Communication-Computation Overlapping for Parallel Multigrid Methods

Preconditioned iterative methods based on the Krylov subspace technique are widely employed in various scientific and technical computing. When utilizing large-scale parallel computing systems, the communication overhead tends to increase with the growth in the number of nodes, making its reduction a crucial challenge. In parallel finite element methods (FEM) and finite volume methods (FVM), halo communication and computation overlapping (CC-Overlapping) are commonly employed, often in conjunction with the dynamic loop scheduling feature of OpenMP. This approach has been primarily applied to sparse matrix-vector products (SpMV) and explicit solvers. Previous studies by the author have proposed reordering techniques for applying CC-Overlapping to processes involving global data dependencies, such as the Conjugate Gradient method preconditioned by Incomplete Cholesky Factorization (ICCG). Successful implementations on massively parallel supercomputers demonstrated high parallel performance, but the application of CC-Overlapping was limited to SpMV. In the present work, the author proposes a method to apply CC-Overlapping to the forward and backward substitutions of the IC(0) smoother of the parallel Conjugate Gradient method preconditioned by Multigrid (MGCG). Using up to 4,096 nodes on Wisteria/BDEC-01 (Odyssey) with A64FX, performance improvement of approximately 40+% was achieved compared to the original implementation, while improvement was 20+% on 1,024 nodes of Oakbridge-CX system with Intel Xeon CPU's.