A new reconstruction method for the inverse source problem from partial boundary measurements

The inverse source problem consists of reconstructing a mass distribution in a geometrical domain from boundary measurements of the associated potential and its normal derivative. In this paper the inverse source problem is reformulated as a topology optimization problem, where the support of the mass distribution is the unknown variable. The Kohn–Vogelius functional is minimized. It measures the misfit between the solutions of two auxiliary problems containing information about the boundary measurements. The Newtonian potential is used to complement the unavailable information on the hidden boundary. The resulting topology optimization algorithm is based on an analytic formula for the variation of the Kohn–Vogelius functional with respect to a class of mass distributions consisting of a finite number of ball-shaped trial anomalies. The proposed reconstruction algorithm is non-iterative and very robust with respect to noisy data. Finally, in order to show the effectiveness of the devised reconstruction algorithm, some numerical experiments in two and three spatial dimensions are presented.