Parallel Quasirandom Walks on the Boundary

Abstract

The Monte Carlo method called “random walks on boundary” has been successfully used for solving boundary-value problems. This method has significant advantages when compared with random walks on spheres, balls or on discrete grids when an exterior Dirichlet or Neumann problem is solved, or when we are interested in computing the solution to a problem at an arbitrary number of points using a single random walk. In this paper we will investigate ways: • to increase the convergence rate of this method by using quasirandom sequences instead of pseudorandom numbers for the construction of the boundary walks, • to find an efficient parallel implementation of this method on a cluster using MPI. In our parallel implementation we use disjoint contiguous blocks of quasirandom numbers extracted from a given quasirandom sequence for each processor. In this case, the increased convergence rate does not come at the cost of less trustworthy answers. We also present some numerical examples confirming both the increased rate of convergence and the good parallel efficiency of the method.