Stochastic simulation algorithms for calculating particle fluxes to small and remote boundary areas: the backward trajectory method

The paper suggests local and global random walk algorithms for calculating small particle fluxes to remote boundary regions. It solves a long standing problem known in Monte Carlo methods: estimating probabilities of diffusion particles reaching small boundary regions far from the position of point sources. This problem is solved using a new stochastic method based on a combination of the global Random Walk on Spheres algorithm (RWS) and backward trajectory simulation. The problems were studied in transient setting with mixed Dirichlet and Neumann boundary conditions. A series of numerical experiments is presented, demonstrating the advantage of the developed algorithms over the conventional direct Random Walk on Spheres method.