Reconstruction of inclusions or cavities in potential problems using the MFS

The Method of Fundamental Solutions - A Meshless Method, Dynamic Publishers, (2008),
http://www.math.usm.edu/cschen/temp/Meshless.pdf#page=66

We discuss the application of the Method of Fundamental Solutions (MFS) to an inverse potential problem that consists of detecting inclusions or cavities using a single boundary measurement on an external boundary. The application of the method of fundamental solutions presented here is closely related to a method introduced by Kirsch and Kress [12], in another context – obstacle detection in an exterior acoustic inverse scattering problem. We briefly address the identifiability questions on the shape reconstruction, presenting a counterexample for the case of Robin boundary conditions. Using fundamental solutions on auxiliary boundary curves we prove density results for separated data on the whole of the boundary or for Cauchy data on the accessible part of the boundary. This justifies the reconstruction of the solution from the Cauchy data using the MFS. Moreover, this connection relates the linear part of the Kirsch-Kress method to the direct MFS, using boundary operator matrices. Numerical examples are presented, showing good reconstructions for the shape and location with this MFS version of the Kirsch-Kress method. Although the method proved to be robust for noisy data, results can be improved using an iterative Quasi-Newton scheme here presented.