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. Weston, Brian; Nourgaliev, Robert; Delplanque, Jean-Pierre; Barker, Andrew T.: Preconditioning a Newton-Krylov solver for all-speed melt pool flow physics (2019)
  2. Adler, James H.; Lashuk, Ilya; MacLachlan, Scott P.: Composite-grid multigrid for diffusion on the sphere. (2018)
  3. Barker, A. T.; Dobrev, V.; Gopalakrishnan, J.; Kolev, T.: A scalable preconditioner for a primal discontinuous Petrov-Galerkin method (2018)
  4. Beaude, Laurence; Beltzung, Thibaud; Brenner, Konstantin; Lopez, Simon; Masson, Roland; Smai, Farid; Thebault, Jean-Frédéric; Xing, Feng: Parallel geothermal numerical model with fractures and multi-branch wells (2018)
  5. Casquero, Hugo; Zhang, Yongjie Jessica; Bona-Casas, Carles; Dalcin, Lisandro; Gomez, Hector: Non-body-fitted fluid-structure interaction: divergence-conforming B-splines, fully-implicit dynamics, and variational formulation (2018)
  6. Esmaily, M.; Jofre, L.; Mani, A.; Iaccarino, G.: A scalable geometric multigrid solver for nonsymmetric elliptic systems with application to variable-density flows (2018)
  7. Gopalakrishnan, J.; Neumüller, M.; Vassilevski, P. S.: The auxiliary space preconditioner for the de Rham complex (2018)
  8. Ibeid, Huda; Yokota, Rio; Pestana, Jennifer; Keyes, David: Fast multipole preconditioners for sparse matrices arising from elliptic equations (2018)
  9. Klawonn, Axel; Lanser, Martin; Rheinbach, Oliver: Nonlinear BDDC methods with approximate solvers (2018)
  10. Liu, Lulu; Keyes, David E.; Krause, Rolf: A note on adaptive nonlinear preconditioning techniques (2018)
  11. Manteuffel, Thomas A.; Ruge, John; Southworth, Ben S.: Nonsymmetric algebraic multigrid based on local approximate ideal restriction ((\ell)AIR) (2018)
  12. Rhebergen, Sander; Wells, Garth N.: Preconditioning of a hybridized discontinuous Galerkin finite element method for the Stokes equations (2018)
  13. Shaydulin, Ruslan; Safro, Ilya: Aggregative coarsening for multilevel hypergraph partitioning (2018)
  14. Sun, Hui; Zhou, Shenggao; Cheng, Li-Tien; Li, Bo: Numerical methods for solvent Stokes flow and solute-solvent interfacial dynamics of charged molecules (2018)
  15. Voronin, Kirill; Lee, Chak Shing; Neumüller, Martin; Sepulveda, Paulina; Vassilevski, Panayot S.: Space-time discretizations using constrained first-order system least squares (CFOSLS) (2018)
  16. Wang, Kun; Liu, Hui; Luo, Jia; Chen, Zhangxin: Efficient CPR-type preconditioner and its adaptive strategies for large-scale parallel reservoir simulations (2018)
  17. Xu, Weiwei; Yang, Haifeng; Yang, Yin; Wang, Yiduo; Zhou, Kun: Stress-aware large-scale mesh editing using a domain-decomposed multigrid solver (2018)
  18. Yue, Xiaoqiang; Shu, Shi; Wang, Junxian; Zhou, Zhiyang: Substructuring preconditioners with a simple coarse space for 2-D 3-T radiation diffusion equations (2018)
  19. Bui, Quan M.; Elman, Howard C.; Moulton, David J.: Algebraic multigrid preconditioners for multiphase flow in porous media (2017)
  20. Côrtes, A. M. A.; Dalcin, L.; Sarmiento, A. F.; Collier, N.; Calo, V. M.: A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem (2017)

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