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An optimal quasi solution for the cauchy problem for laplace equation in the framework of inverse ECG

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 = Γ1 ∪ Γ0 ∪ Σ, where Γ1 and Γ0 are disjoint, regular, and n-dimensional surfaces. Cauchy boundary data is given in Γ0, and null Dirichlet data in Σ, while no data is given in Γ1. 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 Γ0 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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Submitted on : Saturday, November 24, 2018 - 6:56:01 PM
Last modification on : Friday, July 8, 2022 - 10:05:51 AM
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Eduardo Hernández-Montero, Andrés Fraguela-Collar, Jacques Henry. An optimal quasi solution for the cauchy problem for laplace equation in the framework of inverse ECG. Mathematical Modelling of Natural Phenomena, In press, ⟨10.1051/mmnp/2018062⟩. ⟨hal-01933948⟩



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