A note on multigrid preconditioning for fractional PDE-constrained optimization problems
In this note we present a multigrid preconditioning method for solving quadratic optimization problems constrained by a fractional diffusion equation. Multigrid methods within the all-at-once approach to solve the first order optimality Karush–Kuhn–Tucker (KKT) systems are widely popular, but their...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Elsevier
2021-02-01
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| Series: | Results in Applied Mathematics |
| Subjects: | |
| Online Access: | http://www.sciencedirect.com/science/article/pii/S2590037420300431 |
| _version_ | 1818944170318888960 |
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| author | Harbir Antil Andrei Drăgănescu Kiefer Green |
| author_facet | Harbir Antil Andrei Drăgănescu Kiefer Green |
| author_sort | Harbir Antil |
| collection | DOAJ |
| description | In this note we present a multigrid preconditioning method for solving quadratic optimization problems constrained by a fractional diffusion equation. Multigrid methods within the all-at-once approach to solve the first order optimality Karush–Kuhn–Tucker (KKT) systems are widely popular, but their development have relied on the underlying systems being sparse. On the other hand, for most discretizations, the matrix representation of fractional operators is expected to be dense. We develop a preconditioning strategy for our problem based on a reduced approach, namely we eliminate the state constraint using the control-to-state map. Our multigrid preconditioning approach shows a dramatic reduction in the number of CG iterations. We assess the quality of preconditioner in terms of the spectral distance. Finally, we provide a partial theoretical analysis for this preconditioner, and we formulate a conjecture which is clearly supported by our numerical experiments. |
| first_indexed | 2024-12-20T07:38:58Z |
| format | Article |
| id | doaj.art-962762f5c56c4792bcb3cb64a93b466a |
| institution | Directory Open Access Journal |
| issn | 2590-0374 |
| language | English |
| last_indexed | 2024-12-20T07:38:58Z |
| publishDate | 2021-02-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Results in Applied Mathematics |
| spelling | doaj.art-962762f5c56c4792bcb3cb64a93b466a2022-12-21T19:48:10ZengElsevierResults in Applied Mathematics2590-03742021-02-019100133A note on multigrid preconditioning for fractional PDE-constrained optimization problemsHarbir Antil0Andrei Drăgănescu1Kiefer Green2Department of Mathematical Sciences and the Center for Mathematics and Artificial Intelligence (CMAI), George Mason University, Fairfax, VA 22030, USA; Corresponding author.Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USADepartment of Mathematical Sciences and the Center for Mathematics and Artificial Intelligence (CMAI), George Mason University, Fairfax, VA 22030, USAIn this note we present a multigrid preconditioning method for solving quadratic optimization problems constrained by a fractional diffusion equation. Multigrid methods within the all-at-once approach to solve the first order optimality Karush–Kuhn–Tucker (KKT) systems are widely popular, but their development have relied on the underlying systems being sparse. On the other hand, for most discretizations, the matrix representation of fractional operators is expected to be dense. We develop a preconditioning strategy for our problem based on a reduced approach, namely we eliminate the state constraint using the control-to-state map. Our multigrid preconditioning approach shows a dramatic reduction in the number of CG iterations. We assess the quality of preconditioner in terms of the spectral distance. Finally, we provide a partial theoretical analysis for this preconditioner, and we formulate a conjecture which is clearly supported by our numerical experiments.http://www.sciencedirect.com/science/article/pii/S2590037420300431Optimal controlFractional diffusionMultigridPreconditioner |
| spellingShingle | Harbir Antil Andrei Drăgănescu Kiefer Green A note on multigrid preconditioning for fractional PDE-constrained optimization problems Results in Applied Mathematics Optimal control Fractional diffusion Multigrid Preconditioner |
| title | A note on multigrid preconditioning for fractional PDE-constrained optimization problems |
| title_full | A note on multigrid preconditioning for fractional PDE-constrained optimization problems |
| title_fullStr | A note on multigrid preconditioning for fractional PDE-constrained optimization problems |
| title_full_unstemmed | A note on multigrid preconditioning for fractional PDE-constrained optimization problems |
| title_short | A note on multigrid preconditioning for fractional PDE-constrained optimization problems |
| title_sort | note on multigrid preconditioning for fractional pde constrained optimization problems |
| topic | Optimal control Fractional diffusion Multigrid Preconditioner |
| url | http://www.sciencedirect.com/science/article/pii/S2590037420300431 |
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