September  2015, 35(9): 4477-4501. doi: 10.3934/dcds.2015.35.4477

Uniform convergence of the POD method and applications to optimal control

1. 

Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstraße 36, 8010 Graz, Austria, Austria

Received  May 2014 Revised  September 2014 Published  April 2015

We consider proper orthogonal decomposition (POD) based Galerkin approximations to parabolic systems and establish uniform convergence with respect to forcing functions. The result is used to prove convergence of POD approximations to optimal control problems that automatically update the POD basis in order to avoid problems due to unmodeled dynamics in the POD reduced order system. A numerical example illustrates the results.
Citation: Karl Kunisch, Markus Müller. Uniform convergence of the POD method and applications to optimal control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4477-4501. doi: 10.3934/dcds.2015.35.4477
References:
[1]

D. Chapelle, A. Gariah and J. Sainte-Marie, Galerkin approximation with proper orthogonal decomposition: New error estimates and illustrative examples,, ESAIM: Mathematical Modelling and Numerical Analysis, 46 (2012), 731.  doi: 10.1051/m2an/2011053.  Google Scholar

[2]

S. Chaturantabut and D. Sorensen, Nonlinear model reduction via discrete empirical interpolation,, SIAM Journal Scientific Computing, 32 (2010), 2737.  doi: 10.1137/090766498.  Google Scholar

[3]

T. Henri, Réduction de Modèles par des Méthodes de Décomposition Orthogonale Propre,, PhD thesis, (2003).   Google Scholar

[4]

T. Henri and J. Yvon, Convergence estimates of POD-Galerkin methods for parabolic problems,, System modeling and optimization, 166 (2005), 295.  doi: 10.1007/0-387-23467-5_21.  Google Scholar

[5]

D. Hömberg and S. Volkwein, Control of laser surface hardening by a reduced-order approach using POD,, Mathematical and Computer Modelling, 38 (2003), 1003.  doi: 10.1016/S0895-7177(03)90102-6.  Google Scholar

[6]

K. Kunisch and S. Volkwein, Galerkin POD methods for parabolic problems,, Numerische Mathematik, 90 (2001), 117.  doi: 10.1007/s002110100282.  Google Scholar

[7]

K. Kunisch and S. Volkwein, Proper orthogonal decomposition for optimality systems,, ESAIM: M2AN, 42 (2008), 1.  doi: 10.1051/m2an:2007054.  Google Scholar

[8]

J. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971).   Google Scholar

[9]

G. Lube, Theorie und Numerik Instationärer Probleme, 2007,, Lecture Notes, ().   Google Scholar

[10]

R. Meise and D. Vogt, Introduction to Functional Analysis,, Oxford graduate texts in mathematics, (1997).   Google Scholar

[11]

M. Müller, Uniform Convergence of the POD Method and Applications to Optimal Control,, PhD thesis, (2011).   Google Scholar

[12]

L. Schwartz, Analyse Hilbertienne,, Hermann Paris, (1979).   Google Scholar

[13]

H. Triebel, Higher Analysis,, Johann Ambrosius Barth, (1992).   Google Scholar

[14]

H. Triebel, Theory of Function Spaces III,, Birkhäuser Verlag, (2006).   Google Scholar

[15]

F. Tröltzsch and S. Volkwein, POD a-posteriori error estimates for linear-quadratic optimal control problems,, Computational Optimization and Applications, 44 (2009), 83.  doi: 10.1007/s10589-008-9224-3.  Google Scholar

[16]

S. Volkwein, Optimality system POD and a-posteriori error analysis for linear-quadratic problems,, Control and Cybernetics, 40 (2011), 1109.   Google Scholar

[17]

K. Willcox and J. Peraire, Balanced model reduction via the proper orthogonal decomposition,, Balanced Model Reduction via the Proper Orthogonal Decomposition, 40 (2002), 2323.  doi: 10.2514/2.1570.  Google Scholar

[18]

J. Wloka, Partielle Differentialgleichungen,, B.G. Teubner, (1982).   Google Scholar

show all references

References:
[1]

D. Chapelle, A. Gariah and J. Sainte-Marie, Galerkin approximation with proper orthogonal decomposition: New error estimates and illustrative examples,, ESAIM: Mathematical Modelling and Numerical Analysis, 46 (2012), 731.  doi: 10.1051/m2an/2011053.  Google Scholar

[2]

S. Chaturantabut and D. Sorensen, Nonlinear model reduction via discrete empirical interpolation,, SIAM Journal Scientific Computing, 32 (2010), 2737.  doi: 10.1137/090766498.  Google Scholar

[3]

T. Henri, Réduction de Modèles par des Méthodes de Décomposition Orthogonale Propre,, PhD thesis, (2003).   Google Scholar

[4]

T. Henri and J. Yvon, Convergence estimates of POD-Galerkin methods for parabolic problems,, System modeling and optimization, 166 (2005), 295.  doi: 10.1007/0-387-23467-5_21.  Google Scholar

[5]

D. Hömberg and S. Volkwein, Control of laser surface hardening by a reduced-order approach using POD,, Mathematical and Computer Modelling, 38 (2003), 1003.  doi: 10.1016/S0895-7177(03)90102-6.  Google Scholar

[6]

K. Kunisch and S. Volkwein, Galerkin POD methods for parabolic problems,, Numerische Mathematik, 90 (2001), 117.  doi: 10.1007/s002110100282.  Google Scholar

[7]

K. Kunisch and S. Volkwein, Proper orthogonal decomposition for optimality systems,, ESAIM: M2AN, 42 (2008), 1.  doi: 10.1051/m2an:2007054.  Google Scholar

[8]

J. Lions, Optimal Control of Systems Governed by Partial Differential Equations,, Springer-Verlag, (1971).   Google Scholar

[9]

G. Lube, Theorie und Numerik Instationärer Probleme, 2007,, Lecture Notes, ().   Google Scholar

[10]

R. Meise and D. Vogt, Introduction to Functional Analysis,, Oxford graduate texts in mathematics, (1997).   Google Scholar

[11]

M. Müller, Uniform Convergence of the POD Method and Applications to Optimal Control,, PhD thesis, (2011).   Google Scholar

[12]

L. Schwartz, Analyse Hilbertienne,, Hermann Paris, (1979).   Google Scholar

[13]

H. Triebel, Higher Analysis,, Johann Ambrosius Barth, (1992).   Google Scholar

[14]

H. Triebel, Theory of Function Spaces III,, Birkhäuser Verlag, (2006).   Google Scholar

[15]

F. Tröltzsch and S. Volkwein, POD a-posteriori error estimates for linear-quadratic optimal control problems,, Computational Optimization and Applications, 44 (2009), 83.  doi: 10.1007/s10589-008-9224-3.  Google Scholar

[16]

S. Volkwein, Optimality system POD and a-posteriori error analysis for linear-quadratic problems,, Control and Cybernetics, 40 (2011), 1109.   Google Scholar

[17]

K. Willcox and J. Peraire, Balanced model reduction via the proper orthogonal decomposition,, Balanced Model Reduction via the Proper Orthogonal Decomposition, 40 (2002), 2323.  doi: 10.2514/2.1570.  Google Scholar

[18]

J. Wloka, Partielle Differentialgleichungen,, B.G. Teubner, (1982).   Google Scholar

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