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Quasi-best approximation in optimization with PDE constraints

Gaspoz, Fernando ; Kreuzer, Christian ; Veeser, Andreas ; Wollner, Winnifried (2021):
Quasi-best approximation in optimization with PDE constraints. (Publisher's Version)
In: Inverse Problems, 36 (1), IOP Publishing, ISSN 0266-5611, e-ISSN 1361-6420,
DOI: 10.26083/tuprints-00019330,

Available under: CC-BY 3.0 Unported - Creative Commons, Attribution.

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Item Type: Article
Origin: Secondary publication via sponsored Golden Open Access
Status: Publisher's Version
Title: Quasi-best approximation in optimization with PDE constraints
Language: English

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.

Journal or Publication Title: Inverse Problems
Volume of the journal: 36
Issue Number: 1
Publisher: IOP Publishing
Collation: 29 Seiten
Classification DDC: 500 Naturwissenschaften und Mathematik > 510 Mathematik
Divisions: 04 Department of Mathematics > Optimization
Date Deposited: 06 Sep 2021 12:12
Last Modified: 06 Sep 2021 12:12
DOI: 10.26083/tuprints-00019330
Corresponding Links:
URN: urn:nbn:de:tuda-tuprints-193304
URI: https://tuprints.ulb.tu-darmstadt.de/id/eprint/19330
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