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

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

top

## Abstract

top
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

top

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 - Modélisation Mathématique et Analyse Numérique 46.2 (2012): 491-511. <http://eudml.org/doc/273212>.

@article{Kahlbacher2012,
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 - Modélisation Mathématique et Analyse Numérique},
keywords = {optimal control; inexact SQP method; proper orthogonal decomposition; a-posteriori error estimates; bilinear elliptic equation; inexact sequential quadratic programming method},
language = {eng},
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/273212},
volume = {46},
year = {2012},
}

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 - Modélisation Mathématique et Analyse Numérique
PY - 2012
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; inexact sequential quadratic programming method
UR - http://eudml.org/doc/273212
ER -

## References

top
1. [1] W. Alt, The Lagrange-Newton method for infinite-dimensional optimization problems. Numer. Funct. Anal. Optim.11 (1990) 201–224. Zbl0694.49022MR1068833
2. [2] A.C. Antoulas, Approximation of Large-Scale Dynamical Systems. Advances in Design and Control, SIAM, Philadelphia (2005). Zbl1158.93001MR2155615
3. [3] N. Arada, E. Casas and F. Tröltzsch. Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl.23 (2002) 201–229. Zbl1033.65044MR1937089
4. [4] E. Arian, M. Fahl and E.W. Sachs, Trust-region proper orthogonal decomposition for flow control. Technical Report 2000-25, ICASE (2000).
5. [5] J.A. Atwell, J.T. Borggaard and B.B. King, Reduced order controllers for Burgers’ equation with a nonlinear observer. Int. J. Appl. Math. Comput. Sci.11 (2001) 1311–1330. Zbl1051.93045MR1885507
6. [6] P. Benner and E.S. Quintana-Ortí, Model reduction based on spectral projection methods, in Reduction of Large-Scale Systems, Lect. Notes Comput. Sci. Eng. 45, edited by P. Benner, V. Mehrmann and D.C. Sorensen (2005) 5–48. Zbl1106.93015MR2503778
7. [7] P. Deuflhard, Newton Methods for Nonlinear Problems : Affine Invariance and Adaptive Algorithms, Springer Series in Comput. Math. 35 (2004). Zbl1056.65051MR2063044
8. [8] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics. American Mathematical Society, Providence, Rhode Island 19 (2002). Zbl0999.35059MR1625845
9. [9] R.S. Falk, Error estimates for the approximation of a class of variational inequalities. Math. Comput.28 (1974) 963–971. Zbl0297.65061MR391502
10. [10] T. Gänzler, S. Volkwein and M. Weiser, SQP methods for parameter identification problems arising in hyperthermia. Optim. Methods Softw.21 (2006) 869–887. Zbl1113.65067MR2261535
11. [11] M. Hintermüller, On a globalized augmented Lagrangian SQP-algorithm for nonlinear optimal control problems with box constraints, in Fast solution methods for discretized optimization problems, International Series of Numerical Mathematics. edited by K.-H. Hoffmann, R.H.W. Hoppe and V. Schulz, Birkhäuser publishers, Basel 138 (2001) 139–153. Zbl0999.49020MR1941059
12. [12] M. Hinze and S. Volkwein, Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. Comput. Optim. Appl.39 (2008) 319–345. Zbl1191.49040MR2396870
13. [13] A. Kröner and B. Vexler, A priori error estimates for elliptic optimal control problems with bilinear state equation. J. Comput. Appl. Math.230 (2009) 781–802. Zbl1178.65071MR2536007
14. [14] K. Kunisch and S. Volkwein, Proper orthogonal decomposition for optimality systems. ESAIM : M2AN 42 (2008) 1–23. Zbl1141.65050MR2387420
15. [15] H.V. Ly and H.T. Tran, Modeling and control of physical processes using proper orthogonal decomposition. Math. Comput. Model.33 (2001) 223–236. Zbl0966.93018
16. [16] K. Malanowski, C. Büskens and H. Maurer, Convergence of approximations to nonlinear control problems, in Mathematical Programming with Data Perturbation, edited by A.V. Fiacco and M. Dekker. Inc., New York (1997) 253–284. Zbl0883.49025
17. [17] A.T. Patera and G. Rozza, Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations. MIT Pappalardo Graduate Monographs in Mechanical Engineering (2006). Zbl1304.65251
18. [18] S.S. Ravindran, Adaptive reduced order controllers for a thermal flow system using proper orthogonal decomposition. SIAM J. Sci. Comput.28 (2002) 1924–1942. Zbl1026.76015MR1923719
19. [19] M. Read and B. Simon, Methods of Modern Mathematical Physics I : Functional Analysis. Academic Press, Boston (1980). Zbl0459.46001MR751959
20. [20] E.W. Sachs and S. Volkwein, Augmented Lagrange-SQP methods with Lipschitz-continuous Lagrange multiplier updates. SIAM J. Numer. Anal.40 (2002) 233–253. Zbl1027.49027MR1921918
21. [21] L. Sirovich, Turbulence and the dynamics of coherent structures, parts I-III. Quart. Appl. Math. XLV (1987) 561–590. Zbl0676.76047MR910462
22. [22] T. Tonn, K. Urban and S. Volkwein, Comparison of the reduced-basis and POD a-posteriori error estimators for an elliptic linear-quadratic optimal control problem. Math. Comput. Modelling of Dynam. Systems17 (2011) 355-369. Zbl1302.49045MR2823468
23. [23] F. Tröltzsch, Optimal Control of Partial Differential Equations : Theory, Methods and Applications, Graduate Studies in Mathematics. American Mathematical Society 112 (2010). Zbl1195.49001
24. [24] F. Tröltzsch and S. Volkwein, POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl.44 (2009) 83–115. Zbl1189.49050MR2556846
25. [25] M. Vallejos and A. Borzì, Multigrid optimization methods for linear and bilinear elliptic optimal control problems. Computing82 (2008) 31–52. Zbl1156.65068MR2395267
26. [26] S. Volkwein, Mesh-independence of an augmented Lagrangian-SQP method in Hilbert spaces. SIAM J. Control Optimization38 (2000) 767–785. Zbl0945.49024MR1756894

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.