Regularization-robust preconditioners for time-dependent PDE constrained optimization problems
In this article, we motivate, derive and test effective preconditioners to be used with the Minres algorithm for solving a number of saddle point systems, which arise in PDE constrained optimization problems. We consider the distributed control problem involving the heat equation with two different...
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| 格式: | Report |
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SIAM
2011
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| _version_ | 1826299876617486336 |
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| author | Pearson, J Stoll, M Wathen, A |
| author_facet | Pearson, J Stoll, M Wathen, A |
| author_sort | Pearson, J |
| collection | OXFORD |
| description | In this article, we motivate, derive and test effective preconditioners to be used with the Minres algorithm for solving a number of saddle point systems, which arise in PDE constrained optimization problems. We consider the distributed control problem involving the heat equation with two different functionals, and the Neumann boundary control problem involving Poisson's equation and the heat equation. Crucial to the effectiveness of our preconditioners in each case is an effective approximation of the Schur complement of the matrix system. In each case, we state the problem being solved, propose the preconditioning approach, prove relevant eigenvalue bounds, and provide numerical results which demonstrate that our solvers are effective for a wide range of regularization parameter values, as well as mesh sizes and time-steps. |
| first_indexed | 2024-03-07T05:08:38Z |
| format | Report |
| id | oxford-uuid:daca0a93-9de9-4ed3-b9cd-9f709c847fa1 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T05:08:38Z |
| publishDate | 2011 |
| publisher | SIAM |
| record_format | dspace |
| spelling | oxford-uuid:daca0a93-9de9-4ed3-b9cd-9f709c847fa12022-03-27T09:05:44ZRegularization-robust preconditioners for time-dependent PDE constrained optimization problemsReporthttp://purl.org/coar/resource_type/c_93fcuuid:daca0a93-9de9-4ed3-b9cd-9f709c847fa1Mathematical Institute - ePrintsSIAM2011Pearson, JStoll, MWathen, AIn this article, we motivate, derive and test effective preconditioners to be used with the Minres algorithm for solving a number of saddle point systems, which arise in PDE constrained optimization problems. We consider the distributed control problem involving the heat equation with two different functionals, and the Neumann boundary control problem involving Poisson's equation and the heat equation. Crucial to the effectiveness of our preconditioners in each case is an effective approximation of the Schur complement of the matrix system. In each case, we state the problem being solved, propose the preconditioning approach, prove relevant eigenvalue bounds, and provide numerical results which demonstrate that our solvers are effective for a wide range of regularization parameter values, as well as mesh sizes and time-steps. |
| spellingShingle | Pearson, J Stoll, M Wathen, A Regularization-robust preconditioners for time-dependent PDE constrained optimization problems |
| title | Regularization-robust preconditioners for time-dependent PDE constrained optimization problems |
| title_full | Regularization-robust preconditioners for time-dependent PDE constrained optimization problems |
| title_fullStr | Regularization-robust preconditioners for time-dependent PDE constrained optimization problems |
| title_full_unstemmed | Regularization-robust preconditioners for time-dependent PDE constrained optimization problems |
| title_short | Regularization-robust preconditioners for time-dependent PDE constrained optimization problems |
| title_sort | regularization robust preconditioners for time dependent pde constrained optimization problems |
| work_keys_str_mv | AT pearsonj regularizationrobustpreconditionersfortimedependentpdeconstrainedoptimizationproblems AT stollm regularizationrobustpreconditionersfortimedependentpdeconstrainedoptimizationproblems AT wathena regularizationrobustpreconditionersfortimedependentpdeconstrainedoptimizationproblems |