Preconditioning of the Reduced System Associated with the Restricted Additive Schwarz Method

  • François Pacull
  • Damien Tromeur-Dervout
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 104)


It is of interest to solve large scale sparse linear systems on distributed computers, using Krylov subspace methods along with domain decomposition methods. If accurate subdomain solutions are used, the restricted additive Schwarz preconditioner allows a reduction to the interface via the Schur complement, which leads to an unpreconditioned reduced operator for the interface unknowns. Our purpose is to form a preconditioner for this interface operator by approximating it as a low-rank correction of the identity matrix. To this end, we use a sequence of orthogonal vectors and their image under the interface operator, which are both available after some iterations of the generalized minimal residual method.



This work has been supported by the French National Agency of Research, through the ANR-MONU12-0012 H2MNO4 project. The authors wish to express their thanks to Stéphane Aubert for some stimulating conversations and suggestions.


  1. 1.
    E. Brakkee, P. Wilders, A domain decomposition method for the advection-diffusion equation. Technical Report, Delft University of Technology (1994)MATHGoogle Scholar
  2. 2.
    X.-C. Cai, M. Sarkis, A restricted additive Schwarz preconditioner for general sparse linear systems. SIAM J. Sci. Comput. 21(2), 792–797 (electronic) (1999)Google Scholar
  3. 3.
    T.A. Davis, Y. Hu, The University of Florida sparse matrix collection. ACM Trans. Math. Softw. 38(1), Art. 1, 25 (2011)Google Scholar
  4. 4.
    J. Erhel, K. Burrage, B. Pohl, Restarted GMRES preconditioned by deflation. J. Comput. Appl. Math. 69(2), 303–318 (1996)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    F. Pacull, S. Aubert, GMRES acceleration of restricted Schwarz iterations, in Domain Decomposition Methods in Science and Engineering XXI. Lecture Notes in Computational Science and Engineering, vol. 98 (Springer, Berlin, 2014), pp. 725–732Google Scholar
  6. 6.
    Y. Saad, M. Sosonkina, Distributed Schur complement techniques for general sparse linear systems. SIAM J. Sci. Comput. 21(4), 1337–1356 (electronic) (1999/2000)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institut Camille Jordan CNRS UMR 5208Université Lyon 1, Université de LyonVilleurbanne CedexFrance

Personalised recommendations