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

Eduardo Hernandez-Montero, Andres Fraguela Collar, Jacques Henry

Resultado de la investigación: Contribución a una revistaArtículo

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 originalInglés
Número de artículo204
PublicaciónMathematical Modelling of Natural Phenomena
Volumen14
N.º2
DOI
EstadoPublicada - 1 ene 2019

Huella dactilar

Laplace equation
Laplace's equation
Electrocardiography
Inverse problems
Cauchy Problem
Harmonic functions
Derivatives
Cauchy
Null
Regularization
Inverse Problem
Electrocardiogram
Framework
Projection
Completion Problem
Ill-posed Problem
Minimum Distance
Harmonic Functions
Dirichlet
Completion

Citar esto

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abstract = "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.",
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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.

En: Mathematical Modelling of Natural Phenomena, Vol. 14, N.º 2, 204, 01.01.2019.

Resultado de la investigación: Contribución a una revistaArtículo

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AU - Henry, Jacques

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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.

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KW - Factorization method

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