Skip to main content
Book cover

International Conference on Domain Decomposition Methods

DD 2017: Domain Decomposition Methods in Science and Engineering XXIV pp 383–392Cite as

Improving the Parallel Performance of Overlapping Schwarz Methods by Using a Smaller Energy Minimizing Coarse Space

Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE,volume 125)

Abstract

We consider a recent overlapping Schwarz method with an energy-minimizing coarse space of reduced size. In numerical experiments for up to 64,000 cores, we show that the parallel efficiency and the total time to solution is improved significantly, compared to our previous overlapping Schwarz method using an alternative energy-minimizing coarse space.

This is a preview of subscription content, access via your institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-93873-8_36
  • Chapter length: 10 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   149.00
Price excludes VAT (USA)
  • ISBN: 978-3-319-93873-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book
USD   199.99
Price excludes VAT (USA)
Learn about institutional subscriptions
Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. P.R. Amestoy, I.S. Duff, J.-Y. L’Excellent, J. Koster, A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Anal. Appl. 23(1), 15–41 (2001)

    CrossRef  MathSciNet  Google Scholar 

  2. J.G. Calvo, A two-level overlapping Schwarz method for H(curl) in two dimensions with irregular subdomains. Electron. Trans. Numer. Anal. 44, 497–521 (2015)

    MathSciNet  MATH  Google Scholar 

  3. C.R. Dohrmann, O.B. Widlund, Hybrid domain decomposition algorithms for compressible and almost incompressible elasticity. Int. J. Numer. Methods Eng. 82(2), 157–183 (2010)

    MathSciNet  MATH  Google Scholar 

  4. C.R. Dohrmann, O.B. Widlund, An alternative coarse space for irregular subdomains and an overlapping Schwarz algorithm for scalar elliptic problems in the plane. SIAM J. Numer. Anal. 50(5), 2522–2537 (2012)

    CrossRef  MathSciNet  Google Scholar 

  5. C.R. Dohrmann, O.B. Widlund, On the design of small coarse spaces for domain decomposition algorithms. SIAM J. Sci. Comput. 39(4), A1466–A1488 (2017)

    CrossRef  MathSciNet  Google Scholar 

  6. C.R. Dohrmann, A. Klawonn, O.B. Widlund, Domain decomposition for less regular subdomains: overlapping Schwarz in two dimensions. SIAM J. Numer. Anal. 46(4), 2153–2168 (2008)

    CrossRef  MathSciNet  Google Scholar 

  7. C.R. Dohrmann, A. Klawonn, O.B. Widlund, A family of energy minimizing coarse spaces for overlapping Schwarz preconditioners, in Domain Decomposition Methods in Science and Engineering XVII. Lecture Notes in Computer Science Engineering, vol. 60 (Springer, Berlin, 2008), pp. 247–254

    Google Scholar 

  8. M. Dryja, B.F. Smith, O.B. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions. SIAM J. Numer. Anal. 31(6), 1662–1694 (1994)

    CrossRef  MathSciNet  Google Scholar 

  9. A. Heinlein, Parallel overlapping Schwarz preconditioners and multiscale discretizations with applications to fluid-structure interaction and highly heterogeneous problems. PhD thesis, Universität zu Köln, Köln, 2016

    Google Scholar 

  10. A. Heinlein, A. Klawonn, O. Rheinbach, A parallel implementation of a two-level overlapping Schwarz method with energy-minimizing coarse space based on Trilinos. SIAM J. Sci. Comput. 38(6), C713–C747 (2016)

    CrossRef  MathSciNet  Google Scholar 

  11. A. Heinlein, A. Klawonn, O. Rheinbach, Parallel two-level overlapping Schwarz methods in fluid-structure interaction, in Numerical Mathematics and Advanced Applications ENUMATH 2015, ed. by B. Karasözen, M. Manguoğlu, M. Tezer-Sezgin, S. Göktepe, Ö. Uğur (Springer International Publishing, Cham, 2016), pp. 521–530. Also Preprint 2015-15 at http://tu-freiberg.de/fakult1/forschung/preprints

