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Scalable algebraic multilevel preconditioners with application to CFD. (English) Zbl 1398.76068

Tromeur-Dervout, Damien (ed.) et al., Parallel computational fluid dynamics 2008. Parallel numerical methods, software development and applications. Proceedings of the 20th international conference, Lyon, France, May 19–22, 2008. Berlin: Springer (ISBN 978-3-642-14437-0/hbk; 978-3-642-26515-0/pbk978-3-642-14438-7/ebook). Lecture Notes in Computational Science and Engineering 74, 15-27 (2010).
Summary: The solution of large and sparse linear systems is one of the main computational kernels in CFD applications and is often a very time-consuming task, thus requiring the use of effective algorithms on high-performance computers. Preconditioned Krylov solvers are the methods of choice for these systems, but the availability of “good” preconditioners is crucial to achieve efficiency and robustness. In this paper we discuss some issues concerning the design and the implementation of scalable algebraic multilevel preconditioners, that have shown to be able to enhance the performance of Krylov solvers in parallel settings. In this context, we outline the main objectives and the related design choices of MLD2P4, a package of multilevel preconditioners based on Schwarz methods and on the smoothed aggregation technique, that has been developed to provide scalable and easy-to-use preconditioners in the Parallel Sparse BLAS computing framework. Results concerning the application of various MLD2P4 preconditioners within a large eddy simulation of a turbulent channel flow are discussed.
For the entire collection see [Zbl 1201.76012].

MSC:

