### Abstract

Language | English |
---|---|

Pages | 785-808 |

Number of pages | 24 |

Journal | Numerical Linear Algebra with Applications |

Volume | 20 |

DOIs | |

Publication status | Published - 31 Oct 2013 |

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### Keywords

- iterative solvers
- nonstandard inner products
- preconditinoing
- saddle point problems

### Cite this

*Numerical Linear Algebra with Applications*,

*20*, 785-808. https://doi.org/10.1002/nla.1843

}

*Numerical Linear Algebra with Applications*, vol. 20, pp. 785-808. https://doi.org/10.1002/nla.1843

**Combination preconditioning of saddle point systems for positive definiteness.** / Pestana, J.; Wathen, A. J.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Combination preconditioning of saddle point systems for positive definiteness

AU - Pestana, J.

AU - Wathen, A. J.

N1 - This is the peer reviewed version of the following article: Pestana, J., & Wathen, A. J. (2013). Combination preconditioning of saddle point systems for positive definiteness. Numerical Linear Algebra with Applications, 20, 785-808. 10.1002/nla.1843, which has been published in final form at http://dx.doi.org/10.1002/nla.1843. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.

PY - 2013/10/31

Y1 - 2013/10/31

N2 - Amongst recent contributions to preconditioning methods for saddle point systems, standard iterative methods in nonstandard inner products have been usefully employed. Krzyzanowski ( ˙ Numer. Linear Algebra Appl. 2011; 18:123–140) identified a two-parameter family of preconditioners in this context and Stoll and Wathen (SIAM J. Matrix Anal. Appl. 2008; 30:582–608) introduced combination preconditioning, where two preconditioners, self-adjoint with respect to different inner products, can lead to further preconditioners and associated bilinear forms or inner products. Preconditioners that render the preconditioned saddle point matrix nonsymmetric but self-adjoint with respect to a nonstandard inner product always allow a MINREStype method (W-PMINRES) to be applied in the relevant inner product. If the preconditioned matrix is also positive definite with respect to the inner product a more efficient CG-like method (W-PCG) can be reliably used. We establish eigenvalue expressions for Krzyzanowski preconditioners and show that for a specific ˙ choice of parameters, although the Krzyzanowski preconditioned saddle point matrix is self-adjoint with ˙ respect to an inner product, it is never positive definite. We provide explicit expressions for the combination of certain preconditioners and prove the rather counterintuitive result that the combination of two specific preconditioners for which only W-PMINRES can be reliably used leads to a preconditioner for which, for certain parameter choices, W-PCG is reliably applicable. That is, combining two indefinite preconditioners can lead to a positive definite preconditioner. This combination preconditioner outperforms either of the two preconditioners from which it is formed for a number of test problems.

AB - Amongst recent contributions to preconditioning methods for saddle point systems, standard iterative methods in nonstandard inner products have been usefully employed. Krzyzanowski ( ˙ Numer. Linear Algebra Appl. 2011; 18:123–140) identified a two-parameter family of preconditioners in this context and Stoll and Wathen (SIAM J. Matrix Anal. Appl. 2008; 30:582–608) introduced combination preconditioning, where two preconditioners, self-adjoint with respect to different inner products, can lead to further preconditioners and associated bilinear forms or inner products. Preconditioners that render the preconditioned saddle point matrix nonsymmetric but self-adjoint with respect to a nonstandard inner product always allow a MINREStype method (W-PMINRES) to be applied in the relevant inner product. If the preconditioned matrix is also positive definite with respect to the inner product a more efficient CG-like method (W-PCG) can be reliably used. We establish eigenvalue expressions for Krzyzanowski preconditioners and show that for a specific ˙ choice of parameters, although the Krzyzanowski preconditioned saddle point matrix is self-adjoint with ˙ respect to an inner product, it is never positive definite. We provide explicit expressions for the combination of certain preconditioners and prove the rather counterintuitive result that the combination of two specific preconditioners for which only W-PMINRES can be reliably used leads to a preconditioner for which, for certain parameter choices, W-PCG is reliably applicable. That is, combining two indefinite preconditioners can lead to a positive definite preconditioner. This combination preconditioner outperforms either of the two preconditioners from which it is formed for a number of test problems.

KW - iterative solvers

KW - nonstandard inner products

KW - preconditinoing

KW - saddle point problems

U2 - 10.1002/nla.1843

DO - 10.1002/nla.1843

M3 - Article

VL - 20

SP - 785

EP - 808

JO - Numerical Linear Algebra with Applications

T2 - Numerical Linear Algebra with Applications

JF - Numerical Linear Algebra with Applications

SN - 1070-5325

ER -