  12. A. Heinlein, A. Klawonn, O. Rheinbach, Parallel overlapping Schwarz with an energy-minimizing coarse space, in Domain Decomposition Methods in Science and Engineering XXIII, ed. by C.-O. Lee, X.-C. Cai, D.E. Keyes, H.H. Kim, A. Klawonn, E.-J. Park, O.B. Widlund. Lecture Notes in Computer Science Engineering, vol. 116 (Springer International Publishing, Cham, 2017), pp. 353–360

    Google Scholar 

  13. M.A. Heroux, R.A. Bartlett, V.E. Howle, R.J. Hoekstra, J.J. Hu, T.G. Kolda, R.B. Lehoucq, K.R. Long, R.P. Pawlowski, E.T. Phipps, A.G. Salinger, H.K. Thornquist, R.S. Tuminaro, J.M. Willenbring, A. Williams, K.S. Stanley, An overview of the Trilinos project. ACM Trans. Math. Softw. 31(3), 397–423 (2005)

    CrossRef  MathSciNet  Google Scholar 

  14. T.Y. Hou, X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134(1), 169–189 (1997)

    CrossRef  MathSciNet  Google Scholar 

  15. M. Stephan, J. Docter, JUQUEEN: IBM Blue Gene/Q Supercomputer System at the Jülich Supercomputing Centre. J. Large-Scale Res. Facil. 1, A1 (2015)

    CrossRef  Google Scholar 

  16. A. Toselli, O. Widlund, Domain Decomposition Methods—Algorithms and Theory, vol. 34 Springer Series in Computational Mathematics (Springer, Berlin, 2005)

    Google Scholar 

  17. J. Van lent, R. Scheichl, I.G. Graham, Energy-minimizing coarse spaces for two-level Schwarz methods for multiscale PDEs. Numer. Linear Algebra Appl. 16(10), 775–799 (2009)

    Google Scholar 

Download references

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 2094/4-2 and RH 122/3-2. The work of the fourth author was supported by the National Science Foundation Grant DMS-1522736.

The authors gratefully acknowledge the computing time granted by the Center for Computational Sciences and Simulation (CCSS) at Universität Duisburg-Essen and provided on the supercomputer magnitUDE (DFG grants INST 20876/209-1 FUGG, INST 20876/243-1 FUGG) at Zentrum für Informations- und Mediendienste (ZIM).

The authors also gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for providing computing time on the GCS Supercomputer JUQUEEN BG/Q supercomputer [15] at JSC Jülich. 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).

Author information

Authors and Affiliations

  1. Mathematisches Institut, Universität zu Köln, Köln, Germany

    Alexander Heinlein & Axel Klawonn

  2. Institut für Numerische Mathematik und Optimierung, Fakultät für Mathematik und Informatik, Technische Universität Bergakademie Freiberg, Freiberg, Germany

    Oliver Rheinbach

  3. Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    Olof B. Widlund

Authors
  1. Alexander Heinlein
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Axel Klawonn
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Oliver Rheinbach
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Olof B. Widlund
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Alexander Heinlein , Axel Klawonn or Oliver Rheinbach .

Editor information

Editors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Prof. Petter E. Bjørstad

  2. Department of Mathematics, Louisana State University, Baton Rouge, LA, USA

    Prof. Susanne C. Brenner

  3. Laboratoire Analyse, Géométrie et Applications, Université Paris 13, Villetaneuse, France

    Prof. Lawrence Halpern

  4. Department of Applied Mathematics, Kyung Hee University, Yongin, Korea (Republic of)

    Prof. Hyea Hyun Kim

  5. FB Mathematik und Informatik, Freie Universität Berlin, Berlin, Germany

    Prof. Dr. Ralf Kornhuber

  6. Western Norway University of Applied Sciences, Bergen, Norway

    Prof. Talal Rahman

  7. Courant Institute of Mathematical Sciences, New York University, New York, NY, USA

    Prof. Olof B. Widlund

Rights and permissions

Reprints and Permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Heinlein, A., Klawonn, A., Rheinbach, O., Widlund, O.B. (2018). Improving the Parallel Performance of Overlapping Schwarz Methods by Using a Smaller Energy Minimizing Coarse Space. In: , et al. Domain Decomposition Methods in Science and Engineering XXIV . DD 2017. Lecture Notes in Computational Science and Engineering, vol 125. Springer, Cham. https://doi.org/10.1007/978-3-319-93873-8_36

Download citation