# Regularization methods for the construction of preconditioners for saddle point problems

## Main Article Content

## Abstract

For the iterative solution of saddle point problems one needs efficient preconditioners to achieve a fast convergence. Three types of preconditioners are presented which are based on regularization by use of an augmented matrix. They are applicable also for problems with a highly singular pivot block matrix. One of the methods is applicable also for nonsymmetric saddle point problems.

## Article Details

How to Cite

Axelsson, O.
(2020).
Regularization methods for the construction of preconditioners for saddle point problems.

*Proceedings Of The Conference Algoritmy,*, 141 - 150. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/1576/828
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Articles

## References

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[8] O. Axelsson, R. Blaheta and M. Hasal, A Comparison of Preconditioning Methods for Saddle Point Problems with an Application to Porous Media Flow Problems. In: Kozubek T., Blaheta R. (eds) High Performance Computing in Science and Engineering. HPCSE 2015. Lecture Notes in Computer Science, vol 9611. Springer, (2016), pp. 68–84.

[9] A. Klawonn, Block-triangular preconditioners for saddle point problems with a penalty term, SIAM Journal on Scientific Computing, 19 (1998), pp. 172–184.

[10] V. Simoncini, Block triangular preconditioners for symmetric saddle-point problems, Applied Numerical Mathematics, 49 (2004), pp. 63–80.

[11] L. Y. Kolotilina, and A. Y. Yeremin, Factorized sparse approximate inverse preconditionings I. Theory., SIAM Journal on Matrix Analysis and Applications, 14 (1993), pp. 45-58.

[12] O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, 1994.

[13] Z. H. Cao, Augmentation block preconditioners for saddle point-type matrices with singular (1,1) blocks, Numerical Linear Algebra with Applications, 15 (2008), pp. 515–533.

[14] J. Schöberl, and W. Zulehner, Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimization problems, SIAM Journal on Matrix Analysis and Applications, 29 (2007), pp. 752–773.

[15] L.-T. Zhang, Modified block preconditioner for generalized saddle point matrices with highly singular(1,1) blocks, Linear and Multilinear Algebra, 68 (2020), pp. 152–160.

[16] M. Flück, T. Hofer, M. Picasso, J. Rappaz, and G. Steiner, Scientific Computing for Aluminum Production, International Journal of Numerical Analysis & Modeling, 6 (2009), pp. 489–504.

[17] M. Benzi, and J. Liu, Block preconditioning for saddle point systems with indefinite (1,1) block, International Journal of Computer Mathematics, 84 (2007), pp. 1117–1129.

[18] O. Axelsson, and P. S. Vassilevski, A black box generalized conjugate gradient solver with inner iterations and variable-step preconditioning, SIAM Journal on Matrix Analysis and Applications, 12 (1991), pp. 625–644.

[2] W. Zulehner, Analysis of iterative methods for saddle point problems: a unified approach, Mathematics of Computation, 71 (2002), pp. 479–505.

[3] E. G. Dyakonov, Iterative methods with saddle operators, Doklady Akademii Nauk, Russian Academy of Sciences, 292 (1987), pp. 1037–1041.

[4] O. Axelsson, On iterative solvers in structural mechanics, separate displacement orderings and mixed variable methods, Mathematics and Computers in Simulation, 50 (1999), pp. 11–30.

[5] P. J. Phillips and M. F. Wheeler, Overcoming the problem of locking in linear elasticity and poroelasticity: an heuristic approach, Computational Geosciences, 13 (2009), pp. 5–12.

[6] Axelsson, O. Unified analysis of preconditioning methods for saddle point matrices. Numerical Linear Algebra with Applications 22(2015), pp. 233–253.

[7] O. Axelsson, Preconditioners for regularized saddle point matrices, Journal of Numerical Mathematics, 19 (2011), pp. 91–112.

[8] O. Axelsson, R. Blaheta and M. Hasal, A Comparison of Preconditioning Methods for Saddle Point Problems with an Application to Porous Media Flow Problems. In: Kozubek T., Blaheta R. (eds) High Performance Computing in Science and Engineering. HPCSE 2015. Lecture Notes in Computer Science, vol 9611. Springer, (2016), pp. 68–84.

[9] A. Klawonn, Block-triangular preconditioners for saddle point problems with a penalty term, SIAM Journal on Scientific Computing, 19 (1998), pp. 172–184.

[10] V. Simoncini, Block triangular preconditioners for symmetric saddle-point problems, Applied Numerical Mathematics, 49 (2004), pp. 63–80.

[11] L. Y. Kolotilina, and A. Y. Yeremin, Factorized sparse approximate inverse preconditionings I. Theory., SIAM Journal on Matrix Analysis and Applications, 14 (1993), pp. 45-58.

[12] O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, 1994.

[13] Z. H. Cao, Augmentation block preconditioners for saddle point-type matrices with singular (1,1) blocks, Numerical Linear Algebra with Applications, 15 (2008), pp. 515–533.

[14] J. Schöberl, and W. Zulehner, Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimization problems, SIAM Journal on Matrix Analysis and Applications, 29 (2007), pp. 752–773.

[15] L.-T. Zhang, Modified block preconditioner for generalized saddle point matrices with highly singular(1,1) blocks, Linear and Multilinear Algebra, 68 (2020), pp. 152–160.

[16] M. Flück, T. Hofer, M. Picasso, J. Rappaz, and G. Steiner, Scientific Computing for Aluminum Production, International Journal of Numerical Analysis & Modeling, 6 (2009), pp. 489–504.

[17] M. Benzi, and J. Liu, Block preconditioning for saddle point systems with indefinite (1,1) block, International Journal of Computer Mathematics, 84 (2007), pp. 1117–1129.

[18] O. Axelsson, and P. S. Vassilevski, A black box generalized conjugate gradient solver with inner iterations and variable-step preconditioning, SIAM Journal on Matrix Analysis and Applications, 12 (1991), pp. 625–644.