Combination preconditioning of saddle point systems for positive definiteness
Amongst recent contributions to preconditioning methods for saddle point systems, standard iterative methods in nonstandard inner products have been usefully employed. Krzyzanowski (Numerical Linear Algebra with Applications 2011; 18:123-140) identified a two-parameter family of preconditioners in t...
| Những tác giả chính: | , |
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| Định dạng: | Journal article |
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2013
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| _version_ | 1826298374422265856 |
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| author | Pestana, J Wathen, A |
| author_facet | Pestana, J Wathen, A |
| author_sort | Pestana, J |
| collection | OXFORD |
| description | Amongst recent contributions to preconditioning methods for saddle point systems, standard iterative methods in nonstandard inner products have been usefully employed. Krzyzanowski (Numerical Linear Algebra with Applications 2011; 18:123-140) identified a two-parameter family of preconditioners in this context and Stoll and Wathen (SIAM Journal on Matrix Analysis and Applications 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 MINRES-type 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. © 2012 John Wiley and Sons, Ltd. |
| first_indexed | 2024-03-07T04:45:52Z |
| format | Journal article |
| id | oxford-uuid:d33dee8c-32a1-429c-9667-530a0088f1b9 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T04:45:52Z |
| publishDate | 2013 |
| record_format | dspace |
| spelling | oxford-uuid:d33dee8c-32a1-429c-9667-530a0088f1b92022-03-27T08:09:53ZCombination preconditioning of saddle point systems for positive definitenessJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:d33dee8c-32a1-429c-9667-530a0088f1b9Symplectic Elements at Oxford2013Pestana, JWathen, AAmongst recent contributions to preconditioning methods for saddle point systems, standard iterative methods in nonstandard inner products have been usefully employed. Krzyzanowski (Numerical Linear Algebra with Applications 2011; 18:123-140) identified a two-parameter family of preconditioners in this context and Stoll and Wathen (SIAM Journal on Matrix Analysis and Applications 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 MINRES-type 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. © 2012 John Wiley and Sons, Ltd. |
| spellingShingle | Pestana, J Wathen, A Combination preconditioning of saddle point systems for positive definiteness |
| title | Combination preconditioning of saddle point systems for positive definiteness |
| title_full | Combination preconditioning of saddle point systems for positive definiteness |
| title_fullStr | Combination preconditioning of saddle point systems for positive definiteness |
| title_full_unstemmed | Combination preconditioning of saddle point systems for positive definiteness |
| title_short | Combination preconditioning of saddle point systems for positive definiteness |
| title_sort | combination preconditioning of saddle point systems for positive definiteness |
| work_keys_str_mv | AT pestanaj combinationpreconditioningofsaddlepointsystemsforpositivedefiniteness AT wathena combinationpreconditioningofsaddlepointsystemsforpositivedefiniteness |