  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.
