Advertisement

Robust Preconditioners for Saddle Point Problems

  • Owe Axelsson
  • Maya Neytcheva
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2542)

Abstract

We survey preconditioning methods for matrices on saddle point form, as typically arising in constrained optimization problems. Special consideration is given to indefinite matrix preconditioners and a preconditioner which results in a symmetric positive definite matrix, which latter may enable the use of the standard conjugate gradient (CG) method. These methods result in eigenvalues with positive real parts and small or zero imaginary parts. The behaviour of some of these techniques is illustrated on solving a regularized Stokes problem.

Keywords

Conjugate Gradient Method Preconditioned Conjugate Gradient Saddle Point Problem Unit Number Numerical Linear Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Axelsson O.: Preconditioning of indefinite problems by regularization. SIAM Journal on Numerical Analysis, 16 (1979), 58–69.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Axelsson O.: Iterative Solution Methods, Cambridge University Press, Cambridge, 1994.zbMATHGoogle Scholar
  3. 3.
    Axelsson O., Barker V.A., Neytcheva M., Polman B.: Solving the Stokes problem on a massively parallel computer. Mathematical Modelling and Analysis, 4 (2000), 1–22.Google Scholar
  4. 4.
    Axelsson O., Gustafsson I.: An iterative solver for a mixed variable variational formulation of the (first) biharmonic problem. Computer Methods in Applied Mechanics and Engineering, 20 (1979), 9–16.CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Axelsson O., Gustafsson I.: An efficient finite element method for nonlinear diffusion problems. Bulletin Greek Mathematical Society, 22 (1991), 45–61.MathSciNetGoogle Scholar
  6. 6.
    Axelsson O, Makarov M. On a generalized conjugate orthogonal residual method. Numerical Linear Algebra with Applications, 1995; 2:467–480.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Axelsson O., Neytcheva M.: Preconditioning methods for linear systems arising in constrained optimization problems. Submitted to Numerical Linear Algebra with Applications, January 2002.Google Scholar
  8. 8.
    Axelsson O., Vassilevski P.S.: Variable-step multilevel preconditioning methods. I. Selfadjoint and positive definite elliptic problems. Numer. Linear Algebra Appl., 1 (1994), 75–101.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Braess D.: Finite Elements. Theory, fast solvers, and applications in solid mechanics. Cambridge University Press, Cambridge, 2001. (Second edition)zbMATHGoogle Scholar
  10. 10.
    Bramble JH, Pasciak JE.: A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Mathematics of Computation, 1988; 50:1–17.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Elman H.C.: Preconditioning for the steady-state Navier-Stokes equations with low viscosity. SIAM Journal on Scientific Computing, 20 (1999), 1299–1316.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    H.C. Elman and D. Silvester, Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations. SIAM Journal on Scientific Computing, 17 (1996), 33–46.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ewing R.E., Lazarov R., Lu P., Vassilevski P.: Preconditioning indefinite systems arising from mixed finite element discretization of second order elliptic problems. In Axelsson O., Kolotilina L. (eds.): Lecture Notes in Mathematics, 1457, Springer-Verlag, Berlin, 1990.Google Scholar
  14. 14.
    Klawonn A.: Block-triangular preconditioners for saddle point problems with a penalty term. SIAM Journal on Scientific Computing, 1998; 19:172–184.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Luksan L., Vlcek J.: Indefinitely preconditioned inexact Newton method for large sparse equality constrained non-linear programming problems. Numerical Linear Algebra with Applications, 5(1998), 219–247.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Owe Axelsson
    • 1
  • Maya Neytcheva
    • 2
  1. 1.Department of MathematicsUniversity of NijmegenED NijmegenThe Netherlands
  2. 2.Department of Scientific ComputingUppsala UniversityUppsalaSweden

Personalised recommendations