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Solving ill-posed control problems by stabilized finite element methods: an alternative to Tikhonov regularization
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2018 (English)In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 34, no 3, article id 035004Article in journal (Refereed) Published
Abstract [en]

Tikhonov regularization is one of the most commonly used methods for the regularization of ill-posed problems. In the setting of finite element solutions of elliptic partial differential control problems, Tikhonov regularization amounts to adding suitably weighted least squares terms of the control variable, or derivatives thereof, to the Lagrangian determining the optimality system. In this note we show that the stabilization methods for discretely illposed problems developed in the setting of convection-dominated convection-diffusion problems, can be highly suitable for stabilizing optimal control problems, and that Tikhonov regularization will lead to less accurate discrete solutions. We consider some inverse problems for Poisson's equation as an illustration and derive new error estimates both for the reconstruction of the solution from the measured data and reconstruction of the source term from the measured data. These estimates include both the effect of the discretization error and error in the measurements.

Place, publisher, year, edition, pages
IOP PUBLISHING LTD , 2018. Vol. 34, no 3, article id 035004
Keywords [en]
optimal control problem, data assimilation, source identification, finite elements, regularization
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-144929DOI: 10.1088/1361-6420/aaa32bISI: 000423854300001OAI: oai:DiVA.org:umu-144929DiVA, id: diva2:1185371
Available from: 2018-02-23 Created: 2018-02-23 Last updated: 2018-06-09Bibliographically approved

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Larson, Mats G.
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