Monte Carlo Methods for Radiative Transfer with Singular Kernels

Abstract

This work is devoted to Monte Carlo methods for radiative transfer equations with singular kernels, and is motivated by the study of wave propagation in random media with long-range dependence. As opposed to the short-range case where the collision cross section is integrable and leads to a non-zero mean free time, the cross section is not integrable in the long-range situation and yields a vanishing mean free time. For computational efficiency, a particular care is then required in the construction of the stochastic processes used in the Monte Carlo methods. For this, we adapt a method of Asmussen-Rosiński and Cohen-Rosiński based on a small jumps/large jumps decomposition of the generator. We compare this method with another approach based on alpha-stable processes, and show the superiority of the first one. We consider various algorithms for the simulation of the jump distribution, and underline the efficiency of an appropriate stochastic collocation method. Comparisons between integrable and singular kernel solutions are given.