Stochastic Boundary Methods of Fundamental Solutions for solving PDEs

Abstract

Abstract A mesh-free Stochastic Boundary Method (SBM) based on randomized versions of the Method of Fundamental Solutions (MFS) is suggested. The randomization is used in the following steps of MFS: (1) the singular source positions are randomly distributed outside the domain, (2) the large system of linear equations for the weights in the expansion over the fundamental solutions is resolved by a randomized SVD method we introduced in [56] , or the randomized projection method we developed in [54] . We construct also a new method of stochastic boundary method based on the inversion of the Poisson formula representing the solution in a disc (a sphere, in R 3 ). We present a series of applications of the suggested SBM: we combine SBM with the Random Walk on Spheres and Random Walk on Boundary algorithms which results in methods giving the solution in any set of arbitrary points, without introducing any mesh in the domain. The Laplace, biharmonic, and the system of elasticity equations are involved in our analysis. We present some numerical results and give a brief discussion of the performance of the suggested methods. The numerical experiments carried out for the Laplace and Lame equations confirm our conclusion that the best results are obtained with the overdetermined systems generated by MFS where the number of source points is considerably smaller than the number of collocation points.