Math. Model. Nat. Phenom.
Volume 14, Number 2, 2019
Mathematical modelling in cardiology
|Number of page(s)||22|
|Published online||15 February 2019|
An optimal quasi solution for the Cauchy problem for Laplace equation in the framework of inverse ECG★
Facultad de Ciencias Físico-Matemáticas, BUAP, Avenida San Claudio y 18 Sur Colonia San Manuel, Edificio FM1-101B, Ciudad Universitaria,
2 INRIA Bordeaux Sud Ouest, 200 avenue de la Vieille Tour, 33140 Talence, France.
* Corresponding author: firstname.lastname@example.org
Accepted: 6 October 2018
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 ℝn+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 C2(Ω) ∩H1(Ω). The methodology roughly consists of first characterizing the admissible Cauchy data, then finding the minimum distance projection in the L2-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.
Mathematics Subject Classification: 31B20 / 31A25 / 35R30 / 65J22 / 65N21
Key words: Cauchy problem / optimal regularization / quasi solution / factorization method / invariant embedding / ECG inverse problem
© EDP Sciences, 2019
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