Efficient numerical solution of boundary identification problems: MFS with adaptive stochastic optimization

In this article, we study a novel computational technique for the efficient numerical solution of the inverse boundary identification problem with uncertain data in two dimensions. The method essentially relies on a posteriori error indicators consisting of the Tikhonov regularized solutions obtained by the method of fundamental solutions (MFS) and the given data for the problem in hand. For a desired accuracy, the a posteriori error estimator chooses the best possible combination of a complete set of fundamental solutions determined by the location of the sources that are arranged in a particular manner on a pseudo-boundary at each iteration. Also, since we are interested in a stable solution, an adaptive stochastic optimization strategy based on an error-balancing criterion is used, so as to avoid unstable regions where the stability contributions may be relatively large. These ideas are applied to two benchmark problems and are found to produce efficient and accurate results.