A phase-field approach for the interface reconstruction in a nonlinear elliptic problem arising from cardiac electrophysiology
This work provides a theoretically grounded reconstruction algorithm for an inverse problem in cardiac electrophysiology, though the method is incremental as it adapts existing phase-field techniques to a specific application.
The authors developed a phase-field method to reconstruct discontinuous coefficients in a semilinear elliptic equation from boundary data, motivated by cardiac electrophysiology. Numerical results demonstrate effective and robust identification of arbitrarily-shaped inclusions, with convergence properties proven.
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 $Γ$-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 Frèchet 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.