Reducing Multicore Bandwidth Requirements for Combinatorial Multigrid

Memory bandwidth is a major limiting factor in the scalability of parallel algorithms. In this paper, we introduce hierarchical diagonal blocking, a sparse matrix representation which we believe captures most of optimization techniques in a common representation. It can take advantage of symmetry while still being easy to parallelize. It takes advantage of, or actually requires, reordering. It also allows for simple compression of column indices. As applications, we show how to use this highperformance SpMV kernel, together with precision reduction techniques, in a combinatorial multigrid solver to lower the bandwidth consumption without sacrificing the final solution’s quality. We provide extensive empirical evaluation of the effectiveness of the approach on a variety of benchmark matrices, demonstrating substantial speedups on all matrices considered.