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
Artikel, Bibliographie
Kurzbeschreibung (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.
| Typ des Eintrags: | Artikel |
|---|---|
| Erschienen: | 2019 |
| Autor(en): | Gaspoz, F. ; Kreuzer, C. ; Veeser, A. ; Wollner, W. |
| Art des Eintrags: | Bibliographie |
| Titel: | Quasi-best approximation in optimization with PDE constraints |
| Sprache: | Englisch |
| Publikationsjahr: | 19 Dezember 2019 |
| Verlag: | IOP Publishing |
| Titel der Zeitschrift, Zeitung oder Schriftenreihe: | Inverse Problems |
| Jahrgang/Volume einer Zeitschrift: | 36 |
| (Heft-)Nummer: | 1 |
| DOI: | 10.1088/1361-6420/ab47f3 |
| Kurzbeschreibung (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. |
| Zusätzliche Informationen: | Special Issue on Optimal Control and Inverse Problems; Erstveröffentlichung |
| Fachbereich(e)/-gebiet(e): | 04 Fachbereich Mathematik 04 Fachbereich Mathematik > Optimierung |
| Hinterlegungsdatum: | 12 Nov 2020 13:38 |
| Letzte Änderung: | 18 Aug 2022 08:54 |
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