Random Walk on Fixed Spheres for Laplace and Lamé equations

The Random Walk on Fixed Spheres (RWFS) introduced in our paper [25], and furtherdeveloped in [26], is presented in details for Laplace and Lamé equations governing static elasticityproblems. The approach is based on the Poisson type integral formulae written for each disc of a domainconsisting of a family of overlapping discs. The original differential boundary value problem is equiv-alently reformulated in the form of a system of integral equations defined on the intersection surfaces(arches, in 2D, and caps, if generalized to 3D spheres). To solve the obtained system of integral equa-tions, a Random Walk procedure is constructed where the random walks are living on the intersectionsurfaces. Since the spheres are fixed, it is convenient to construct also discrete random walk methodsfor solving the system of linear equations approximating the system of integral equations. We develophere two classes of special Monte Carlo iterative methods for solving these systems of linear algebraicequations which are constructed as a kind of randomized versions of the Chebyshev iteration method andSuccessive Over Relaxation (SOR) method. It is found that in this class of randomized SOR methods,the Gauss-Seidel method has a minimal variance. In [25] we have concluded that in the case of classicalpotential theory, the Random Walk on Fixed Spheres considerably improves the convergence rate of thestandard Random Walk on Spheres method. More interesting, we succeeded there to extend the algo-rithm to the system of Lamé equations which cannot be solved by the conventional Random Walk onSpheres method. We present here a series of numerical experiments for 2D domains consisting of 5, 10,and 17 discs, and analyze the dependence of the variance on the number of discs and elastic constants.Further generalizations to Neumann and Dirichlet-Neumann boundary conditions are possible, see [23]