几乎各向同性的高维空间分数阶扩散方程的分块快速正则Hermite分裂预处理方法

一类空间分数阶扩散方程经过有限差分离散后所得到的离散线性方程组的系数矩阵是两个对角矩阵与Toeplitz型矩阵的乘积之和.在本文中,对于几乎各向同性的二维或三维空间分数阶扩散方程的离散线性方程组,采用预处理Krylov子空间迭代方法,我们利用其系数矩阵的特殊结构和具体性质构造了一类分块快速正则Hermite分裂预处理子.通过理论分析,我们证明了所对应的预处理矩阵的特征值大部分都聚集于1的附近.数值实验也表明,这类分块快速正则Hermite分裂预处理子可以明显地加快广义极小残量(GMRES)方法和稳定化的双共轭梯度(BiCGSTAB)方法等Krylov子空间迭代方法的收敛速度.

ON BLOCK FAST REGULARIZED HERMITIAN SPLITTING PRECONDITIONING METHODS FOR SOLVING DISCRETIZED ALMOST-ISOTROPIC HIGH-DIMENSIONAL SPATIAL FRACTIONAL DIFFUSION EQUATIONS

The finite-difference discretization of a class of spatial fractional diffusion equations gives the discrete linear system whose coefficient matrix is in the form of a sum of two diagonal-times-Toeplitz-like matrices. In this paper, for the discrete linear system of two- or three-dimensional discretized almost-istropic spatial fractional diffusion equation, we solve it by using the preconditioned Krylov subspace iteration methods, so we propose a block fast regularized Hermitian splitting preconditioner. From theoretical analysis, we prove that most of the eigenvalues of the corresponding preconditioned matrix are clustered around 1. Numerical experiments also demonstrate that the block fast regularized Hermitian splitting preconditioner can significantly accelerate the convergence rates of the Krylov subspace iteration methods such as generalized minimal residual (GMRES) and bi-conjugate gradient stabilized (BiCGSTAB) methods.