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 (n+1) domain Omega. The piecewise regular boundary of Omega is defined as the union partial derivative Omega = Gamma (1) Gamma (0) Sigma, where Gamma (1) and Gamma (0) are disjoint, regular, and n-dimensional surfaces. Cauchy boundary data is given in Gamma (0), and null Dirichlet data in Sigma, while no data is given in Gamma (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 Gamma (0) corresponding to an harmonic function in C-2(Omega) H-1(Omega). 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.