A meshless fading regularization algorithm for solving the Cauchy problem for the three-dimensional Helmholtz equation

We investigate the Cauchy problem associated with the Helmholtz equation in three dimensions, namely the numerical reconstruction of the primary field (Dirichlet data) and its normal derivative (Neumann data) on a part of the boundary from the knowledge of overprescribed noisy measurements taken on the remaining boundary part. This inverse problem is solved by combining the fading regularization method with the method of fundamental solutions (MFS). A stopping regularizing/stabilizing criterion is also proposed. Two numerical examples are investigated in order to validate the proposed method in terms of its accuracy, convergence, stability and efficiency.