Block low-rank single precision coarse grid solvers for extreme scale multigrid methods

Extreme scale simulation requires fast and scalable algorithms, such as multigrid methods. To achieve asymptotically optimal complexity, it is essential to employ a hierarchy of grids. The cost to solve the coarsest grid system can often be neglected in sequential computings, but cannot be ignored in massively parallel executions. In this case, the coarsest grid can be large and its efficient solution becomes a challenging task. We propose solving the coarse grid system using modern, approximate sparse direct methods and investigate the expected gains compared with traditional iterative methods. Since the coarse grid system only requires an approximate solution, we show that we can leverage block low-rank techniques, combined with the use of single precision arithmetic, to significantly reduce the computational requirements of the direct solver. In the case of extreme scale computing, the coarse grid system is too large for a sequential solution, but too small to permit massively parallel efficiency. We show that the agglomeration of the coarse grid system to a subset of processors is necessary for the sparse direct solver to achieve performance. We demonstrate the efficiency of the proposed method on a Stokes-type saddle point system solved with a monolithic Uzawa multigrid method. In particular, we show that the use of an approximate sparse direct solver for the coarse grid system can outperform that of a preconditioned minimal residual iterative method. This is demonstrated for the multigrid solution of systems of order up to $10^{11}$ degrees of freedom on a petascale supercomputer using 43,200 processes.