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.
- iterative solvers
- nonstandard inner products
- saddle point problems
Pestana, J., & Wathen, A. J. (2013). Combination preconditioning of saddle point systems for positive definiteness. Numerical Linear Algebra with Applications, 20, 785-808. https://doi.org/10.1002/nla.1843