Detection of conductivity inclusions in a semilinear elliptic problem arising from cardiac electrophysiology

Elena Beretta, Luca Ratti, Marco Verani

Research output: Contribution to journalArticle

Abstract

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 planar bounded domain. The problem is motivated by an 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, employing a Ginzburg-Landau-type energy and assessing the Γ-convergence of the relaxed functional to the original one. After computing the optimality conditions of the phase-field optimization problem and introducing a discrete finite element formulation, we propose an iterative algorithm and prove convergence properties. Several numerical results are reported, assessing the effectiveness and the robustness of the algorithm 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.

Original languageEnglish (US)
Pages (from-to)1975-2002
Number of pages28
JournalCommunications in Mathematical Sciences
Volume16
Issue number7
DOIs
StatePublished - Jan 1 2018

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Cardiac Electrophysiology
Semilinear Elliptic Problem
Phase Field
Optimality Conditions
Conductivity
Inclusion
Shape Derivative
Quadratic Functional
Discontinuous Coefficients
Computing
Ginzburg-Landau
Semilinear Elliptic Equations
Perimeter
Convergence Properties
Minimization Problem
Iterative Algorithm
Bounded Domain
Regularization
Finite Element
Robustness

Keywords

  • Inverse problem
  • Phase-field relaxation
  • Semilinear elliptic equation

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Detection of conductivity inclusions in a semilinear elliptic problem arising from cardiac electrophysiology. / Beretta, Elena; Ratti, Luca; Verani, Marco.

In: Communications in Mathematical Sciences, Vol. 16, No. 7, 01.01.2018, p. 1975-2002.

Research output: Contribution to journalArticle

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