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

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.

24 pages, 3 figures

References in corpus (1)

• A fast algorithm for radiative transport in isotropic media

Cited by in corpus (10)

• Meta-learning Pseudo-differential Operators with Deep Neural Networks
• A Nonlinear Three-Moment Model for Radiative Transfer in Spherical Symmetry
• A Nonlinear Hyperbolic Model for Radiative Transfer Equation in Slab Geometry
• A multiscale neural network based on hierarchical nested bases
• Separability of the kernel function in an integral formulation for anisotropic radiative transfer equation
• A Nonlinear Moment Model for Radiative Transfer Equation
• A Nonlinear Moment Model for Radiative Transfer Equation in Slab Geometry
• Quantitative PAT with simplified $P_N$ approximation
• Coarse-Proxy Reduced Basis Methods for Integral Equations
• A fast algorithm for time-dependent radiative transport equation based on integral formulation