TY - JOUR
T1 - A Block Fast Regularized Hermitian Splitting Preconditioner for Two-Dimensional Discretized Almost Isotropic Spatial Fractional Diffusion Equations
AU - Liu , Yao-Ning
AU - Muratova , Galina V.
JO - East Asian Journal on Applied Mathematics
VL - 2
SP - 213
EP - 232
PY - 2022
DA - 2022/02
SN - 12
DO - http://doi.org/10.4208/eajam.070621.300821 
UR - https://global-sci.org/intro/article_detail/eajam/20251.html
KW - Preconditioning, spatial fractional diffusion equation, Toeplitz matrix, two-dimensional problem.
AB - <p style="text-align: justify;">Block fast regularized Hermitian splitting preconditioners for matrices arising
in approximate solution of two-dimensional almost-isotropic spatial fractional diffusion
equations are constructed. The matrices under consideration can be represented as the
sum of two terms, each of which is a nonnegative diagonal matrix multiplied by a block
Toeplitz matrix having a special structure. We prove that excluding a small number
of outliers, the eigenvalues of the preconditioned matrix are located in a complex disk
of radius $r&lt;1$ and centered at the point $z_0=1$. Numerical experiments show that
such structured preconditioners can significantly improve computational efficiency of
the Krylov subspace iteration methods such as the generalized minimal residual and
bi-conjugate gradient stabilized methods. Moreover, if the corresponding equation is
almost isotropic, the methods constructed outperform many other existing preconditioners.</p>