A Monte Carlo Method for 3D Radiative Transfer Equations with Multifractional Singular Kernels

TL;DR

This paper introduces a novel Monte Carlo method for 3D radiative transfer equations with complex, non-integrable scattering kernels, addressing computational challenges posed by vanishing mean free times and fractional operators.

Contribution

It presents a new Monte Carlo approach inspired by finance techniques to efficiently handle multifractional singular kernels in radiative transfer equations.

Findings

Effective reduction of computational cost through jump decomposition

Numerical simulations demonstrating method performance

Complete error analysis provided

Abstract

We propose in this work a Monte Carlo method for three dimensional scalar radiative transfer equations with non-integrable, space-dependent scattering kernels. Such kernels typically account for long-range statistical features, and arise for instance in the context of wave propagation in turbulent atmosphere, geophysics, and medical imaging in the peaked-forward regime. In contrast to the classical case where the scattering cross section is integrable, which results in a non-zero mean free time, the latter here vanishes. This creates numerical difficulties as standard Monte Carlo methods based on a naive regularization exhibit large jump intensities and an increased computational cost. We propose a method inspired by the finance literature based on a small jumps - large jumps decomposition, allowing us to treat the small jumps efficiently and reduce the computational burden. We demonstrate the performance of the approach with numerical simulations and provide a complete error analysis. The multifractional terminology refers to the fact that the high frequency contribution of the scattering operator is a fractional Laplace-Beltrami operator on the unit sphere with space-dependent index.