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Abstrakty
In this paper an abstract linear-quadratic optimal control problem governed by an evolution equation is considered. To solve this problem numerically a reduced-order approach based on proper orthogonal decomposition (POD) is applied. The error between the POD suboptimal control and the optimal control of the original problem is controlled by an a-posteriori error analysis. However, if the POD basis has bad approximation properties, a huge number of POD basis function is required to solve the reduced-order problem with the desired accuracy. To overcome this problem, optimality system POD (OS-POD) is utilized, where the POD basis is chosen with respect to the optimization criteria.
Czasopismo
Rocznik
Tom
Strony
1109--1124
Opis fizyczny
Bibliogr. 23 poz., wykr.
Twórcy
autor
- Universitat Konstanz, Fachbereich Mathematik und Statistik Universitatsstraße 10, D-78457 Konstanz, Germany, Stefan.Volkwein@uni-konstanz.de
Bibliografia
- Afanasiev, K. and Hinze, M. (2001) Adaptive control of a wake flow using proper orthogonal decomposition. Lecture Notes in Pure and Applied Mathematics, 216, 317-332.
- Arian, E., Fahl, M. and Sachs, E.W. (2000) Trust-region proper orthogonal decomposition for flow control. Technical Report 2000-25, ICASE.
- Benner, P. and Quintana-Ortí,E.S. (2005) Model reduction based on spectra projection methods. In: P. Benner, V. Mehrmann, D.C. Sorensen, eds., Reduction of Large-Scale Systems, Lecture Notes in Computational Science and Engineering, 45, 5-48.
- Dautray, R. and Lions, J.L. (1992) Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5: Evolution Problems I. Springer-Verlag, Berlin.
- Grepl, M.A. and Kärcher, M. (2011) Reduced basis a posteriori error bounds for parametrized linear-quadratic elliptic optimal control problems. C. R. Acad. Sci. Paris, Ser. I, 349, 873-877.
- Hintermüller, M., Ito, K. and Kunisch, K. (2003) The prima-dual active set strategy as a semi-smooth Newton method. SIAM J. Opt., 13, 865-888.
- Hintermüller, M., Kopacka, I. and Volkwein, S. (2008) Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints. ESAIM: Control, Optimization and Calculus of Variations, 15, 626-652.
- Hintermüller,M. and Ulbrich,M. (2004) A mesh-independence result for semismooth Newton methods. Math. Program. Ser. B, 101, 151-184.
- Hinze, M. and Volkwein, S. (2008) Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. Computat. Optim. and Appl., 39, 319-345.
- Holmes, P., Lumley, J.L. and Berkooz, G. (1996) Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge Monographs on Mechanics, Cambridge University Press.
- Kahlbacher,M. and Volkwein, S. (2012) POD a-posteriori error based inexact SQP method for bilinear elliptic optimal control problems. ESAIM: Mathematical Modelling and Numerical Analysis, 46, 491-511.
- Kammann, E. (2010) Modellreduktion und Fehlerabschätzung bei parabolischen Optimalsteuerproblemen. Diploma thesis, Institute ofMathematics, Berlin University of Technology.
- Kelley, C.T. (1999) Iterative Methods for Optimization. Frontiers in Applied Mathematics. SIAM, Philadelphia.
- Kunisch, K. and Volkwein, S. (2002) Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal., 40, 492-515.
- Kunisch,K. and Volkwein, S. (2008) Proper orthogonal decomposition for optimality systems. ESAIM: Mathematical Modelling and Numerical Analysis, 42, 1-23.
- Lions, J.L. (1971) Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin.
- Manzoni, A., Quarteroni, A. and Rozza, G. (2011) Shape optimization for viscous flows by reduced basis methods and free-form deformation. International Journal for Numerical Methods in Fluids (to appear).
- Müller, M. (2011) Uniform Convergence of the POD method and Applications to Optimal Control. Ph.D thesis, University of Graz.
- Nocedal, J. and Wright, S.J. (2006) Numerical Optimization. Springer Series in Operation Research, Second Edition.
- Read, M. and Simon, B. (1980) Methods of Modern Mathematical Physics I: Functional Analysis Academic Press, Boston.
- Sachs, E. and Volkwein, S. (2010) POD-Galerkin approximations in PDE constrained optimization. GAMM-Mitt., 33, 194-208.
- Tonn,T., Urban,K. and Volkwein, S. (2011) Comparison of the reducedbasis and POD a-posteriori error estimators for an elliptic linear quadratic optimal control problem. Mathematical and Computer Modelling of Dynamical Systems, Special Issue: Model order reduction of parameterized problems, 17, 355-369.
- Tröltzsch, F. and Volkwein, S. (2009) POD a-posteriori error estimates for linear-quadratic optimal control problms. Comput. Optim. Appl., 44, 83-115.
Typ dokumentu
Bibliografia
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