Reconstruction of a source domain for the Poisson equation

The inverse and ill-posed problem of reconstructing the unknown support of a source in the Poisson equation from a single pair of Cauchy boundary data is investigated. The solution is sought as a linear combination of fundamental solutions for the Laplace’s equation, as in the method of fundamental solutions (MFS), but prior to this, the free-term source is removed by the method of particular solutions. The numerical solution which minimizes the gap between the measured and the computed data is achieved using the Matlab toolbox routine lsqnonlin. Since the inverse problem is ill-posed, the least-squares functional which is minimized needs to be augmented with regularizing penalty terms for a stable estimation of the MFS coefficients and the radial function. Numerical investigations are undertaken for retrieving the shape of the source support