On efficient reconstruction of boundary data with optimal placement of the source points in the MFS: application to inverse Stefan problems: application to inverse Stefan problems

Current practice in the use of the method of fundamental solutions (MFS) for inverse Stefan problems typically involves setting the source and collocation points at some distance, h,  from the boundaries of the domain in which the solution is required, and then varying their number, N, so that the obtained solution fulfils a desired tolerance, Tol, when a random noise level δ is added to the boundary conditions. This leads to an open question: can h and N be chosen simultaneously so that N is minimized, thereby leading to a lower computational expense in the solution of the inverse problem? Here, we develop a novel, simple and practical algorithm to help answer this question. The algorithm is used to study the effect of Tol and δ on both h and N. Its effectiveness is shown through three test problems and numerical experiments show promising results: for example, even with δ as high as 5% and Tol as low as 10^-3, we are able to find satisfactory solutions for N as low as 8.