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

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