Comparative Performance Analysis of Coarse Solvers for Algebraic Multigrid on Multicore and Manycore Architectures

Druinsky, Alex; Ghysels, Pieter; Li, Xiaoye S.; Marques, Osni; Williams, Samuel; Barker, Andrew T.; Kalchev, Delyan Z.; Vassilevski, Panayot S.

doi:10.1007/978-3-319-32149-3_12

Cited by 2 publications

(2 citation statements)

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“…The MUMPS solver is based on a matrix format called BLR, 20,21 where the matrix is partitioned into blocks in a checkerboard fashion, and blockwise low-rank approximations are exploited to significantly reduce the theoretical complexity 36 and practical cost of the factorization and solve phases. Although even lower theoretical complexities can be achieved by using multilevel 37 or hierarchical 10,38 approximations, the flexible BLR format has proven to be very efficient in the context of a general purpose, fully-featured sparse solver such as MUMPS. 20,39 Additionally, running MUMPS in single precision arithmetic decreases the overall memory usage and time consumption of the solver.…”

Section: Blr Approximationmentioning

confidence: 99%

“…An approximate direct solver is acceptable for the purpose of solving the MG coarse grid problem. 1,8 Although several low-rank techniques have been proposed in the past, for example, as preconditioner or solver on the coarse grid in AMG, [9][10][11] to the best of our knowledge, this article represents the first attempt to apply them to extreme scale geometric multigrid solvers.…”

Section: Introductionmentioning

confidence: 99%

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

Buttari

Huber²,

Leleux

et al. 2021

Numerical Linear Algebra App

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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 ecient 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 signicantly 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 eciency. 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 eciency of the proposed method on a Stokes-type saddle point system. We employ 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.

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