Light Weight Coarse Grid Aggregation for Smoothed Aggregation Algebraic Multigrid Solver

The smoothed aggregation algebraic multigrid (SA-AMG) method is one of the most efficient linear solvers for large sized problems. It creates smaller-sized matrices, which is called "coarse level matrices", from the problem matrix and the solution converges quickly using the multilevel structure. However, when SA-AMG method is executed with a large number of processes, the coarse level matrix size can become smaller than the number of executing processes. For example, supercomputer Fugaku in Japan has about 150,000 computing nodes, and hundreds of thousands processes are often launched for large sized problems. If the number of active processes is not reduced appropriately in coarse levels, coarse level matrices are over-distributed to processes, and it causes difficulty in generating the next coarser level matrix, which leads to ill-convergence. Thus coarse level matrix re-allocation adjustment becomes important, which reduces the number of active processes to maintain computational granularity. We've studied the coarse level matrix re-allocation adjustment by coarse grid aggregation that agglomerates neighboring process domains and shrinks the degree of parallelism in accordance with the size of the coarse-level domain. In this study, we propose a light-weight coarse grid aggregation method. The proposed method determines which adjacent process domains are grouped together mainly at the computational cost of several adjacent and collective communications. In the numerical test, the proposed method was applied to 1012 DOF distributed sparse matrix problem, which is generated from heterogeneous diffusion coefficient Poisson problem, with approximately 2.3 million CPU cores. It shows that it can be applicable and a good weak scaling performance of 1.3 times longer execution time for the 64 times larger sized problem.