A new reconstruction method for the inverse potential problem

The inverse potential problem consists in reconstructing an unknown measure with support in a geometrical domain from a single boundary measurement. In order to deal with this severely ill-posed inverse problem, we rewrite it as an optimization problem where a Kohn–Vogelius-type functional measuring the misfit between the solutions of two auxiliary problems is minimized. One auxiliary problem contains information on the boundary measurement while the other one corresponds to the boundary excitation. The solutions of the auxiliary problems coincide once the inverse problem is solved. In order to minimize the Kohn–Vogelius criterion, its total variation with respect to a set of ball-shaped perturbations on the measure is explicitly evaluated. Then, a new method for solving the inverse potential problem based on the expression obtained is devised. Finally, some numerical results are presented in order to show the effectiveness of the proposed reconstruction algorithm.