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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Axelsson O.: Preconditioning of indefinite problems by regularization. SIAM Journal on Numerical Analysis, 16 (1979), 58–69.
Axelsson O.: Iterative Solution Methods, Cambridge University Press, Cambridge, 1994.
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.
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.
Axelsson O., Gustafsson I.: An efficient finite element method for nonlinear diffusion problems. Bulletin Greek Mathematical Society, 22 (1991), 45–61.
Axelsson O, Makarov M. On a generalized conjugate orthogonal residual method. Numerical Linear Algebra with Applications, 1995; 2:467–480.
Axelsson O., Neytcheva M.: Preconditioning methods for linear systems arising in constrained optimization problems. Submitted to Numerical Linear Algebra with Applications, January 2002.
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.
Braess D.: Finite Elements. Theory, fast solvers, and applications in solid mechanics. Cambridge University Press, Cambridge, 2001. (Second edition)
Bramble JH, Pasciak JE.: A preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems. Mathematics of Computation, 1988; 50:1–17.
Elman H.C.: Preconditioning for the steady-state Navier-Stokes equations with low viscosity. SIAM Journal on Scientific Computing, 20 (1999), 1299–1316.
H.C. Elman and D. Silvester, Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations. SIAM Journal on Scientific Computing, 17 (1996), 33–46.
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.
Klawonn A.: Block-triangular preconditioners for saddle point problems with a penalty term. SIAM Journal on Scientific Computing, 1998; 19:172–184.
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Axelsson, O., Neytcheva, M. (2003). Robust Preconditioners for Saddle Point Problems. In: Dimov, I., Lirkov, I., Margenov, S., Zlatev, Z. (eds) Numerical Methods and Applications. NMA 2002. Lecture Notes in Computer Science, vol 2542. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36487-0_17
Download citation
DOI: https://doi.org/10.1007/3-540-36487-0_17
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-00608-4
Online ISBN: 978-3-540-36487-0
eBook Packages: Springer Book Archive