### 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 language | English (US) |
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Pages (from-to) | 1975-2002 |

Number of pages | 28 |

Journal | Communications in Mathematical Sciences |

Volume | 16 |

Issue number | 7 |

DOIs | |

State | Published - Jan 1 2018 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications in Mathematical Sciences*,

*16*(7), 1975-2002. https://doi.org/10.4310/CMS.2018.V16.N7.A10

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

Research output: Contribution to journal › Article

*Communications in Mathematical Sciences*, vol. 16, no. 7, pp. 1975-2002. https://doi.org/10.4310/CMS.2018.V16.N7.A10

}

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 - 2018/1/1

Y1 - 2018/1/1

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

UR - http://www.scopus.com/inward/record.url?scp=85064486819&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85064486819&partnerID=8YFLogxK

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 -