Acceleration of iterative solution of series of systems due to better initial guess

Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 74)


Efficient choice of the initial guess for the iterative solution of series of systems is considered. The series of systems are typical for unsteady nonlinear fluid flow problems. The history of iterative solution at previous time steps is used for computing a better initial guess. This strategy is applied for two iterative linear system solvers (GCR and GMRES). A reduced model technique is developed for implicitly discretized nonlinear evolution problems. The technique computes a better initial guess for the inexact Newton method. The methods are successfully tested in parallel CFD simulations. The latter approach is suitable for GRID computing as well.

Key words

Krylov method Newton method Proper Orthogonal Decomposition client-server architecture grid computing 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Eisenstat and H. Walker, Globally convergent inexact Newton methods. SIAM J. Optim. 4, 1994, 393-422.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    S. Eisenstat and H. Walker, Choosing the forcing terms in an inexact Newton method. SIAM J. Sci. Comput., 17, 1996, 16-32.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    G. Golub and G. Meurant, Matrices, moments and quadrature II: how to compute the norm of the error in iterative methods. BIT, 37-3, 1997, 687-705.CrossRefMathSciNetGoogle Scholar
  4. 4.
    P. Gosselet and Ch. Rey, On a selective reuse of Krylov subspaces in Newton-Krylov approaches for nonlinear elasticity. Domain decomposition methods in science and engineering, Natl. Auton. Univ. Mex., Mexico, 2003, 419-426.Google Scholar
  5. 5.
    C. T. Kelley, Iterative Methods for Optimization. Frontiers in Applied Mathematics 18. SIAM, Philadelphia, 1999Google Scholar
  6. 6.
    I. Keshtiban, F. Belblidia and M. Webster Compressible flow solvers for Low Mach number flows – a review. Technical report CSR2, Institute of Non-Newtonian Fluid Mechanics, University of Wales, Swansea, UK, 2004.Google Scholar
  7. 7.
    G. Meurant, Computer solution of large linear systems. Amsterdam, North-Holland, 1999.MATHGoogle Scholar
  8. 8.
    G. Meurant, The computation of bounds for the norm of the error in the conjugate gradient algorithm.Numerical Algorithms, 16, 1998, 77-87.Google Scholar
  9. 9.
    M. Pernice and H. Walker, NITSOL: a Newton iterative solver for nonlinear systems. SIAM J. Sci. Comput., 19, 1998, 302-318.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    M.Rathinam and L.Petzold, Dynamic iteration using reduced order models: a method for simulation of large scale modular systems. SIAM J. Numer.Anal., 40, 2002, 1446-1474.Google Scholar
  11. 11.
    M. Rathinam and L. Petzold, A new look at proper orthogonal decomposition. SIAM J. Numer.Anal., 41, 2003, 1893-1925.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    F. Risler and Ch. Rey, Iterative accelerating algorithms with Krylov subspaces for the solution to large-scale nonlinear problems. Numer. Algorithms 23, 2000, 1-30.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    M. Schefer and S. Turek, Benchmark Computations of Laminar Flow around a Cylinder. In: Flow Simulation with High-Performance Computers II (E.H.Hirschel ed.), Notes on Numerical Fluid Mechanics,52, Vieweg, 1996, 547-566.Google Scholar
  14. 14.
    D. Tromeur-Dervout and Y. Vassilevski, POD acceleration of fully implicit solver for unsteady nonlinear flows and its application on GRID architecture. Proc. Int. Conf. PCFD05, 2006, 157-160.Google Scholar
  15. 15.
    D. Tromeur-Dervout and Y. Vassilevski, Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations. J.Comput.Phys. 219, 2006, 210-227.Google Scholar
  16. 16.
    D. Tromeur-Dervout and Y. Vassilevski, POD acceleration of fully implicit solver for unsteady nonlinear flows and its application on grid architecture.Adv. Eng. Softw. 38, 2007, 301-311.Google Scholar
  17. 17.
    D. Tromeur-Dervout, Résolution des Equations de Navier-Stokes en Formulation Vitesse Tourbillon sur Systèmes multiprocesseurs à Mémoire Distribuée. Thesis, Univ. Paris VI, 1993.Google Scholar

Copyright information

© Springer Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.CNRS, Institut Camille JordanUniversity of LyonVilleurbanne CedexFrance
  2. 2.Institute of Numerical MathematicsMoscowRussia

Personalised recommendations