A single-sided all-at-once preconditioning for linear system from a non-local evolutionary equation with weakly singular kernels

In Lin et al. (2021) [21] and Zhao et al. (2023) [37], two-sided preconditioning techniques are proposed for non-local evolutionary equations, which possesses (i) mesh-size independent theoretical bound of condition number of the two-sided preconditioned matrix; (ii) small and stable iteration numbers in numerical tests. In this paper, we modify the two-sided preconditioning by multiplying the left-sided and the right-sided preconditioners together as a single-sided preconditioner. Such a single-sided preconditioner essentially derives from approximating the spatial matrix with a fast diagonalizable matrix and keeping the temporal matrix unchanged. Clearly, the matrix-vector multiplication of the single-sided preconditioning is faster to compute than that of the two-sided one, since the single-sided preconditioned matrix has a simpler structure. More importantly, we show theoretically that the single-sided preconditioned generalized minimal residual (GMRES) method has a convergence rate no worse than the two-sided preconditioned one. As a result, the one-sided preconditioned GMRES solver requires less computational time than the two-sided preconditioned GMRES solver in total. Numerical results are reported to show the efficiency of the proposed single-sided preconditioning technique.