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 rank(l)≤2‎. ‎we introduce an approximation for the spd part s‎, ‎which is called ps‎, ‎and then we show that the preconditioner p=ps+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.