Parallel-in-time preconditioners for the Sinc-Nystr\"{o}m method

TL;DR

This paper introduces parallel-in-time preconditioners for the Sinc-Nyström method, enabling efficient solution of large dense systems from high-order discretizations of PDEs with mesh-independent convergence.

Contribution

The paper proposes a novel low-rank perturbation preconditioner that clusters eigenvalues and allows parallel processing, improving computational efficiency for the Sinc-Nyström method.

Findings

Eigenvalues of preconditioned system are highly clustered.

Preconditioner enables parallel computation across time points.

Numerical results show mesh-independent convergence rates.

Abstract

The Sinc-Nystr\"{o}m method is a high-order numerical method based on Sinc basis functions for discretizing evolutionary differential equations in time. But in this method we have to solve all the time steps in one-shot (i.e. all-at-once), which results in a large-scale nonsymmetric dense system that is expensive to handle. In this paper, we propose and analyze preconditioner for such dense system arising from both the parabolic and hyperbolic PDEs. The proposed preconditioner is a low-rank perturbation of the original matrix and has two advantages. First, we show that the eigenvalues of the preconditioned system are highly clustered with some uniform bounds which are independent of the mesh parameters. Second, the preconditioner can be used parallel for all the Sinc time points via a block diagonalization procedure. Such a parallel potential owes to the fact that the eigenvector matrix of the diagonalization is well conditioned. In particular, we show that the condition number of the eigenvector matrix only mildly grows as the number of Sinc time points increases, and thus the roundoff error arising from the diagonalization procedure is controllable. The effectiveness of our proposed PinT preconditioners is verified by the observed mesh-independent convergence rates of the preconditioned GMRES in reported numerical examples.