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 i...

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Veröffentlicht in:	Numerical linear algebra with applications Jg. 29; H. 1
Hauptverfasser:	Buttari, Alfredo, Huber, Markus, Leleux, Philippe, Mary, Theo, Rüde, Ulrich, Wohlmuth, Barbara
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Oxford Wiley Subscription Services, Inc 01.01.2022 Wiley
Schlagworte:	Algorithms Analysis of PDEs Approximation Asymptotic methods block low‐rank Computer Science Distributed, Parallel, and Cluster Computing efficient coarse level solver hierarchical hybrid grids high‐performance computing Iterative methods Mathematics Multigrid methods MUMPS Saddle points Solvers sparse direct solver sparse direct solver block low-rank efficient coarse level solver hierarchical hybrid grids geometric multigrid MUMPS multifrontal high-performance computing
ISSN:	1070-5325, 1099-1506
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Abstract	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 1011 degrees of freedom on a petascale supercomputer using 43,200 processes.
AbstractList	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 1011 degrees of freedom on a petascale supercomputer using 43,200 processes. 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 degrees of freedom on a petascale supercomputer using 43,200 processes. 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. 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 1+e11 degrees of freedom on a petascale supercomputer using 43 200 processes.
Author	Mary, Theo Wohlmuth, Barbara Leleux, Philippe Buttari, Alfredo Huber, Markus Rüde, Ulrich
Author_xml	– sequence: 1 givenname: Alfredo orcidid: 0000-0003-3207-7021 surname: Buttari fullname: Buttari, Alfredo email: alfredo.buttari@irit.fr organization: CNRS – sequence: 2 givenname: Markus surname: Huber fullname: Huber, Markus email: huber@ma.tum.de organization: Technical Universtiy Munich – sequence: 3 givenname: Philippe orcidid: 0000-0002-3760-4698 surname: Leleux fullname: Leleux, Philippe email: leleux@cerfacs.fr organization: CERFACS – sequence: 4 givenname: Theo orcidid: 0000-0001-9949-4634 surname: Mary fullname: Mary, Theo email: theo.mary@lip6.fr organization: Sorbonne Université – sequence: 5 givenname: Ulrich surname: Rüde fullname: Rüde, Ulrich email: Ulrich.ruede@fau.de organization: Friedrich‐Alexander Universität – sequence: 6 givenname: Barbara surname: Wohlmuth fullname: Wohlmuth, Barbara email: wohlmuth@ma.tum.de organization: Technical Universtiy Munich
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Keywords	sparse direct solver block low-rank efficient coarse level solver hierarchical hybrid grids geometric multigrid MUMPS multifrontal high-performance computing
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Snippet	Extreme scale simulation requires fast and scalable algorithms, such as multigrid methods. To achieve asymptotically optimal complexity, it is essential to... Extreme scale simulation requires fast and scalable algorithms, such as multigrid methods. To achieve asymptotically optimal complexity it is essential to...
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SubjectTerms	Algorithms Analysis of PDEs Approximation Asymptotic methods block low‐rank Computer Science Distributed, Parallel, and Cluster Computing efficient coarse level solver hierarchical hybrid grids high‐performance computing Iterative methods Mathematics Multigrid methods MUMPS Saddle points Solvers sparse direct solver
Title	Block low‐rank single precision coarse grid solvers for extreme scale multigrid methods
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