A parallel-in-time two-sided preconditioning for all-at-once system from a non-local evolutionary equation with weakly singular kernel

TL;DR

This paper introduces a novel parallel-in-time preconditioning method for all-at-once systems derived from non-local evolutionary equations with weakly singular kernels, achieving efficient and stable iterative solutions.

Contribution

It develops the first fast, exact two-sided preconditioning technique with a uniformly bounded condition number for such non-local equations with variable coefficients.

Findings

Preconditioned system has a condition number bounded independently of matrix size.

The proposed method enables fast, efficient iterative solutions in a parallel-in-time framework.

Numerical results confirm the effectiveness and efficiency of the preconditioning approach.

Abstract

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.