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

Martin Kahlbacher; Stefan Volkwein

ESAIM: Mathematical Modelling and Numerical Analysis (2011)

  • Volume: 46, Issue: 2, page 491-511
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Kahlbacher, Martin, and Volkwein, Stefan. "POD a-posteriori error based inexact SQP method for bilinear elliptic optimal control problems∗." ESAIM: Mathematical Modelling and Numerical Analysis 46.2 (2011): 491-511. <http://eudml.org/doc/222119>.

@article{Kahlbacher2011,
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.},
author = {Kahlbacher, Martin, Volkwein, Stefan},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Optimal control; inexact SQP method; proper orthogonal decomposition; a-posteriori error estimates; bilinear elliptic equation; optimal control; inexact sequential quadratic programming method; a-posteriori error estimates},
language = {eng},
month = {12},
number = {2},
pages = {491-511},
publisher = {EDP Sciences},
title = {POD a-posteriori error based inexact SQP method for bilinear elliptic optimal control problems∗},
url = {http://eudml.org/doc/222119},
volume = {46},
year = {2011},
}

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
DA - 2011/12//
PB - EDP Sciences
VL - 46
IS - 2
SP - 491
EP - 511
AB - 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.
LA - eng
KW - Optimal control; inexact SQP method; proper orthogonal decomposition; a-posteriori error estimates; bilinear elliptic equation; optimal control; inexact sequential quadratic programming method; a-posteriori error estimates
UR - http://eudml.org/doc/222119
ER -

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