Scaling Structured Multigrid to 500K+ Cores Through Coarse-Grid Redistribution

Reisner, Andrew; Olson, Luke N.; Moulton, J. David

doi:10.1137/17m1146440

Cited by 12 publications

(5 citation statements)

References 28 publications

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“…Here, we clearly see another increase in the cost per linear iteration, especially in the three-stage methods where the time required increases by about 50% for = 7. Improved performance would almost certainly be seen by duplicating the coarse-grid solve on each node, as considered in [58][59][60]. We leave these performance enhancements for future work.…”

Section: Two-dimensional Time-dependent Stokesmentioning

confidence: 99%

Monolithic multigrid for implicit Runge-Kutta discretizations of incompressible fluid flow

Abu-Labdeh¹,

MacLachlan²,

Farrell³

2022

Preprint

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Most research on preconditioners for time-dependent PDEs has focused on implicit multi-step or diagonallyimplicit multi-stage temporal discretizations. In this paper, we consider monolithic multigrid preconditioners for fully-implicit multi-stage Runge-Kutta (RK) time integration methods. These temporal discretizations have very attractive accuracy and stability properties, but they couple the spatial degrees of freedom across multiple time levels, requiring the solution of very large linear systems. We extend the classical Vanka relaxation scheme to implicit RK discretizations of saddle point problems. We present numerical results for the incompressible Stokes, Navier-Stokes, and resistive magnetohydrodynamics equations, in two and three dimensions, confirming that these relaxation schemes lead to robust and scalable monolithic multigrid methods for a challenging range of incompressible fluid-flow models.

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Section: Two-dimensional Time-dependent Stokesmentioning

confidence: 99%

Monolithic multigrid for implicit Runge-Kutta discretizations of incompressible fluid flow

Abu-Labdeh¹,

MacLachlan²,

Farrell³

2022

Preprint

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“…Guided by a predictive performance model, the algorithm provides robust redistribution decisions for structured multilevel solvers. It used a two-dimensional diffusion problem to show that the algorithm provides significant performance gains over previous methods using agglomeration by processor [1].…”

Section: Related Workmentioning

confidence: 99%

Assessment of English Teaching Post Competency Relying on K-Means Clustering Computing Algorithm

Meng

2022

Mathematical Problems in Engineering

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Post competency assessment has become an important part of modern recruitment, and teaching is one of the indispensable occupations. Assessing the competence of teachers in teaching positions has become a hot topic. This paper takes English teachers as the research object and uses K-means clustering algorithm to evaluate the competence of teachers. This paper first analyzes the references related to this research and finds suitable materials for this research and then mainly describes in detail the main point of English teaching post competency. Next, this paper uses the formula to illustrate the K-means clustering algorithm used in this paper. Finally, it uses experiments to verify the influencing factors of the four dimensions of English job competency. And the final survey results show that the influencing factors have different significant differences on the four dimensions. And the experiment proves the effectiveness and robustness of the proposed algorithm in evaluating the competence of English teaching positions. In addition, from the perspective of the composition of professional titles, it is consistent with the age structure, accounting for 63.3% of the total. Associate senior researchers are the backbone of colleges and universities. This research provides a theoretical basis and reference for measurement tools for college English teacher recruitment and selection, vocational training, performance evaluation, salary reform, and other related educational practices.

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“…Thus the coarsest grid problems are redistributed. The remaining processors can either perform redundant computations or the unneeded processors stay idle [31,30]. To make dierent coarse grid solvers available and thus help improve the HHG multigrid eciency, we link an external software library to the HHG framework.…”

Section: Coarse Level Strategiesmentioning

confidence: 99%

“…The remaining processors can either perform redundant computations or the unneeded processors stay idle. 17,18 To make different coarse grid solvers available and thus help improve the HHG multigrid efficiency, we link an external software library to the HHG framework. The efficiency of these external solvers is also limited with excessive number of processors so that agglomerating the coarse grid data to a suitable number of processors is essential.…”

Section: Coarse Level Strategiesmentioning

confidence: 99%

“…The coarser the problem, the smaller the number of processes involved, in order to avoid an unnecessary large volume of communication. While in this article, we base our agglomeration strategy on heuristics, a performance model is used in [30] for a structured MG scenario to predict the best agglomeration strategy. In [31], the GMG solver is constructed by starting from a coarse grid.…”

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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Scaling Structured Multigrid to 500K+ Cores Through Coarse-Grid Redistribution

Cited by 12 publications

References 28 publications

Monolithic multigrid for implicit Runge-Kutta discretizations of incompressible fluid flow

Monolithic multigrid for implicit Runge-Kutta discretizations of incompressible fluid flow

Assessment of English Teaching Post Competency Relying on K-Means Clustering Computing Algorithm

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

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