Observability and the inverse problem in electrocardiography
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The problem of observability of a system governed by a partial differential equation is considered. This problem arises in electrocardiography, and the goal is to reconstruct the electrical potentials on the surface of the heart from the information obtained noninvasively on the torso surface. The formulation of the problem leads to a Cauchy problem for the Laplace's equation, i.e., a harmonic function is sought in some region, given that the values of the function and its normal derivative are specified only on some part of the boundary. The aim of this dissertation is to investigate the feasibility of recovering the solution, given only discrete boundary measurements. First, the existence and the uniqueness of the solution are shown based on some general assumptions about the geometry and the function representing the electrical potentials of the surface of the heart. Then, a spherical model representing the heart-torso model is introduced and an analytical solution to the problem is obtained. Then, a regularization method is developed and some error estimates of the solution are obtained based on some a priori assumptions about the solution. Finally, the approximation on the surface of a sphere, by means of a numerical integration is considered, and the dependence of the solution on the location and the number of measurements is investigated.