Fast algorithms for integral formulations of steady-state radiative transfer equation

TL;DR

This paper develops fast algorithms for solving the steady-state radiative transfer equation using integral formulations, Fourier and spherical harmonic transforms, and advanced matrix factorizations, enabling efficient solutions in various media.

Contribution

It introduces a novel approach combining Fourier and spherical harmonic transforms with recursive skeletonization for efficient solutions of the radiative transfer equation.

Findings

Algorithms are efficient for homogeneous media.

Recursive skeletonization improves inhomogeneous medium solutions.

Numerical results confirm the method's effectiveness across regimes.

Abstract

We investigate integral formulations and fast algorithms for the steady-state radiative transfer equation with isotropic and anisotropic scattering. When the scattering term is a smooth convolution on the unit sphere, a model reduction step in the angular domain using the Fourier transformation in 2D and the spherical harmonic transformation in 3D significantly reduces the number of degrees of freedoms. The resulting Fourier coefficients or spherical harmonic coefficients satisfy a Fredholm integral equation of the second kind. We study the uniqueness of the equation and proved an a priori estimate. For a homogeneous medium, the integral equation can be solved efficiently using the FFT and iterative methods. For an inhomogeneous medium, the recursive skeletonization factorization method is applied instead. Numerical simulations demonstrate the efficiency of the proposed algorithms in both homogeneous and inhomogeneous cases and for both transport and diffusion regimes.