An adaptive boundary algorithm to reconstruct initial and boundary data using the method of fundamental solutions for the inverse Cauchy-Stefan problem

In this paper, a recent algorithm, based around the method of fundamental solutions (MFS), for reconstructing boundary data in inverse Stefan problems is extended and applied to inverse Cauchy–Stefan problems, wherein initial data must also be reconstructed. A key feature of the algorithm is that it is adaptive and iterates to find the optimal locations of the source points that are required by the method. Tikhonov regularization is used to take care of the ill-conditioned matrix that the MFS generates, with the algorithm being able to determine the optimal regularization parameter automatically. The effects of accuracy and random noise on the optimal location and number of source points are also evaluated. In addition, we consider a nonlinear variant of the inverse problem where one has to identify the moving boundary along with the missing initial data. Numerical experiments, carried out on five different benchmark examples, show promising results.