Natural preconditioning and iterative methods for saddle point systems
The solution of quadratic or locally quadratic extremum problems subject to linear(ized) constraints gives rise to linear systems in saddle point form. This is true whether in the continuous or the discrete setting, so saddle point systems arising from the discretization of partial differential equa...
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| Format: | Journal article |
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Society for Industrial and Applied Mathematics
2015
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| _version_ | 1826307062739501056 |
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| author | Pestana, J Wathen, AJ |
| author_facet | Pestana, J Wathen, AJ |
| author_sort | Pestana, J |
| collection | OXFORD |
| description | The solution of quadratic or locally quadratic extremum problems subject to linear(ized) constraints gives rise to linear systems in saddle point form. This is true whether in the continuous or the discrete setting, so saddle point systems arising from the discretization of partial differential equation problems, such as those describing electromagnetic problems or incompressible flow, lead to equations with this structure, as do, for example, interior point methods and the sequential quadratic programming approach to nonlinear optimization. This survey concerns iterative solution methods for these problems and, in particular, shows how the problem formulation leads to natural preconditioners which guarantee a fast rate of convergence of the relevant iterative methods. These preconditioners are related to the original extremum problem and their effectiveness---in terms of rapidity of convergence---is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matrix on which iteration convergence depends. |
| first_indexed | 2024-03-07T06:57:21Z |
| format | Journal article |
| id | oxford-uuid:fe93a977-3f2c-406a-9f92-6f211c642c6d |
| institution | University of Oxford |
| last_indexed | 2024-03-07T06:57:21Z |
| publishDate | 2015 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:fe93a977-3f2c-406a-9f92-6f211c642c6d2022-03-27T13:37:49ZNatural preconditioning and iterative methods for saddle point systemsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:fe93a977-3f2c-406a-9f92-6f211c642c6dSymplectic Elements at OxfordSociety for Industrial and Applied Mathematics2015Pestana, JWathen, AJThe solution of quadratic or locally quadratic extremum problems subject to linear(ized) constraints gives rise to linear systems in saddle point form. This is true whether in the continuous or the discrete setting, so saddle point systems arising from the discretization of partial differential equation problems, such as those describing electromagnetic problems or incompressible flow, lead to equations with this structure, as do, for example, interior point methods and the sequential quadratic programming approach to nonlinear optimization. This survey concerns iterative solution methods for these problems and, in particular, shows how the problem formulation leads to natural preconditioners which guarantee a fast rate of convergence of the relevant iterative methods. These preconditioners are related to the original extremum problem and their effectiveness---in terms of rapidity of convergence---is established here via a proof of general bounds on the eigenvalues of the preconditioned saddle point matrix on which iteration convergence depends. |
| spellingShingle | Pestana, J Wathen, AJ Natural preconditioning and iterative methods for saddle point systems |
| title | Natural preconditioning and iterative methods for saddle point systems |
| title_full | Natural preconditioning and iterative methods for saddle point systems |
| title_fullStr | Natural preconditioning and iterative methods for saddle point systems |
| title_full_unstemmed | Natural preconditioning and iterative methods for saddle point systems |
| title_short | Natural preconditioning and iterative methods for saddle point systems |
| title_sort | natural preconditioning and iterative methods for saddle point systems |
| work_keys_str_mv | AT pestanaj naturalpreconditioninganditerativemethodsforsaddlepointsystems AT wathenaj naturalpreconditioninganditerativemethodsforsaddlepointsystems |