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.
Mots clés : optimal control, inexact SQP method, proper orthogonal decomposition, a-posteriori error estimates, bilinear elliptic equation
@article{M2AN_2012__46_2_491_0, author = {Kahlbacher, Martin and Volkwein, Stefan}, title = {POD \protect\emph{a-posteriori }error based inexact {SQP} method for bilinear elliptic optimal control problems}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {491--511}, publisher = {EDP-Sciences}, volume = {46}, number = {2}, year = {2012}, doi = {10.1051/m2an/2011061}, zbl = {1272.49059}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2011061/} }
TY - JOUR AU - Kahlbacher, Martin AU - Volkwein, Stefan TI - POD a-posteriori error based inexact SQP method for bilinear elliptic optimal control problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 491 EP - 511 VL - 46 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2011061/ DO - 10.1051/m2an/2011061 LA - en ID - M2AN_2012__46_2_491_0 ER -
%0 Journal Article %A Kahlbacher, Martin %A Volkwein, Stefan %T POD a-posteriori error based inexact SQP method for bilinear elliptic optimal control problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 491-511 %V 46 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2011061/ %R 10.1051/m2an/2011061 %G en %F M2AN_2012__46_2_491_0
Kahlbacher, Martin; Volkwein, Stefan. POD a-posteriori error based inexact SQP method for bilinear elliptic optimal control problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 2, pp. 491-511. doi : 10.1051/m2an/2011061. http://www.numdam.org/articles/10.1051/m2an/2011061/
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