Milano, Italy)

Marco Verani (Dipartimento di Matematica, Politecnico di Milano, Italy)


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 $\Gamma$-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.


inverse problem, semilinear elliptic equation, phase-field relaxation

2010 Mathematics Subject Classification

35J61, 35R30, 65N21

Received 31 January 2018

Accepted 12 July 2018

Published 7 March 2019