A Note on Parallel Preconditioning for All-At-Once Evolutionary PDEs
For researchers solving evolutionary PDEs, this work provides a practical parallel implementation of an existing preconditioner, showing scalability but offering only incremental improvements.
This paper presents parallel numerical results for a preconditioner for time-dependent PDEs, demonstrating strong scaling, and extends the preconditioner via a Neumann series approach for efficient parallel execution.
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