Reconstruction of a volumetric source domain

Inverse and ill-posed problems which consist of reconstructing the unknown support of a three-dimensional volumetric source from a single pair of exterior boundary Cauchy data are investigated. The underlying dependent variable may satisfy the Laplace, Poisson, Helmholtz or modified Helmholtz equations. In the case of constant physical properties, the solutions of these elliptic PDEs are sought as linear combinations of fundamental solutions, as in the method of fundamental solutions (MFS). The unknown source domain is parametrized by the radial coordinate, as a function of the spherical angles. The resulting least-squares functional estimating the gap between the measured and the computed data is regularized and minimized using the lsqnonlin toolbox routine in Matlab. Numerical results are presented and discussed for both exact and noisy data, confirming the accuracy and stability of reconstruction.