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

¹ Universität Graz, Institut für Mathematik und Wissenschaftliches Rechnen, Heinrichstraße 36, 8010 Graz, Austria
martin.kahlbacher@hotmail.com
² Universität Konstanz, Fachbereich Mathematik und Statistik, Universitätsstraße 10, 78457 Konstanz, Germany
Stefan.Volkwein@uni-konstanz.de

Received: 23 March 2010
Revised: 24 May 2011

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.