BoomerAMG: A parallel algebraic multigrid solver and preconditioner. Driven by the need to solve linear systems arising from problems posed on extremely large, unstructured grids, there has been a recent resurgence of interest in algebraic multigrid (AMG). AMG is attractive in that it holds out the possibility of multigrid-like performance on unstructured grids. The sheer size of many modern physics and simulation problems has led to the development of massively parallel computers, and has sparked much research into developing algorithms for them. Parallelizing AMG is a difficult task, however. While much of the AMG method parallelizes readily, the process of coarse-grid selection, in particular, is fundamentally sequential in nature. We have previously introduced a parallel algorithm [cf. A. J. Cleary, R. D. Falgout, V. E. Henson and J. E. Jones, Coarse grid selection for parallel algebraic multigrid, in: A. Ferriera, J. Rollin, H. Simon, S.-H. Teng (eds.), Proceedings of the Fifth International Symposium on Solving Irregularly Structured Problems in Parallel, Lecture Notes in Computer Science, Vol. 1457, Springer, New York (1998)] for the selection of coarse-grid points, based on modifications of certain parallel independent set algorithms and the application of heuristic designed to insure the quality of the coarse grids, and shown results from a prototype serial version of the algorithm. In this paper we describe an implementation of a parallel AMG code, using the algorithm of A. J. Cleary, R. D. Falgout and V. E. Henson [loc. cit.] as well as other approaches to parallelizing the coarse-grid selection. We consider three basic coarsening schemes and certain modifications to the basic schemes, designed to address specific performance issues. We present numerical results for a broad range of problem sizes and descriptions, and draw conclusion regarding the efficacy of the method. Finally, we indicate the current directions of the research.

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  1. Tao, Weifeng; Cai, Dongsheng; Yan, Xiaoyang; Ken-Ichi, Nishikawa; Lembege, Bertrand: Scalability analysis of parallel particle-in-cell codes on computational grids (2008)
  2. D’Ambra, Pasqua; di Serafino, Daniela; Filippone, Salvatore: On the development of PSBLAS-based parallel two-level Schwarz preconditioners (2007)
  3. MacLachlan, S.; Saad, Yousef: A greedy strategy for coarse-grid selection (2007)
  4. Scheichl, R.; Vainikko, E.: Additive Schwarz with aggregation-based coarsening for elliptic problems with highly variable coefficients (2007)
  5. Barchanski, Andreas; Clemens, Markus; De Gersem, Herbert; Weiland, Thomas: Efficient calculation of current densities in the human body induced by arbitrarily shaped, low-frequency magnetic field sources (2006)
  6. De Sterck, Hans; Yang, Ulrike Meier; Heys, Jeffrey J.: Reducing complexity in parallel algebraic multigrid preconditioners (2006)
  7. Falgout, Robert D.; Jones, Jim E.; Meier Yang, Ulrike: Pursuing scalability for hypre’s conceptual interfaces. (2005)
  8. Kirby, Robert C.; Knepley, Matthew; Logg, Anders; Scott, L. Ridgway: Optimizing the evaluation of finite element matrices (2005)
  9. De Sterck, H.; Manteuffel, Thomas A.; McCormick, Stephen F.; Olson, Luke: Least-squares finite element methods and algebraic multigrid solvers for linear hyperbolic PDEs (2004)
  10. Masson, Roland; Quandalle, Philippe; Requena, Stéphane; Scheichl, Robert: Parallel preconditioning for sedimentary basin simulations (2004)
  11. Adams, Mark; Brezina, Marian; Hu, Jonathan; Tuminaro, Ray: Parallel multigrid smoothing: Polynomial versus Gauss--Seidel (2003)
  12. Li, Zhongze; Saad, Yousef; Sosonkina, Masha: pARMS: a parallel version of the algebraic recursive multilevel solver. (2003)
  13. Benzi, Michele: Preconditioning techniques for large linear systems: A survey (2002)
  14. Falgout, R. D.; Yang, U. Meier: hypre: A library of high performance preconditioners (2002)
  15. Henson, Van Emden; Meier Yang, Ulrike: \textitBoomerAMG: A parallel algebraic multigrid solver and preconditioner (2002)

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