Critical Points for Least-Squares Estimation of Dipolar Sources in Inverse Problems for Poisson Equation

In this work, we study some aspects of the solvability of the minimization of a non-convex least-squares criterion involved in dipolar source recovery issues, using boundary values of a solution to a Poisson problem in a domain of dimension 3. This Poisson problem arises in particular from the quasi-static approximation of Maxwell equations with localized sources modeled as dipoles. We establish the uniqueness of the minimizer of the criterion for general geometries and the uniqueness of its critical point for the Euclidean geometry, that is when the boundary is a plane. This has consequences on the numerical approach, for the convergence of the computed solution to the global minimizer. Related inverse potential problems have applications in biomedical imaging issues pertaining to neurosciences, and in paleomagnetism issues pertaining to geosciences. There, solutions to such inverse problems are used to recover electric currents in the brain, or rock magnetizations, from measurements of the induced electric potential or magnetic field.