  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.
