### Resumen

The inverse ECG problem is set as a boundary data completion for the Laplace equation: at each time the potential is measured on the torso and its normal derivative is null. One aims at reconstructing the potential on the heart. A new regularization scheme is applied to obtain an optimal regularization strategy for the boundary data completion problem. We consider the ℝ
^{n+1}
domain Ω. The piecewise regular boundary of Ω is defined as the union ∂Ω = Γ
_{1}
∪ Γ
_{0}
∪ Σ, where Γ
_{1}
and Γ
_{0}
are disjoint, regular, and n-dimensional surfaces. Cauchy boundary data is given in Γ
_{0}
, and null Dirichlet data in Σ, while no data is given in Γ
_{1}
. This scheme is based on two concepts: admissible output data for an ill-posed inverse problem, and the conditionally well-posed approach of an inverse problem. An admissible data is the Cauchy data in Γ
_{0}
corresponding to an harmonic function in C
^{2}
(Ω) ∩ H
^{1}
(Ω). The methodology roughly consists of first characterizing the admissible Cauchy data, then finding the minimum distance projection in the L
^{2}
-norm from the measured Cauchy data to the subset of admissible data characterized by given a priori information, and finally solving the Cauchy problem with the aforementioned projection instead of the original measurement.

Idioma original | Inglés |
---|---|

Número de artículo | 204 |

Publicación | Mathematical Modelling of Natural Phenomena |

Volumen | 14 |

N.º | 2 |

DOI | |

Estado | Publicada - 1 ene 2019 |

### Huella dactilar

### Citar esto

*Mathematical Modelling of Natural Phenomena*,

*14*(2), [204]. https://doi.org/10.1051/mmnp/2018062

}

*Mathematical Modelling of Natural Phenomena*, vol. 14, n.º 2, 204. https://doi.org/10.1051/mmnp/2018062

**An optimal quasi solution for the Cauchy problem for Laplace equation in the framework of inverse ECG.** / Hernandez-Montero, Eduardo; Fraguela Collar, Andres; Henry, Jacques.

Resultado de la investigación: Contribución a una revista › Artículo

TY - JOUR

T1 - An optimal quasi solution for the Cauchy problem for Laplace equation in the framework of inverse ECG

AU - Hernandez-Montero, Eduardo

AU - Fraguela Collar, Andres

AU - Henry, Jacques

PY - 2019/1/1

Y1 - 2019/1/1

N2 - The inverse ECG problem is set as a boundary data completion for the Laplace equation: at each time the potential is measured on the torso and its normal derivative is null. One aims at reconstructing the potential on the heart. A new regularization scheme is applied to obtain an optimal regularization strategy for the boundary data completion problem. We consider the ℝ n+1 domain Ω. The piecewise regular boundary of Ω is defined as the union ∂Ω = Γ 1 ∪ Γ 0 ∪ Σ, where Γ 1 and Γ 0 are disjoint, regular, and n-dimensional surfaces. Cauchy boundary data is given in Γ 0 , and null Dirichlet data in Σ, while no data is given in Γ 1 . This scheme is based on two concepts: admissible output data for an ill-posed inverse problem, and the conditionally well-posed approach of an inverse problem. An admissible data is the Cauchy data in Γ 0 corresponding to an harmonic function in C 2 (Ω) ∩ H 1 (Ω). The methodology roughly consists of first characterizing the admissible Cauchy data, then finding the minimum distance projection in the L 2 -norm from the measured Cauchy data to the subset of admissible data characterized by given a priori information, and finally solving the Cauchy problem with the aforementioned projection instead of the original measurement.

AB - The inverse ECG problem is set as a boundary data completion for the Laplace equation: at each time the potential is measured on the torso and its normal derivative is null. One aims at reconstructing the potential on the heart. A new regularization scheme is applied to obtain an optimal regularization strategy for the boundary data completion problem. We consider the ℝ n+1 domain Ω. The piecewise regular boundary of Ω is defined as the union ∂Ω = Γ 1 ∪ Γ 0 ∪ Σ, where Γ 1 and Γ 0 are disjoint, regular, and n-dimensional surfaces. Cauchy boundary data is given in Γ 0 , and null Dirichlet data in Σ, while no data is given in Γ 1 . This scheme is based on two concepts: admissible output data for an ill-posed inverse problem, and the conditionally well-posed approach of an inverse problem. An admissible data is the Cauchy data in Γ 0 corresponding to an harmonic function in C 2 (Ω) ∩ H 1 (Ω). The methodology roughly consists of first characterizing the admissible Cauchy data, then finding the minimum distance projection in the L 2 -norm from the measured Cauchy data to the subset of admissible data characterized by given a priori information, and finally solving the Cauchy problem with the aforementioned projection instead of the original measurement.

KW - Cauchy problem

KW - ECG inverse problem

KW - Factorization method

KW - Invariant embedding

KW - Optimal regularization

KW - Quasi solution

UR - http://www.scopus.com/inward/record.url?scp=85062028412&partnerID=8YFLogxK

U2 - 10.1051/mmnp/2018062

DO - 10.1051/mmnp/2018062

M3 - Artículo

AN - SCOPUS:85062028412

VL - 14

JO - Mathematical Modelling of Natural Phenomena

JF - Mathematical Modelling of Natural Phenomena

SN - 0973-5348

IS - 2

M1 - 204

ER -