On the choice of preconditioner for minimum residual methods for non-Hermitian matrices
We consider the solution of left preconditioned linear systems P- 1Cx=P-1c, where P,CECn×n are non-Hermitian, cECn, and C, P, and P-1C are diagonalisable with spectra symmetric about the real line. We prove that, when P and C are self-adjoint with respect to the same Hermitian sesquilinear form, the...
| Hoofdauteurs: | , |
|---|---|
| Formaat: | Journal article |
| Taal: | English |
| Gepubliceerd in: |
2013
|
| _version_ | 1826268770479374336 |
|---|---|
| author | Pestana, J Wathen, A |
| author_facet | Pestana, J Wathen, A |
| author_sort | Pestana, J |
| collection | OXFORD |
| description | We consider the solution of left preconditioned linear systems P- 1Cx=P-1c, where P,CECn×n are non-Hermitian, cECn, and C, P, and P-1C are diagonalisable with spectra symmetric about the real line. We prove that, when P and C are self-adjoint with respect to the same Hermitian sesquilinear form, the convergence of a minimum residual method in a particular nonstandard inner product applied to the preconditioned linear system is bounded by a term that depends only on the spectrum of P-1C. The inner product is related to the spectral decomposition of P. When P is self-adjoint with respect to a nearby Hermitian sesquilinear form to C, the convergence of a minimum residual method in this nonstandard inner product applied to the preconditioned linear system is bounded by a term involving the eigenvalues of P-1C and a constant factor. The size of this factor is related to the nearness of the Hermitian sesquilinear forms. Numerical experiments indicate that for certain matrices eigenvalue-dependent convergence is observed both for the nonstandard method and for standard GMRES. © 2013 Elsevier B.V. All rights reserved. |
| first_indexed | 2024-03-06T21:14:42Z |
| format | Journal article |
| id | oxford-uuid:3f5f13bb-8e11-4f4f-883f-49b063b4bc2b |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T21:14:42Z |
| publishDate | 2013 |
| record_format | dspace |
| spelling | oxford-uuid:3f5f13bb-8e11-4f4f-883f-49b063b4bc2b2022-03-26T14:31:40ZOn the choice of preconditioner for minimum residual methods for non-Hermitian matricesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:3f5f13bb-8e11-4f4f-883f-49b063b4bc2bEnglishSymplectic Elements at Oxford2013Pestana, JWathen, AWe consider the solution of left preconditioned linear systems P- 1Cx=P-1c, where P,CECn×n are non-Hermitian, cECn, and C, P, and P-1C are diagonalisable with spectra symmetric about the real line. We prove that, when P and C are self-adjoint with respect to the same Hermitian sesquilinear form, the convergence of a minimum residual method in a particular nonstandard inner product applied to the preconditioned linear system is bounded by a term that depends only on the spectrum of P-1C. The inner product is related to the spectral decomposition of P. When P is self-adjoint with respect to a nearby Hermitian sesquilinear form to C, the convergence of a minimum residual method in this nonstandard inner product applied to the preconditioned linear system is bounded by a term involving the eigenvalues of P-1C and a constant factor. The size of this factor is related to the nearness of the Hermitian sesquilinear forms. Numerical experiments indicate that for certain matrices eigenvalue-dependent convergence is observed both for the nonstandard method and for standard GMRES. © 2013 Elsevier B.V. All rights reserved. |
| spellingShingle | Pestana, J Wathen, A On the choice of preconditioner for minimum residual methods for non-Hermitian matrices |
| title | On the choice of preconditioner for minimum residual methods for non-Hermitian matrices |
| title_full | On the choice of preconditioner for minimum residual methods for non-Hermitian matrices |
| title_fullStr | On the choice of preconditioner for minimum residual methods for non-Hermitian matrices |
| title_full_unstemmed | On the choice of preconditioner for minimum residual methods for non-Hermitian matrices |
| title_short | On the choice of preconditioner for minimum residual methods for non-Hermitian matrices |
| title_sort | on the choice of preconditioner for minimum residual methods for non hermitian matrices |
| work_keys_str_mv | AT pestanaj onthechoiceofpreconditionerforminimumresidualmethodsfornonhermitianmatrices AT wathena onthechoiceofpreconditionerforminimumresidualmethodsfornonhermitianmatrices |