Solution of Inverse Geometric Problems Using a Non-Iterative MFS

In most of the method of fundamental solutions (MFS) approaches employed so far for the solution of inverse geometric problems, the MFS implementation typically leads to non-linear systems which were solved by standard nonlinear iterative least squares software. In the current approach, we apply a three-step non-iterative MFS technique for identifying a rigid inclusion from internal data measurements, which consists of: (i) a direct problem to calculate the solution at the set of measurement points, (ii) the solution of an ill-posed linear problem to determine the solution on a known virtual boundary and (iii) the solution of a direct problem in the virtual domain which leads to the identification of the unknown curve using the ${\rm MATLAB}^®$ functions contour in 2D and isosurface in 3D. The results of several numerical experiments for steady-state heat conduction and linear elasticity in two and three dimensions are presented and analyzed.