Algebraic multigrid domain and range decomposition (AMGDD / AMGRD)*
In modern largescale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMGDD) and algebraic multigrid range decomposition (AMGRD), that replace traditional AMG Vcycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods in Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 14111443], they differ primarily in that they are purely algebraic: AMGRD and AMGDD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMGRD and AMGDD is their novel residualmore »
 Authors:

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 Univ. of California at San Diego, La Jolla, CA (United States)
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Univ. of Colorado, Boulder, CO (United States)
 Publication Date:
 Report Number(s):
 LLNLJRNL666751
Journal ID: ISSN 10648275
 Grant/Contract Number:
 AC5207NA27344
 Type:
 Accepted Manuscript
 Journal Name:
 SIAM Journal on Scientific Computing
 Additional Journal Information:
 Journal Volume: 37; Journal Issue: 5; Journal ID: ISSN 10648275
 Publisher:
 SIAM
 Research Org:
 Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
 Sponsoring Org:
 USDOE
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; iterative methods; multigrid; algebraic multigrid; parallel; scalability; domain decomposition
 OSTI Identifier:
 1245718
Bank, R., Falgout, R. D., Jones, T., Manteuffel, T. A., McCormick, S. F., and Ruge, J. W.. Algebraic multigrid domain and range decomposition (AMGDD / AMGRD)*. United States: N. p.,
Web. doi:10.1137/140974717.
Bank, R., Falgout, R. D., Jones, T., Manteuffel, T. A., McCormick, S. F., & Ruge, J. W.. Algebraic multigrid domain and range decomposition (AMGDD / AMGRD)*. United States. doi:10.1137/140974717.
Bank, R., Falgout, R. D., Jones, T., Manteuffel, T. A., McCormick, S. F., and Ruge, J. W.. 2015.
"Algebraic multigrid domain and range decomposition (AMGDD / AMGRD)*". United States.
doi:10.1137/140974717. https://www.osti.gov/servlets/purl/1245718.
@article{osti_1245718,
title = {Algebraic multigrid domain and range decomposition (AMGDD / AMGRD)*},
author = {Bank, R. and Falgout, R. D. and Jones, T. and Manteuffel, T. A. and McCormick, S. F. and Ruge, J. W.},
abstractNote = {In modern largescale supercomputing applications, algebraic multigrid (AMG) is a leading choice for solving matrix equations. However, the high cost of communication relative to that of computation is a concern for the scalability of traditional implementations of AMG on emerging architectures. This paper introduces two new algebraic multilevel algorithms, algebraic multigrid domain decomposition (AMGDD) and algebraic multigrid range decomposition (AMGRD), that replace traditional AMG Vcycles with a fully overlapping domain decomposition approach. While the methods introduced here are similar in spirit to the geometric methods developed by Brandt and Diskin [Multigrid solvers on decomposed domains, in Domain Decomposition Methods in Science and Engineering, Contemp. Math. 157, AMS, Providence, RI, 1994, pp. 135155], Mitchell [Electron. Trans. Numer. Anal., 6 (1997), pp. 224233], and Bank and Holst [SIAM J. Sci. Comput., 22 (2000), pp. 14111443], they differ primarily in that they are purely algebraic: AMGRD and AMGDD trade communication for computation by forming global composite “grids” based only on the matrix, not the geometry. (As is the usual AMG convention, “grids” here should be taken only in the algebraic sense, regardless of whether or not it corresponds to any geometry.) Another important distinguishing feature of AMGRD and AMGDD is their novel residual communication process that enables effective parallel computation on composite grids, avoiding the alltoall communication costs of the geometric methods. The main purpose of this paper is to study the potential of these two algebraic methods as possible alternatives to existing AMG approaches for future parallel machines. As a result, this paper develops some theoretical properties of these methods and reports on serial numerical tests of their convergence properties over a spectrum of problem parameters.},
doi = {10.1137/140974717},
journal = {SIAM Journal on Scientific Computing},
number = 5,
volume = 37,
place = {United States},
year = {2015},
month = {10}
}