An adaptive greedy technique for inverse boundary determination problem

In this paper, the method of fundamental solutions (MFS) is employed for determining an unknown portion of the boundary from the Cauchy data specified on parts of the boundary. We propose a new numerical method with adaptive placement of source points in the MFS to solve the inverse boundary determination problem. Since the MFS source points placement here is not trivial due to the unknown boundary, we employ an adaptive technique to choose a sub-optimal arrangement of source points on various fictitious boundaries. Afterwards, the standard Tikhonov regularization method is used to solve ill-conditional matrix equation, while the regularization parameter is chosen by the L-curve criterion. The numerical studies of both open and closed fictitious boundaries are considered. It is shown that the proposed method is effective and stable even for data with relatively high noise levels.