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Using Algebraic Multigrid in Inexact BDDC Domain Decomposition Methods

  • Axel KlawonnEmail author
  • Martin LanserEmail author
  • Oliver RheinbachEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 125)

Abstract

A highly scalable implementation of an inexact BDDC (Balancing Domain Decomposition by Constraints) method is presented, and scalability results for linear elasticity problems in two and three dimensions for up to 131,072 computational cores of the JUQUEEN BG/Q are shown. In this method, the inverse action of the partially coupled stiffness matrix is replaced by V-cycles of an AMG (algebraic multigrid) method. The use of classical AMG for systems of PDEs, based on a nodal coarsening approach is compared with a recent AMG method using an explicit interpolation of the rigid body motions (global matrix approach; GM). It is illustrated, that for systems of PDEs an appropriate AMG interpolation is mandatory for fast convergence, i.e., using exact interpolation of rigid body modes in elasticity.

Notes

Acknowledgements

This work was supported in part by the German Research Foundation (DFG) through the Priority Programme 1648 “Software for Exascale Computing” (SPPEXA) under grants KL 2094/4-1, KL 2094/4-2, RH 122/2-1, and RH 122/3-2. The authors also gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for providing computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, www.lrz.de) and JUQUEEN [11] at Jülich Supercomputing Centre (JSC, www.fz-juelich.de/ias/jsc). GCS is the alliance of the three national supercomputing centres HLRS (Universität Stuttgart), JSC (Forschungszentrum Jülich), and LRZ (Bayerische Akademie der Wissenschaften), funded by the German Federal Ministry of Education and Research (BMBF) and the German State Ministries for Research of Baden-Württemberg (MWK), Bayern (StMWFK) and Nordrhein-Westfalen (MIWF).

References

  1. 1.
    S. Badia, A.F. Martín, J. Principe, On the scalability of inexact balancing domain decomposition by constraints with overlapped coarse/fine corrections. Parallel Comput. 50, 1–24 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    A.H. Baker, A. Klawonn, T. Kolev, M. Lanser, O. Rheinbach, U.M. Yang, Scalability of classical algebraic multigrid for elasticity to half a million parallel tasks, in Software for Exascale Computing - SPPEXA 2013–2015, ed. by H.-J. Bungartz, P. Neumann, W.E. Nagel (Springer International Publishing, Cham, 2016), pp. 113–140CrossRefGoogle Scholar
  3. 3.
    S. Balay, W.D. Gropp, L.C. McInnes, B.F. Smith, Efficient management of parallelism in object oriented numerical software libraries, in Modern Software Tools in Scientific Computing, ed. by E. Arge, A.M. Bruaset, H.P. Langtangen (Birkhauser, Boston, 1997), pp. 163–202CrossRefGoogle Scholar
  4. 4.
    C.R. Dohrmann, An approximate BDDC preconditioner. Numer. Linear Algebra Appl. 14(2), 149–168 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    V.E. Henson, U.M. Yang, Boomeramg: a parallel algebraic multigrid solver and preconditioner. Appl. Numer. Math. 41, 155–177 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Klawonn, O. Rheinbach, Inexact FETI-DP methods. Int. J. Numer. Methods Eng. 69(2), 284–307 (2007)MathSciNetCrossRefGoogle Scholar
  7. 7.
    A. Klawonn, M. Lanser, O. Rheinbach, Toward extremely scalable nonlinear domain decomposition methods for elliptic partial differential equations. SIAM J. Sci. Comput. 37(6), C667–C696 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. Klawonn, M. Lanser, O. Rheinbach, Nonlinear BDDC methods with inexact solvers. Technical report (August 2017, submitted for publication)Google Scholar
  9. 9.
    J. Li, O.B. Widlund, On the use of inexact subdomain solvers for BDDC algorithms. Comput. Methods Appl. Mech. Eng. 196, 1415–1428 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    J. Ruge, K. Stüben, Efficient solution of finite difference and finite element equations by algebraic multigrid (AMG), in The Institute of Mathematics and Its Applications Conference Series, ed. by J.D. Paddon, H. Holstein, vol. 3 (Clarenden Press, Oxford, 1985), pp. 169–212Google Scholar
  11. 11.
    M. Stephan, J. Docter, JUQUEEN: IBM Blue Gene/Q SUpercomputer System at the Jülich Supercomputing Centre. J. Large Scale Res Facil. 1, A1 (2015)CrossRefGoogle Scholar
  12. 12.
    A. Toselli, O. Widlund, Domain Decomposition Methods – Algorithms and Theory. Springer Series in Computational Mathematics, vol. 34 (Springer, Berlin, 2004)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität zu KölnKölnGermany
  2. 2.Institut für Numerische Mathematik und Optimierung, Fakultät für Mathematik und InformatikTechnische Universität Bergakademie FreibergFreibergGermany

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