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

TL;DR

This paper introduces a novel reconstruction algorithm utilizing topological gradients to detect small internal inhomogeneities in a domain governed by a semilinear elliptic PDE, with applications in cardiac electrophysiology.

Contribution

It develops a new reconstruction method based on topological gradients for inverse boundary problems involving semilinear elliptic equations, extending previous theoretical results.

Findings

Numerical tests demonstrate the method's effectiveness in accurately detecting inhomogeneities.

The algorithm shows good stability and robustness in various test scenarios.

Abstract

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 $\partial \Omega$. Exploiting theoretical results recently achieved in [11], 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.