A Note on Parallel Preconditioning for All-At-Once Evolutionary PDEs

TL;DR

This paper demonstrates the strong parallel scalability of a preconditioning method for time-dependent PDEs based on circulant matrices, and extends it using a Neumann series for improved efficiency.

Contribution

It provides parallel numerical results confirming scalability and introduces an extended preconditioner via Neumann series for better parallel performance.

Findings

Strong parallel scaling demonstrated in numerical experiments

Extended preconditioner via Neumann series improves efficiency

Open-source implementation available in C++ and MPI

Abstract

McDonald, Pestana and Wathen (SIAM J. Sci. Comput. 40(2), pp. A2012-A1033, 2018) present a method for preconditioning of time-dependent PDEs via approximation by a nearby time-periodic problem, that is, they employ circulant-related matrices as preconditioners for the non-symmetric block Toeplitz matrices which arise from an all-at-once formulation. They suggest that such an approach might be efficiently implemented in parallel. In this short article, we present parallel numerical results for their preconditioner which exhibit strong scaling. We also extend their preconditioner via a Neumann series approach, which also allows for efficient parallel execution. Our simple implementation (in C++ and MPI) is available at the Git repository PARALAAOMPI. https://github.com/anthonyjamesgoddard/PARALAAOMPI