Parallel superconvergent multigrid

Frederickson, P O; McBryan, O A

Title: Parallel superconvergent multigrid

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Abstract

We describe a class of multiscale algorithms for the solution of large sparse linear systems that are particularly well adapted to massively parallel supercomputers. While standard multigrid algorithms are unable to effectively use all processors when computing on coarse grids, the new algorithms utilize the same number of processors at all times. The basic idea is to solve many coarse scale problems simultaneously, combining the results in an optimal way to provide an improved fine scale solution. As a result, convergence rates are much faster than for standard multigrid methods - we have obtained V-cycle convergence rates as good as .0046 with one smoothing application per cycle, and .0013 with two smoothings. On massively parallel machines the improved convergence rate is attained at no extra computational cost since processors that would otherwise be sitting idle are utilized to provide the better convergence. On serial machines the algorithm is slower because of the extra time spent on multiple coarse scales, though in certain cases the improved convergence rate may justify this - particularly in cases where other methods do not converge. In constant coefficient situations the algorithm is easily analyzed theoretically using Fourier methods on a single grid. The fact thatmore »« less

Authors:: Frederickson, P O; McBryan, O A

Publication Date:: Thu Jan 01 00:00:00 EST 1987

Research Org.:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States); Cornell Univ., Ithaca, NY (USA). Theory Center

OSTI Identifier:: 5972484

Report Number(s):: LA-UR-87-2329; CONF-8704205-1
ON: DE87013153

DOE Contract Number:: AC02-76ER03077; W-7405-ENG-36

Resource Type:: Conference

Resource Relation:: Conference: 3. conference on multigrid methods, Copper Mountain, CO, USA, 6 Apr 1987; Other Information: Portions of this document are illegible in microfiche products

Country of Publication:: United States

Language:: English

Subject:: 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; LINEAR PROGRAMMING; SUPERCONVERGENCE RELATIONS; ALGORITHMS; FOURIER ANALYSIS; NUMERICAL SOLUTION; OPTIMIZATION; PARALLEL PROCESSING; SUPERCOMPUTERS; THREE-DIMENSIONAL CALCULATIONS; COMPUTERS; DIGITAL COMPUTERS; MATHEMATICAL LOGIC; PROGRAMMING; 990210* - Supercomputers- (1987-1989); 990230 - Mathematics & Mathematical Models- (1987-1989)

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                    Frederickson, P O, and McBryan, O A. Parallel superconvergent multigrid.  United States: N. p., 1987. 
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                    Frederickson, P O, & McBryan, O A. Parallel superconvergent multigrid.  United States.

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                    Frederickson, P O, and McBryan, O A. 1987.  
        "Parallel superconvergent multigrid".  United States.  https://www.osti.gov/servlets/purl/5972484.

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@article{osti_5972484,

  title        = {Parallel superconvergent multigrid},

  author       = {Frederickson, P O and McBryan, O A},

  abstractNote = {We describe a class of multiscale algorithms for the solution of large sparse linear systems that are particularly well adapted to massively parallel supercomputers. While standard multigrid algorithms are unable to effectively use all processors when computing on coarse grids, the new algorithms utilize the same number of processors at all times. The basic idea is to solve many coarse scale problems simultaneously, combining the results in an optimal way to provide an improved fine scale solution. As a result, convergence rates are much faster than for standard multigrid methods - we have obtained V-cycle convergence rates as good as .0046 with one smoothing application per cycle, and .0013 with two smoothings. On massively parallel machines the improved convergence rate is attained at no extra computational cost since processors that would otherwise be sitting idle are utilized to provide the better convergence. On serial machines the algorithm is slower because of the extra time spent on multiple coarse scales, though in certain cases the improved convergence rate may justify this - particularly in cases where other methods do not converge. In constant coefficient situations the algorithm is easily analyzed theoretically using Fourier methods on a single grid. The fact that only one grid is involved substantially simplifies convergence proofs. A feature of the algorithms is the use of a matched pair of operators: an approximate inverse for smoothing and a superinterpolation operator to move the correction from coarse to fine scales, chosen to optimize the rate of convergence.},

  doi          = {},

  url          = {https://www.osti.gov/biblio/5972484},
  journal      = {},
number       = ,

  volume       = ,

  place        = {United States},

  year         = {Thu Jan 01 00:00:00 EST 1987},

  month        = {Thu Jan 01 00:00:00 EST 1987}

}

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