Efficient Parallel Multigrid Method on Intel Xeon Phi Clusters

The parallel multigrid method is expected to play an important role in scientific computing on exa-scale supercomputer systems for solving large-scale linear equations with sparse matrices. Because solving sparse linear systems is a very memory-bound process, efficient method for storage of coefficient matrices is a crucial issue. In the previous works, authors implemented sliced ELL method to parallel conjugate gradient solvers with multigrid preconditioning (MGCG) for the application on 3D groundwater flow through heterogeneous porous media (pGW3D-FVM), and excellent performance has been obtained on large-scale multicore/manycore clusters. In the present work, authors introduced SELL-C-sigma to the MGCG solver, and evaluated the performance of the solver with various types of OpenMP/MPI hybrid parallel programing models on the Oakforest-PACS (OFP) system at JCAHPC using up to 1,024 nodes of Intel Xeon Phi. Because SELL-C-sigma is suitable for wide-SIMD architecture, such as Xeon Phi, improvement of the performance over the sliced ELL was more than 20%. This is one of the first examples of SELL-C-sigma applied to forward/backward substitutions in ILU-type smoother of multigrid solver. Furthermore, effects of IHK/McKernel has been investigated, and it achieved 11% improvement on 1,024 nodes.