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

¹ Facultad de Ciencias Físico-Matemáticas, BUAP, Avenida San Claudio y 18 Sur Colonia San Manuel, Edificio FM1-101B, Ciudad Universitaria, C.P. 72570, Puebla, Mexico.
² INRIA Bordeaux Sud Ouest, 200 avenue de la Vieille Tour, 33140 Talence, France.

^* Corresponding author: eduardo.hdz.mto@gmail.com

Received: 5 October 2018
Accepted: 6 October 2018

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 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.