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

• Published in: Journal of computational physics Vol. 380; no. C; pp. 191 - 211
• Main Authors: Fan, Yuwei, An, Jing, Ying, Lexing
• Format: Journal Article
• Language: English
• Published: Cambridge Elsevier Inc 01.03.2019; Elsevier Science Ltd; Elsevier
• Subjects: Algorithms; Anisotropic scattering; Computational physics; Computer simulation; Convolution; Fast algorithm; FFT; Formulations; Fourier transforms; Fredholm equations; Fredholm integral equation; Inhomogeneous media; Integral equations; Iterative methods; Mathematical models; Model reduction; Radiative transfer; Recursive methods; Recursive skeletonization; Scattering; Spherical harmonics; Steady state; Radiative transfer; Anisotropic scattering; Fast algorithm; Recursive skeletonization; Fredholm integral equation; FFT
• ISSN: 0021-9991; 1090-2716; 1090-2716
• DOI: 10.1016/j.jcp.2018.12.014

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. •Proposed a novel integral equation based model reduction of steady-state RTE.•Constructed FFT-based and RSF-based algorithms for the integral equation.•Proved uniqueness of the integral equation and presented an a priori estimate.•Demonstrate numerically the efficiency of the proposed algorithms.