76F65 Direct numerical and large eddy simulation of turbulence
65F08 Preconditioners for iterative methods
65Y05 Parallel numerical computation
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[1] 1.Andrea Aprovitola, Pasqua D’Ambra, Filippo Denaro, Daniela di Serafino, and Salvatore Filippone. Application of parallel algebraic multilevel domain decomposition preconditioners in large eddy simulations of wall-bounded turbulent flows: first experiments. Technical Report RT-ICAR-NA-07-02, ICAR-CNR, Naples, Italy, 2007. · Zbl 1280.76010
[2] 2.Andrea Aprovitola and Filippo M. Denaro. On the application of congruent upwind discretizations for large eddy simulations. \(J. Comput. Phys.\), 194(1):329-343, 2004. · Zbl 1136.76376
[3] 3.Andrea Aprovitola and Filippo M. Denaro. A non-diffusive, divergence-free, finite volume-based double projection method on non-staggered grids. \(Internat. J. Numer. Methods Fluids\), 53(7):1127-1172, 2007. · Zbl 1370.76089
[4] 4.Marian Brezina and Petr Vaněk. A black-box iterative solver based on a two-level Schwarz method. \(Computing\), 63(3):233-263, 1999. · Zbl 0951.65133
[5] 5.Peter N. Brown and Homer F. Walker. GMRES on (nearly) singular systems. \(SIAM J. Matrix Anal. Appl.\), 18(1):37-51, 1997. · Zbl 0876.65019
[6] 6.Alfredo Buttari, Pasqua D’Ambra, Daniela di Serafino, and Salvatore Filippone. Extending PSBLAS to build parallel schwarz preconditioners. In K. Madsen J. Dongarra and J. Wasniewski, editors, \(Applied Parallel Computing\), volume 3732 of \(Lecture Notes in Computer Science\), pages 593-602, Berlin/Heidelberg, 2006. Springer.
[7] 7.Alfredo Buttari, Pasqua D’Ambra, Daniela di Serafino, and Salvatore Filippone. 2LEV-D2P4: a package of high-performance preconditioners for scientific and engineering applications. \(Appl. Algebra Engrg. Comm. Comput.\), 18(3):223-239, 2007. · Zbl 1122.65046
[8] 8.Xiao-Chuan Cai and Yousef Saad. Overlapping domain decomposition algorithms for general sparse matrices. \(Numer. Linear Algebra Appl.\), 3(3):221-237, 1996. · Zbl 0851.65083
[9] 9.Xiao-Chuan Cai and Marcus Sarkis. A restricted additive Schwarz preconditioner for general sparse linear systems. \(SIAM J. Sci. Comput.\), 21(2):792-797, 1999. · Zbl 0944.65031
[10] 10.Xiao-Chuan Cai and Olof B. Widlund. Domain decomposition algorithms for indefinite elliptic problems. \(SIAM J. Sci. Statist. Comput.\), 13(1):243-258, 1992. · Zbl 0746.65085
[11] 11.Tony F. Chan and Tarek P. Mathew. Domain decomposition algorithms. In \(Acta numerica, 1994\), Acta Numer., pages 61-143. Cambridge Univ. Press, Cambridge, 1994. · Zbl 0809.65112
[12] 12.Pasqua D’Ambra, Daniela di Serafino, and Salvatore Filippone. On the development of PSBLAS-based parallel two-level Schwarz preconditioners. \(Appl. Numer. Math.\), 57(11-12):1181-1196, 2007. · Zbl 1123.65029
[13] 13.Pasqua D’Ambra, Daniela di Serafino, and Salvatore Filippone. \(MLD2P4 User’s and Reference Guide\), September 2008. Available from · Zbl 1364.65276
[14] 14.Timothy A. Davis. Algorithm 832: UMFPACK V4.3—an unsymmetric-pattern multifrontal method. \(ACM Trans. Math. Software\), 30(2):196-199, 2004. · Zbl 1072.65037
[15] 15.James W. Demmel, Stanley C. Eisenstat, John R. Gilbert, Xiaoye S. Li, and Joseph W. H. Liu. A supernodal approach to sparse partial pivoting. \(SIAM J. Matrix Anal. Appl.\), 20(3):720-755, 1999. · Zbl 0931.65022
[16] 16.James W. Demmel, John R. Gilbert, and Xiaoye S. Li. An asynchronous parallel supernodal algorithm for sparse Gaussian elimination. \(SIAM J. Matrix Anal. Appl.\), 20(4):915-952, 1999. · Zbl 0939.65036
[17] 17.Iain S. Duff, Michele Marrone, Giuseppe Radicati, and Carlo Vittoli. Level 3 basic linear algebra subprograms for sparse matrices: a user-level interface. \(ACM Trans. Math. Software\), 23(3):379-401, 1997. · Zbl 0903.65041
[18] 18.Evridiki Efstathiou and Martin J. Gander. Why restricted additive Schwarz converges faster than additive Schwarz. \(BIT\), 43(suppl.):945-959, 2003. · Zbl 1045.65027
[19] 19.Salvatore Filippone and Alfredo Buttari. \(PSBLAS: User’s and Reference Guide\), 2008. Available from
[20] 20.Salvatore Filippone and Michele Colajanni. PSBLAS: A library for parallel linear algebra computation on sparse matrices. \(ACM Trans. Math. Software\), 26(4):527-550, 2000. See also · Zbl 1365.65128
[21] 21.Michael W. Gee, Christofer M. Siefert, Jonathan J. Hu, Ray S. Tuminaro, and Marzio G. Sala. ML 5.0 smoothed aggregation user’s guide. Technical Report SAND2006-2649, Sandia National Laboratories, Albuquerque, NM, and Livermore, CA, 2006.
[22] 22.Van Emden Henson and Ulrike Meier Yang. BoomerAMG: A parallel algebraic multigrid solver and preconditioner. \(Appl. Numer. Math.\), 41:155-177, 2000. · Zbl 0995.65128
[23] 23.Randall J. LeVeque. \(Finite volume methods for hyperbolic problems\). Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002. · Zbl 1010.65040
[24] 24.Paul T. Lin, Marzio G. Sala, John N. Shadid, and Ray S. Tuminaro. Performance of fully-coupled algebraic multilevel domain decomposition preconditioners for incompressible flow and transport. \(Int. J. Numer. Meth. Eng.\), 67:208-225, 2006. · Zbl 1110.76315
[25] 25.Gérard Meurant. Numerical experiments with algebraic multilevel preconditioners. \(Electron. Trans. Numer. Anal.\), 12:1-65 (electronic), 2001. · Zbl 0974.65040
[26] 26.Yousef Saad. \(Iterative methods for sparse linear systems\). Society for Industrial and Applied Mathematics, Philadelphia, PA, second edition, 2003. · Zbl 1031.65046
[27] 27.Yousef Saad and Masha Sosonkina. pARMS: A package for the parallel iterative solution of general large sparse linear systems user’s guide. Technical Report UMSI2004-8, Minnesota Supercomputing Institute, Minneapolis, MN, 2004.
[28] 28.Pierre Sagaut. \(Large eddy simulation for incompressible flows. An introduction\). Scientific Computation. Springer-Verlag, Berlin, third edition, 2005. · Zbl 0964.76002
[29] 29.Barry F. Smith, Petter E. Bjørstad, and William D. Gropp. \(Domain decomposition. Parallel multilevel methods for elliptic partial differential equations\). Cambridge University Press, Cambridge, 1996. · Zbl 0857.65126
[30] 30.Marc Snir, Steve Otto, Steven Huss-Lederman, David W. Walker, and Jack J. Dongarra. \(MPI: The Complete Reference. Vol. 1 - The MPI Core\). Scientific and Engineering Computation. The MIT Press, Cambridge, MA, second edition, 1998.
[31] 31.Klaus Stüben. A review of algebraic multigrid. \(J. Comput. Appl. Math.\), 128(1-2):281-309, 2001. · Zbl 0979.65111
[32] 32.Ray S. Tuminaro and Charles Tong. Parallel smoothed aggregation multigrid: Aggregation strategies on massively parallel machines. In \(Proceedings of the 2000 ACM/IEEE conference on Supercomputing\), Dallas, TX, 2000.
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