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POD a-posteriori error based inexact SQPmethod 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. Thisproblem is solved by the sequential quadratic programming (SQP) method in aninfinite-dimensional framework. In each level of this iterative method the solution oflinear-quadratic subproblem is computed by a Galerkin projection using proper orthogonaldecomposition (POD). Thus, an approximate (inexact) solution of the subproblem isdetermined. Based on a POD a-posteriori error estimator developed byTröltzsch and Volkwein [Comput. Opt. Appl. 44 (2009) 83–115]the difference of the suboptimal to the (unknown) optimal solution of the linear-quadraticsubproblem is estimated. Hence, the inexactness of the discrete solution is controlled insuch a way that locally superlinear or even quadratic rate of convergence of the SQP isensured. 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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References

Alt, W., The Lagrange-Newton method for infinite-dimensional optimization problems. Numer. Funct. Anal. Optim. 11 (1990) 201–224. Google Scholar
A.C. Antoulas, Approximation of Large-Scale Dynamical Systems. Advances in Design and Control, SIAM, Philadelphia (2005).
Arada, N., Casas, E. and Tröltzsch, F.. Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23 (2002) 201–229. Google Scholar
E. Arian, M. Fahl and E.W. Sachs, Trust-region proper orthogonal decomposition for flow control. Technical Report 2000-25, ICASE (2000).
Atwell, J.A., Borggaard, J.T. and King, B.B., Reduced order controllers for Burgers’ equation with a nonlinear observer. Int. J. Appl. Math. Comput. Sci. 11 (2001) 1311–1330. Google Scholar
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.
P. Deuflhard, Newton Methods for Nonlinear Problems : Affine Invariance and Adaptive Algorithms, Springer Series in Comput. Math. 35 (2004).
L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics. American Mathematical Society, Providence, Rhode Island 19 (2002).
Falk, R.S., Error estimates for the approximation of a class of variational inequalities. Math. Comput. 28 (1974) 963–971. Google Scholar
Gänzler, T., Volkwein, S. and Weiser, M., SQP methods for parameter identification problems arising in hyperthermia. Optim. Methods Softw. 21 (2006) 869–887. Google Scholar
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.
Hinze, M. and Volkwein, S., Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. Comput. Optim. Appl. 39 (2008) 319–345. Google Scholar
Kröner, A. and Vexler, B., A priori error estimates for elliptic optimal control problems with bilinear state equation. J. Comput. Appl. Math. 230 (2009) 781–802. Google Scholar
Kunisch, K. and Volkwein, S., Proper orthogonal decomposition for optimality systems. ESAIM : M2AN 42 (2008) 1–23. Google Scholar
Ly, H.V. and Tran, H.T., Modeling and control of physical processes using proper orthogonal decomposition. Math. Comput. Model. 33 (2001) 223–236. Google Scholar
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.
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).
Ravindran, S.S., Adaptive reduced order controllers for a thermal flow system using proper orthogonal decomposition. SIAM J. Sci. Comput. 28 (2002) 1924–1942. Google Scholar
M. Read and B. Simon, Methods of Modern Mathematical Physics I : Functional Analysis. Academic Press, Boston (1980).
Sachs, E.W. and Volkwein, S., Augmented Lagrange-SQP methods with Lipschitz-continuous Lagrange multiplier updates. SIAM J. Numer. Anal. 40 (2002) 233–253. Google Scholar
Sirovich, L., Turbulence and the dynamics of coherent structures, parts I-III. Quart. Appl. Math. XLV (1987) 561–590. Google Scholar
Tonn, T., Urban, K. and Volkwein, S., Comparison of the reduced-basis and POD a-posteriori error estimators for an elliptic linear-quadratic optimal control problem. Math. Comput. Modelling of Dynam. Systems 17 (2011) 355-369. Google Scholar
F. Tröltzsch, Optimal Control of Partial Differential Equations : Theory, Methods and Applications, Graduate Studies in Mathematics. American Mathematical Society 112 (2010).
Tröltzsch, F. and Volkwein, S., POD a-posteriori error estimates for linear-quadratic optimal control problems. Comput. Optim. Appl. 44 (2009) 83–115. Google Scholar
Vallejos, M. and Borzì, A., Multigrid optimization methods for linear and bilinear elliptic optimal control problems. Computing 82 (2008) 31–52. Google Scholar
Volkwein, S., Mesh-independence of an augmented Lagrangian-SQP method in Hilbert spaces. SIAM J. Control Optimization 38 (2000) 767–785. Google Scholar