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.