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http://yadda.icm.edu.pl:80/baztech/element/bwmeta1.element.baztech-article-BATC-0009-0029

Czasopismo

Control and Cybernetics

Tytuł artykułu

Regularization error estimates and discrepancy principle for optimal control problems with inequality constraints

Autorzy Wachsmuth, D.  Wachsmuth, G. 
Treść / Zawartość
Warianty tytułu
Języki publikacji EN
Abstrakty
EN In this article we study the regularization of optimization problems by Tikhonov regularization. The optimization problems are subject to pointwise inequality constraints in L²(Ω). We derive a-priori regularization error estimates if the regularization parameter as well as the noise level tend to zero. We rely on an assumption that is a combination of a source condition and of a structural assumption on the active sets. Moreover, we introduce a strategy to choose the regularization parameter in dependence of the noise level. We prove convergence of this parameter choice rule with optimal order.
Słowa kluczowe
EN source condition   discrepancy principle   nonsmooth optimization   convex constraints   sparsity   regularization error estimates  
Wydawca Systems Research Institute, Polish Academy of Sciences
Czasopismo Control and Cybernetics
Rocznik 2011
Tom Vol. 40, no 4
Strony 1125--1158
Opis fizyczny Bibliogr. 14 poz.
Twórcy
autor Wachsmuth, D.
autor Wachsmuth, G.
  • Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences Altenbergerstraße 69, A-4040 Linz, Austria, daniel.wachsmuth@oeaw.ac.at
Bibliografia
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Felgenhauer, U. (2003) On stability of Bang-bang type controls. SIAM J. Control Optim., 41(6), 1843-1867.
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Hofmann, B., Düvelmeyer, D. and Krumbiegel, K. (2006) Approximative source conditions in Tikhonov regularization- new analytical results and some numerical studies. Mathematical Modeling and Analysis, 11(1), 41-56.
Hofmann, B. and Mathé, P. (2007) Analysis of profile functions for general linear regularization methods. SIAM Journal on Numerical Analysis, 45(3), 1122-1141 (electronic).
Morozov, V.A. (1993) Regularization methods for ill-posed problems. CRC Press, Boca Raton, FL. (translated from the 1987 Russian original).
Neubauer,A. (1988) Tikhonov-regularization of ill-posed linear operator equations on closed convex sets. J. Approx. Theory, 53(3), 304-320.
Stadler, G. (2009) Elliptic optimal control problems with L1-control cost and applications for the placement of control devices. Computational Optimization and Applications, 44, 159-181.
Tröltzsch, F. (2010) Optimal Control of Partial Differentail Equations. Graduate Studies in Mathematics, 112. American Mathematical Society, Providence (translated from the 2005 German original by J. Sprekels).
Wachsmuth, D. and Wachsmuth, G. (2011) Convergence and regularization results for optimal control problems with Sparsity Functional. ESAIM Control Optim. Calc. Var., 17(3), 858-886.
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