  In this paper, we study a parallel-in-time (PinT) algorithm for all-at-once
system from a non-local evolutionary equation with weakly singular kernel where
the temporal term involves a non-local convolution with a weakly singular
kernel and the spatial term is the usual Laplacian operator with variable
coefficients. We propose to use a two-sided preconditioning technique for the
all-at-once discretization of the equation. Our preconditioner is constructed
by replacing the variable diffusion coefficients with a constant coefficient to
obtain a constant-coefficient all-at-once matrix. We split a square root of the
constant Laplacian operator out of the constant-coefficient all-at-once matrix
as a right preconditioner and take the remaining part as a left preconditioner,
which constitutes our two-sided preconditioning. Exploiting the
diagonalizability of the constant-Laplacian matrix and the triangular Toeplitz
structure of the temporal discretization matrix, we obtain efficient
representations of inverses of the right and the left preconditioners, because
of which the iterative solution can be fast updated in a PinT manner.
Theoretically, the condition number of the two-sided preconditioned matrix is
proven to be uniformly bounded by a constant independent of the matrix size. To
the best of our knowledge, for the non-local evolutionary equation with
variable coefficients, this is the first attempt to develop a PinT
preconditioning technique that has fast and exact implementation and that the
corresponding preconditioned system has a uniformly bounded condition number.
Numerical results are reported to confirm the efficiency of the proposed
two-sided preconditioning technique.
