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

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 Ω, where the coefficients of the equation are altered, starting from observations of the solution of the equation on the boundary ∂Ω. 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.

An inverse problem for a semilinear parabolic equation arising from cardiac electrophysiology

“…Following the approach of [8], [14], but taking into account the time-dependence of the problem, we introduce the mismatch functional…”

“…We use these results to set a reconstruction procedure for detecting the inclusion. To this aim, as in [8], we propose a reconstruction algorithm based on topological optimization, where a suitable quadratic functional is minimized to detect the position and the size of the inclusion (see also [14]). Numerical results obtained on an idealized left ventricle geometry assess the feasibility of the proposed procedure.…”

On the determination of ischemic regions in the monodomain model of cardiac electrophysiology from boundary measurements

“…A similar approach has also been used in Thermal Imaging (see, e.g., [4]). Here, we are able to extend in a non trivial way the results obtained previously in [6] and [9] for simplified versions of the monodomain model making use of fine regularity for the solutions to nonlinear reaction diffusion systems. In particular, we establish a rigorous expansion for the perturbed transmembrane potential and use this to implement an effective reconstruction algorithm.…”

A new approach to the intracardiac inverse problem using Laplacian distance kernel

“…The electrical source estimation from regularized inversions is an ill-posed problem, thus different approaches have been proposed to deal with it [ 23 – 27 ]. Most of the proposed algorithms, including the truncated singular value decomposition (TSVD), the regularized Least Squares (LS), the curvature of the L-curve from Tikhonov method, and the minimum relative entropy, perform the inversion of the transfer matrix of the problem, which causes resolution reduction to their solutions [ 28 – 30 ].…”