Approximate inverse preconditioners for linear systems arising from spatial balanced fractional diffusion equations

We consider the preconditioned iterative methods for the linear systems arising from the finite volume discretization of spatial balanced fractional diffusion equations where the fractional differential operators are comprised of both Riemann-Liouville and Caputo fractional derivatives. The coefficient matrices of the linear systems consist of the sum of tridiagonal matrix and Toeplitz-times-diagonal-times-Toeplitz matrix. We propose using symmetric approximate inverse preconditioners to solve such linear systems. We show that the spectra of the preconditioned matrices are clustered around 1. Numerical examples, for both one and two dimensional problems, are given to demonstrate the efficiency of the new preconditioners.