Cyclic Reduction Tridiagonal Solvers on GPUs Applied to Mixed-Precision Multigrid

We have previously suggested mixed precision iterative solvers specifically tailored to the iterative solution of sparse linear equation systems as they typically arise in the finite element discretization of partial differential equations. These schemes have been evaluated for a number of hardware platforms, in particular, single-precision GPUs as accelerators to the general purpose CPU. This paper reevaluates the situation with new mixed precision solvers that run entirely on the GPU: We demonstrate that mixed precision schemes constitute a significant performance gain over native double precision. Moreover, we present a new implementation of cyclic reduction for the parallel solution of tridiagonal systems and employ this scheme as a line relaxation smoother in our GPU-based multigrid solver. With an alternating direction implicit variant of this advanced smoother, we can extend the applicability of the GPU multigrid solvers to very ill-conditioned systems arising from the discretization on anisotropic meshes, that previously had to be solved on the CPU. The resulting mixed-precision schemes are always faster than double precision alone, and outperform tuned CPU solvers consistently by almost an order of magnitude.