Computational identification of the source domain in an inverse problem of potential theory

The inverse potential problem consists in determining the density of the volume potential from measurements outside the sources. Its ill-posedness is due both to the non-uniqueness of the solution and to the instability of the solution with respect to measurement errors. The inverse problem is solved under additional assumptions about the sources using regularizing algorithms. In this work, an inverse problem is posed for identifying the domain that contains the sources. The computational algorithm is based on approximating the volume potential by the single-layer potential on the boundary of the domain containing the sources. The inverse problem is considered in the class of a priori constraints of nonnegativity of the potential density. Residual minimization in the class of nonnegative solutions is performed using the classical Nonnegative Least Squares algorithm. The capabilities of the proposed approach are illustrated by numerical experiments for a two-dimensional test problem with an analytically prescribed potential on the observation surface.