Scalable implementation of multigrid methods using partial semi-aggregation of coarse grids

Multigrid methods are efficient and fast algorithms for solving elliptic equations. However, they suffer from the degradation of parallel efficiency on coarser levels: communication costs are much higher than computational costs when the number of computing core becomes massive. This study presents a coarse-grid partial semi-aggregation (CGPSA) method, where coarse grids in each direction are independently aggregated step-by-step across multiple levels. This multilevel aggregation provides a hierarchical communication structure. Thus, communication overheads are distributed into multiple levels and computational workloads are processed by multiple processes on coarser levels. Independent coarse-grid aggregation along each dimension also enhances the flexibility of multigrid method; it can be applied to the problem with a non-cubic geometry and anisotropic sizes of grids and parallel processes. Benchmark results on a large-scale cluster (up to 65k cores) are provided with a detailed performance profiling. The results show the improvement of parallel performance and scalability with the CGPSA method, which focuses on reducing communication overheads at coarser levels. They also show that the changes in the order of semi-aggregation levels can further improve performance, which implies that the proposed CGPSA method presents further scope for performance optimization.