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

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.