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The Cauchy problem for an elliptic partial differential equation is ill-posed. In this paper, we study a numerical method for solving the Cauchy problem. The numerical method is based on a reformulation of the Cauchy problem through an optimal control approach coupled with a regularization term which is included to treat the severe ill-conditioning of the corresponding discretized formulation. We prove convergence of the numerical method and present theoretical results for the limiting behaviors of the numerical solution as the regularization parameter approaches zero. Results from some numerical examples are reported.

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Bioluminescence tomography (BLT) is a rapidly developing new area of molecular imaging. The goal of BLT is to produce a quantitative reconstruction of a bioluminescent source distribution within a living mouse from bioluminescent signals measured on the body surface of the mouse. While in most BLT studies so far the optical parameters of the key anatomical regions are assumed known from the literature or diffuse optical tomography (DOT), these parameters cannot be very accurate in general. In this paper, we propose and study a new BLT approach that optimizes optical parameters when an underlying bioluminescent source distribution is reconstructed to match the measured data. We prove the solution existence and the convergence of numerical methods. Also, we present numerical results to illustrate the utility of our approach and evaluate its performance.