Pestana, Jennifer and Wathen, A. J. (2011) Combination preconditioning of saddle point systems for positive definiteness. Technical Report. Numer. Linear Algebra Appl.. (Submitted)

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Abstract
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 twoparameter 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, selfadjoint 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 selfadjoint with respect to a nonstandard inner product always allow a MINREStype method (WPMINRES) 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 CGlike method (WPCG) 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 selfadjoint 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 WPMINRES can be reliably used leads to a preconditioner for which, for certain parameter choices, WPCG 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.
Item Type:  Technical Report (Technical Report) 

Subjects:  H  N > Numerical analysis 
Research Groups:  Numerical Analysis Group 
ID Code:  1406 
Deposited By:  Lotti Ekert 
Deposited On:  26 Oct 2011 08:20 
Last Modified:  29 May 2015 19:06 
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