An all-at-once preconditioner for evolutionary partial differential equations

TL;DR

This paper introduces a generalized block circulant preconditioner for evolutionary PDEs that maintains efficiency and improves convergence by incorporating a small parameter, ensuring eigenvalues cluster at 1 and enabling size-independent linear convergence.

Contribution

The paper extends existing block circulant preconditioners by adding a small parameter, preserving key properties and enhancing performance for solving evolutionary PDE systems.

Findings

Eigenvalues of the preconditioned matrix cluster at 1 for small epsilon

The generalized preconditioner maintains diagonalizability and low-rank structure

GMRES convergence rate is independent of system size with the new preconditioner

Abstract

In [McDonald, Pestana and Wathen, \textit{SIAM J. Sci. Comput.}, 40 (2018), pp. A1012--A1033], a block circulant preconditioner is proposed for all-at-once linear systems arising from evolutionary partial differential equations, in which the preconditioned matrix is proven to be diagonalizable and to have identity-plus-low-rank decomposition in the case of the heat equation. In this paper, we generalize the block circulant preconditioner by introducing a small parameter $\epsilon>0$ into the top-right block of the block circulant preconditioner. The implementation of the generalized preconditioner requires the same computational complexity as that of the block circulant one.Theoretically, we prove that (i) the generalization preserves the diagonalizability and the identity-plus-low-rank decomposition; (ii) all eigenvalues of the new preconditioned matrix are clustered at 1 for sufficiently small $\epsilon$; (iii) GMRES method for the preconditioned system has a linear convergence rate independent of size of the linear system when $\epsilon$ is taken to be smaller than or comparable to square root of time-step size. Numerical results are reported to confirm the efficiency of the proposed preconditioner and to show that the generalization improves the performance of block circulant preconditioner.