Improving the Efficiency and Theoretical Understanding of Time-Parallel Multigrid Methods (Aušra Pogoželskytė)

13.05.2025 14:00 – 15:00

Clusters worldwide have millions of cores. It is important to develop algorithms that can fully use those resources. Parallel-in-time algorithms parallelize simulations by taking advantage of additional cores to start computing the solution at a later time, even if the solution at the current time has not been computed yet. We present three algorithms: one which is easy to use, Parareal, and a variant with more layers of computation, Multigrid Reduction-in-Time, as well as a more performant one, Space-Time Multigrid. We analyze their convergence properties to make them more efficient and to better understand them. In particular, the convergence of Multigrid Reduction-in-Time was long thought to be linear but we show it is superlinear like the convergence of Parareal. To preserve performance as we increase the size of the problem, we can use coarsening in space. Only numerical results existed for this approach. We provide a theoretical analysis which shows that very oscillating information is badly transported by the algorithm, which could be problematic depending on the application. We present a new interpretation of Parareal as Runge-Kutta method. This outlines some of its inefficiencies and opens the door for better understanding of the method. Finally, we optimize parameters in Space-Time Multigrid to make it converge faster.

Lieu

Conseil Général 7-9, Room 1-05,

entrée libre