Skip to main content

Robust Multilevel Restricted Schwarz Preconditioners and Applications

  • Conference paper
  • 1586 Accesses

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 55))

Abstract

We introduce a multi-level restricted Schwarz preconditioner with a special coarse-to-fine interpolation and show numerically that the new preconditioner works extremely well for some difficult large systems of linear equations arising from some optimization problems constrained by the incompressible Navier-Stokes equations. Performance of the preconditioner is reported for parameters including number of processors, mesh sizes and Reynolds numbers.

This is a preview of subscription content, log in via an institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. S. Balay, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, B. F. Smith, and H. Zhang, PETSc users manual, Argonne National Laboratory, http://www.mcs.anl.gov/petsc, 2004.

    Google Scholar 

  2. G. Biros and O. Ghattas, Parallel Lagrange-Newton-Krylov-Schur methods for pde-constrained optimization, part I: The Krylov-Schur solver, SIAMJ. Sci. Comput., 27 (2005), pp. 687–713.

    Article  MATH  Google Scholar 

  3. G. Biros, Parallel Lagrange-Newton-Krylov-Schur methods for pde-constrained optimization, part II: The Lagrange-Newton solver and its application to optimal control of steady viscous flows, SIAM J. Sci. Comput., 27 (2005), pp. 714–739.

    Article  MATH  Google Scholar 

  4. W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, SIAM, Philadelphia, second ed., 2000.

    MATH  Google Scholar 

  5. X.-C. Cai, M. Dryja, and M. Sarkis, Restricted additive Schwarz preconditioners with harmonic overlap for symmetric positive definite linear systems, SIAM J. Numer. Anal., 41 (2003), pp. 1209–1231.

    Article  MATH  Google Scholar 

  6. X.-C. Cai and D. E. Keyes, Nonlinearly preconditioned inexact Newton algorithms, SIAM J. Sci. Comput., 24 (2002), pp. 183–200.

    Article  MATH  Google Scholar 

  7. X.-C. Cai and M. Sarkis, A restricted additive Schwarz preconditioner for general sparse linear systems, SIAM J. Sci. Comput., 21 (1999), pp. 792–797.

    Article  MATH  Google Scholar 

  8. M. Dryja and O. B. Widlund, Domain decomposition algorithms with small overlap, SIAM J. Sci.Comput., 15 (1994), pp. 604–620.

    Article  MATH  Google Scholar 

  9. M. D. Gunzburger, Perspectives in Flow Control and Optimization, SIAM, Philadelphia, 2002.

    Google Scholar 

  10. A. D. Ioffe and V. M. Tihomirov, Theory of Extremal Problems, North-Holland Publishing Company, first ed., 1979. Translation from Russian edition, (c) 1974 NAUKA, Moscow.

    Google Scholar 

  11. E. E. Prudencio, Parallel Fully Coupled Lagrange-Newton-Krylov-Schwarz Algorithms and Software for Optimization Problems Constrained by Partial Differential Equations, PhD thesis, Department of Computer Science, University of Colorado at Boulder, 2005.

    Google Scholar 

  12. E. E. Prudencio, R. Byrd, and X.-C. Cai, Parallel full space SQP Lagrange-Newton-Krylov-Schwarz algorithms for pde-constrained optimization problems, SIAM J. Sci. Comput., 27 (2006), pp. 1305–1328.

    Article  MATH  Google Scholar 

  13. A. Toselli and O. B. Widlund, Domain Decomposition Methods — Algorithms and Theory, vol. 34 of Series in Computational Mathematics, Springer, 2005.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer

About this paper

Cite this paper

Prudencio, E.E., Cai, XC. (2007). Robust Multilevel Restricted Schwarz Preconditioners and Applications. In: Widlund, O.B., Keyes, D.E. (eds) Domain Decomposition Methods in Science and Engineering XVI. Lecture Notes in Computational Science and Engineering, vol 55. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-34469-8_14

Download citation

Publish with us

Policies and ethics