快速算法用于稳态辐射传输方程的积分形式
Fast algorithms for integral formulations of steady-state radiative transfer equation
AI总结：
本文研究了稳态辐射传输方程的积分形式及快速算法，通过傅里叶变换和球谐变换减少自由度，提出了解决均匀和非均匀介质的有效方法，验证了算法在传输和扩散情形下的效率。
摘要：
我们研究了稳态辐射传输方程中各向同性和各向异性散射的积分形式及快速算法。当散射项是单位球上的光滑卷积时，在角域中使用二维傅里叶变换和三维球谐变换进行模型降阶显著减少自由度。所得的傅里叶系数或球谐系数满足二自由度积分方程。我们研究了方程的唯一性并证明了先验估计。对于均匀介质，可以使用FFT和迭代方法高效求解积分方程。对于非均匀介质，应用递归骨架分解法。数值模拟展示了所提算法在均匀和非均匀情况以及传输和扩散情形下的效率。
英文摘要：
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.
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