TY - JOUR
T1 - Detection of conductivity inclusions in a semilinear elliptic problem arising from cardiac electrophysiology
AU - Beretta, Elena
AU - Ratti, Luca
AU - Verani, Marco
PY - 2018
Y1 - 2018
N2 - 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.
AB - 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.
KW - Inverse problem
KW - Phase-field relaxation
KW - Semilinear elliptic equation
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U2 - 10.4310/CMS.2018.V16.N7.A10
DO - 10.4310/CMS.2018.V16.N7.A10
M3 - Article
AN - SCOPUS:85064486819
VL - 16
SP - 1975
EP - 2002
JO - Communications in Mathematical Sciences
JF - Communications in Mathematical Sciences
SN - 1539-6746
IS - 7
ER -