Optimal preconditioners for nonsymmetric multilevel Toeplitz systems with application to solving non-local evolutionary partial differential equations

TL;DR

This paper introduces a new preconditioning method for nonsymmetric multilevel Toeplitz systems, particularly those from evolutionary PDEs, achieving mesh-independent convergence with efficient implementation and broad applicability.

Contribution

We develop a novel symmetric positive definite multilevel Tau preconditioner for nonsymmetric Toeplitz systems, enabling optimal, mesh-independent convergence in solving non-local evolutionary PDEs.

Findings

Preconditioner achieves mesh-independent convergence.

Numerical examples demonstrate effectiveness.

Applicable to a wide range of non-local PDEs.

Abstract

Preconditioning for multilevel Toeplitz systems has long been a focal point of research in numerical linear algebra. In this work, we develop a novel preconditioning method for a class of nonsymmetric multilevel Toeplitz systems, which includes the all-at-once systems that arise from evolutionary partial differential equations. These systems have recently garnered considerable attention in the literature. To further illustrate our proposed preconditioning strategy, we specifically consider the application of solving a wide range of non-local, time-dependent partial differential equations in a parallel-in-time manner. For these equations, we propose a symmetric positive definite multilevel Tau preconditioner that is not only efficient to implement but can also be adapted as an optimal preconditioner. In this context, the proposed preconditioner is optimal in the sense that it enables mesh-independent convergence when using the preconditioned generalized minimal residual method. Numerical examples are provided to critically analyze the results and underscore the effectiveness of our preconditioning strategy.