Random walk algorithms with iterative refinement for calculating particle fluxes to small and remote boundary areas

This work addresses a well-known problem in Monte Carlo methods: The estimation of small probabilities for diffusion particles to reach small boundary regions located far from point sources. The problem is solved using a new stochastic method based on a combination of the global Random Walk on Spheres (RWS) algorithm and an iterative refinement applied directly to the underlying differential problem. The problems are studied in stationary formulation with mixed Dirichlet and Neumann boundary conditions. A series of numerical experiments demonstrates the superiority of the proposed algorithms over other approaches, in particular over methods based on fundamental solutions.