  In this paper, we study a $\tau$-matrix approximation based preconditioner
for the linear systems arising from discretization of unsteady state Riesz
space fractional diffusion equation with non-separable variable coefficients.
The structure of coefficient matrices of the linear systems is identity plus
summation of diagonal-times-multilevel-Toeplitz matrices. In our
preconditioning technique, the diagonal matrices are approximated by scalar
identity matrices and the Toeplitz matrices are approximated by {\tau}-matrices
(a type of matrices diagonalizable by discrete sine transforms). The proposed
preconditioner is fast invertible through the fast sine transform (FST)
algorithm. Theoretically, we show that the GMRES solver for the preconditioned
systems has an optimal convergence rate (a convergence rate independent of
discretization stepsizes). To the best of our knowledge, this is the first
preconditioning method with the optimal convergence rate for the
variable-coefficients space fractional diffusion equation. Numerical results
are reported to demonstrate the efficiency of the proposed method.
