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POD a-posteriori error based inexact SQP method for bilinear elliptic optimal control problems∗

Published online by Cambridge University Press:  19 December 2011

Martin Kahlbacher  and
Stefan Volkwein
Martin Kahlbacher
Affiliation:
Universität Graz, Institut für Mathematik und Wissenschaftliches Rechnen, Heinrichstraße 36, 8010 Graz, Austria. martin.kahlbacher@hotmail.com
Stefan Volkwein
Affiliation:
Universität Konstanz, Fachbereich Mathematik und Statistik, Universitätsstraße 10, 78457 Konstanz, Germany; Stefan.Volkwein@uni-konstanz.de
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Abstract

An optimal control problem governed by a bilinear elliptic equation is considered. This problem is solved by the sequential quadratic programming (SQP) method in an infinite-dimensional framework. In each level of this iterative method the solution of linear-quadratic subproblem is computed by a Galerkin projection using proper orthogonal decomposition (POD). Thus, an approximate (inexact) solution of the subproblem is determined. Based on a POD a-posteriori error estimator developed by Tröltzsch and Volkwein [Comput. Opt. Appl. 44 (2009) 83–115] the difference of the suboptimal to the (unknown) optimal solution of the linear-quadratic subproblem is estimated. Hence, the inexactness of the discrete solution is controlled in such a way that locally superlinear or even quadratic rate of convergence of the SQP is ensured. Numerical examples illustrate the efficiency for the proposed approach.

Keywords

Optimal controlinexact SQP methodproper orthogonal decompositiona-posteriori error estimatesbilinear elliptic equation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 2 , March 2012 , pp. 491 - 511
Copyright
© EDP Sciences, SMAI 2011

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