Using Sparse Tiling with Symmetric Multigrid

Good data locality is an important aspect of obtaining scalable performance for multigrid methods. However, locality can be difficult to achieve, especially when working with unstructured grids and sparse matrices whose structure is not known until runtime. Our previous work developed full sparse tiling, a runtime reordering and rescheduling technique for improving locality. We applied full sparse tiling to Gauss-Seidel, an iterative smoother that dominates the computation time of multigrid, and showed that, in principle, good locality could be achieved. In this paper, we consider whether sparse tiling techniques improve performance in practice and under the more general assumption that each level of the multigrid hierarchy is represented by a sparse matrix. We also expand the applicability of sparse tiling techniques to Gauss-Seidel with reverse ordering, which is used for symmetric multigrid. We demonstrate speedups both for full sparse tiling and for cache block sparse tiling, a competing technique.