Toeplitz-like preconditioner for linear systems from spatial fractional diffusion equations

‎The article deals with constructing Toeplitz-like preconditioner for linear systems arising from finite difference discretization of the spatial fractional diffusion equations‎. ‎The coefficient matrices of these linear systems have an $S+L$ structure‎, ‎where $S$ is a symmetric positive definite (SPD) matrix and $L$ satisfies $mbox{rank}(L)leq 2$‎. ‎We introduce an approximation for the SPD part $S$‎, ‎which is called $P_S$‎, ‎and then we show that the preconditioner $P=P_S+L$ has the Toeplitz-like structure and its displacement rank is 6‎.  ‎The analysis shows that the eigenvalues of the corresponding preconditioned matrix are clustered around 1. Numerical experiments exhibit that the Toeplitz-like preconditioner can significantly improve the convergence properties of the applied iteration method.