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Additive Schwarz with Variable Weights

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Domain Decomposition Methods in Science and Engineering XXI

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 98))

Abstract

For Additive Schwarz preconditioning of nonsymmetric systems, it is proposed to use weights that change from one iteration to the next. At each iteration, weights for all earlier iterations are implicitly chosen to minimize the current residual. This strategy fits the paradigm of the recently proposed multipreconditioned GMRES. Numerical experiments illustrating the potential of the proposed method are presented.

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Notes

  1. 1.

    We point out that this is completely different than the approach in [4], where the weights are zeros and ones, and the emphasis is on asynchronous iterations.

  2. 2.

    This algorithm extends previous work on using a combination of preconditioners—e.g., flexible GMRES [7] with alternating preconditioners, as described by Rui et al. [6] in the method they call multipreconditioned GMRES—by making an ‘optimal’ choice of weights. See [5] for a discussion.

References

  1. Ayuso de Dios, B., Baker, A.T., Vassilevski, P.S.: A combined preconditioning strategy for nonsymmetric systems (2012). arxiv:1208.4544v1

    Google Scholar 

  2. Bridson, R., Greif, C.: A multipreconditioned conjugate gradient algorithm. SIAM J. Matrix Anal. Appl. 27(4), 1056–1068 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cai, X.C., Sarkis, M.: A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM J. Sci. Comput. 21, 239–247 (1999)

    Article  MathSciNet  Google Scholar 

  4. Frommer, A., Schwandt, H., Szyld, D.B.: Asynchronous weighted additive Schwarz methods. Electron. Trans. Numer. Anal. 5, 48–61 (1997)

    MATH  MathSciNet  Google Scholar 

  5. Greif, C., Rees, T., Szyld, D.B.: MPGMRES: a generalized minimum residual method with multiple preconditioners. Tech. Rep. 11-12-23, Department of Mathematics, Temple University (2011). Revised September 2012. Also available as Technical Report TR-2011-12, Department of Computer Science, University of British Columbia

    Google Scholar 

  6. Rui, P.L., Yong, H., Chen, R.S.: Multipreconditioned GMRES method for electromagnetic wave scattering problems. Microw. Opt. Technol. Lett. 50(1), 150–152 (2008)

    Article  Google Scholar 

  7. Saad, Y.: A flexible inner-outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14, 461–469 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)

    Book  MATH  Google Scholar 

  9. Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7, 856–869 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  10. Simoncini, V., Szyld, D.B.: Recent computational developments in Krylov subspace methods for linear systems. Numer. Linear Algebra Appl. 14, 1–59 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

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Acknowledgements

The work of the first author was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), and that of the third author in part by the U.S. National Science Foundation under grant DMS-1115520.

Author information

Authors and Affiliations

  1. Department of Computer Science, University of British Columbia, Vancouver, BC, Canada

    Chen Greif

  2. Scientific Computing Department, STFC Rutherford Appleton Laboratory, Chilton, Didcot, UK

    Tyrone Rees

  3. Department of Mathematics, Temple University, Philadelphia, PA, USA

    Daniel B. Szyld

Authors
  1. Chen Greif
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  2. Tyrone Rees
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  3. Daniel B. Szyld
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Corresponding author

Correspondence to Chen Greif .

Editor information

Editors and Affiliations

  1. INRIA, Rennes, France

    Jocelyne Erhel

  2. Section de Mathématiques, Université de Genève, Genève, Switzerland

    Martin J. Gander

  3. Laboratoire Analyse, Géometrie & Applications, Université Paris XIII, Villetaneuse, France

    Laurence Halpern

  4. INRIA, Rennes, France

    Géraldine Pichot

  5. Laboratoire LMNO, Université de Caen, Caen, France

    Taoufik Sassi

  6. New York University Courant Institute, New York, New York, USA

    Olof Widlund

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Greif, C., Rees, T., Szyld, D.B. (2014). Additive Schwarz with Variable Weights. In: Erhel, J., Gander, M., Halpern, L., Pichot, G., Sassi, T., Widlund, O. (eds) Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering, vol 98. Springer, Cham. https://doi.org/10.1007/978-3-319-05789-7_75

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