Liu, Yaoning On block fast regularized Hermitian splitting preconditioning methods for solving discretized almost-isotropic high-dimensional spatial fractional diffusion equations. (Chinese. English summary) Zbl 1513.65055 Math. Numer. Sin. 44, No. 2, 187-205 (2022). Summary: 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. MSC: 65F08 Preconditioners for iterative methods 65F10 Iterative numerical methods for linear systems Keywords:Toeplitz matrix; circulant matrix; fractional diffusion equation; regularized Hermitian splitting preconditioner; preconditioned Krylov subspace iteration method PDFBibTeX XMLCite \textit{Y. Liu}, Math. Numer. Sin. 44, No. 2, 187--205 (2022; Zbl 1513.65055) Full Text: DOI