ParaDIAG: Parallel-in-Time Algorithms Based on the Diagonalization Technique

In 2008, Maday and Ronquist introduce{d} an interesting new approach for the direct parallel-in-time (PinT) solution of time-dependent PDEs. The idea is to diagonalize the time stepping matrix, keeping the matrices for the space discretization unchanged, and then to solve all time steps in parallel. Since then, several variants appeared, and we call these closely related algorithms ParaDIAG algorithms. ParaDIAG algorithms in the literature can be classified into two groups: ParaDIAG-I: direct standalone solvers, and ParaDIAG-II: iterative solvers.We will explain the basic features of each group in this note. To have concrete examples, we will introduce ParaDIAG-I and ParaDIAG-II for the advection-diffusion equation. We will also introduce ParaDIAG-II for the wave equation and an optimal control problem for the wave equation. We show the main known theoretical results in each case, and also provide Matlab codes for testing. The goal of the Matlab codes (this http URL) is to help the interested reader understand the key features of the ParaDIAG algorithms, without intention to be highly tuned for efficiency and/or low memory use. We also provide speedup measurements of ParaDIAG algorithms for a 2D linear advection-diffusion equation. These results are obtained on the Tianhe-1 supercomputer in China, which is a multi-array, configurable and cooperative parallel system, and we compare these results to the performance of parareal and MGRiT, two widely used PinT algorithms. In a forthcoming update of this note, we will provide more material on ParaDIAG algorithms, in particular further Matlab codes and parallel computing results,also for more realistic applications.