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Robust Multilevel Restricted Schwarz Preconditioners and Applications

  • Ernesto E. Prudencio
  • Xiao-Chuan Cai
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (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.

Keywords

Lagrange Multiplier Mesh Point Sharp Jump Coarse Discontinuity Multilevel Preconditioner 
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References

  1. 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. 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.zbMATHCrossRefGoogle Scholar
  3. 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.zbMATHCrossRefGoogle Scholar
  4. 4.
    W. L. Briggs, V. E. Henson, and S. F. McCormick, A Multigrid Tutorial, SIAM, Philadelphia, second ed., 2000.zbMATHGoogle Scholar
  5. 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.zbMATHCrossRefGoogle Scholar
  6. 6.
    X.-C. Cai and D. E. Keyes, Nonlinearly preconditioned inexact Newton algorithms, SIAM J. Sci. Comput., 24 (2002), pp. 183–200.zbMATHCrossRefGoogle Scholar
  7. 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.zbMATHCrossRefGoogle Scholar
  8. 8.
    M. Dryja and O. B. Widlund, Domain decomposition algorithms with small overlap, SIAM J. Sci.Comput., 15 (1994), pp. 604–620.zbMATHCrossRefGoogle Scholar
  9. 9.
    M. D. Gunzburger, Perspectives in Flow Control and Optimization, SIAM, Philadelphia, 2002.Google Scholar
  10. 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. 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. 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.zbMATHCrossRefGoogle Scholar
  13. 13.
    A. Toselli and O. B. Widlund, Domain Decomposition Methods — Algorithms and Theory, vol. 34 of Series in Computational Mathematics, Springer, 2005.Google Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • Ernesto E. Prudencio
    • 1
  • Xiao-Chuan Cai
    • 2
  1. 1.Advanced Computations DepartmentStanford Linear Accelerator CenterUSA
  2. 2.Department of Computer ScienceUniversity of Colorado at BoulderBoulderUSA

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