A reconstruction algorithm based on topological gradient for an inverse problem related to a semilinear elliptic boundary value problem

In this paper we develop a reconstruction algorithm for the solution of an inverse boundary value problem dealing with a semilinear elliptic partial differential equation of interest in cardiac electrophysiology. The goal is the detection of small inhomogeneities located inside a domain Omega, where the coefficients of the equation are altered, starting from observations of the solution of the equation on the boundary partial derivative Omega. Exploiting theoretical results recently achieved in [13], we implement a reconstruction procedure based on the computation of the topological gradient of a suitable cost functional. Numerical results obtained for several test cases finally assess the feasibility and the accuracy of the proposed technique.