Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators

In this paper we combine the method of fundamental solutions with various regularization techniques to solve Cauchy problems of elliptic differential operators. The main idea is to approximate the unknown solution by a linear combination of fundamental solutions whose singularities are located outside the solution domain. To solve effectively the discrete ill-posed resultant matrix, we use three regularization strategies under three different choices for the regularization parameter. Several examples on problems with smooth and non-smooth geometries in 2D and 3D spaces using under-, equally, and over-specified Cauchy data on an accessible boundary are given. Numerical results indicate that the generalized cross-validation and LL-curve choice rulers for Tikhonov regularization and damped singular value decomposition strategy are most effective when using the same numbers of collocation and source points. It has also been observed that the use of more Cauchy data will greatly improve the accuracy of the approximate solution.