Gaspoz, F. ; Kreuzer, C. ; Veeser, A. ; Wollner, W. (2019)
Quasi-best approximation in optimization with PDE constraints.
In: Inverse Problems, 36 (1)
doi: 10.1088/1361-6420/ab47f3
Article, Bibliographie
This is the latest version of this item.
Abstract
We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we prove that the combined error in the state and adjoint state of the variational discretization is bounded by the best approximation error in the underlying discrete spaces. The constant in this bound depends on the inverse square root of the Tikhonov regularization parameter. Furthermore, if the operators of control action and observation are compact, this quasi-best approximation constant becomes independent of the Tikhonov parameter as the mesh size tends to 0 and we give quantitative relationships between mesh size and Tikhonov parameter ensuring this independence. We also derive generalizations of these results when the control variable is discretized or when it is taken from a convex set.
| Item Type: | Article |
|---|---|
| Erschienen: | 2019 |
| Creators: | Gaspoz, F. ; Kreuzer, C. ; Veeser, A. ; Wollner, W. |
| Type of entry: | Bibliographie |
| Title: | Quasi-best approximation in optimization with PDE constraints |
| Language: | English |
| Date: | 19 December 2019 |
| Publisher: | IOP Publishing |
| Journal or Publication Title: | Inverse Problems |
| Volume of the journal: | 36 |
| Issue Number: | 1 |
| DOI: | 10.1088/1361-6420/ab47f3 |
| Corresponding Links: | |
| Abstract: | We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we prove that the combined error in the state and adjoint state of the variational discretization is bounded by the best approximation error in the underlying discrete spaces. The constant in this bound depends on the inverse square root of the Tikhonov regularization parameter. Furthermore, if the operators of control action and observation are compact, this quasi-best approximation constant becomes independent of the Tikhonov parameter as the mesh size tends to 0 and we give quantitative relationships between mesh size and Tikhonov parameter ensuring this independence. We also derive generalizations of these results when the control variable is discretized or when it is taken from a convex set. |
| Additional Information: | Special Issue on Optimal Control and Inverse Problems; Erstveröffentlichung |
| Divisions: | 04 Department of Mathematics 04 Department of Mathematics > Optimization |
| Date Deposited: | 12 Nov 2020 13:38 |
| Last Modified: | 03 Jul 2024 02:48 |
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| Suche nach Titel in: | TUfind oder in Google |
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Quasi-best approximation in optimization with PDE constraints. (deposited 06 Sep 2021 12:12)
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