Abstract
In general, reduced-order model (ROM) solutions obtained using proper orthogonal decomposition (POD) at a single parameter cannot approximate the solutions at other parameter values accurately. In this paper, parameter sensitivity analysis is performed for POD reduced order optimal control problems (OCPs) governed by linear diffusion-convection-reaction equations. The OCP is discretized in space and time by discontinuous Galerkin (dG) finite elements. We apply two techniques, extrapolating and expanding the POD basis, to assess the accuracy of the reduced solutions for a range of parameters. Numerical results are presented to demonstrate the performance of these techniques to analyze the sensitivity of the OCP with respect to the ratio of the convection to the diffusion terms.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
- Optimal Control Problem
- Proper Orthogonal Decomposition
- Proper Orthogonal Decomposition Mode
- Parameter Sensitivity Analysis
- Proper Orthogonal Decomposition Basis
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
T. Akman, H. Yücel, B. Karasözen, A priori error analysis of the upwind symmetric interior penalty galerkin (SIPG) method for the optimal control problems governed by unsteady convection diffusion equations. Comput. Optim. Appl. 57(3), 703–729 (2014)
D.N. Arnold, F. Brezzi, B. Cockburn, L.D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39(5), 1749–1779 (2001/2002)
A. Hay, J.T. Borggaard, D. Pelletier, Local improvements to reduced-order models using sensitivity analysis of the proper orthogonal decomposition. J. Fluid Mech. 629, 41–72 (2009)
M. Hinze, S. Volkwein, Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control, in Dimension Reduction of Large-Scale Systems, ed. by P. Benner, V. Mehrmann, D.C. Sorensen. Lecture Notes in Computational Science and Engineering, vol. 45 (Springer, Berlin, 2005), pp. 261–306
M. Hinze, S. Volkwein, Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. Comput. Optim. Appl. 39(3), 319–345 (2008)
A. Quarteroni, G. Rozza, A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Ind. 1, Art. 3, 44 (2011)
V. Thomée, Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, vol. 25, 2nd edn. (Springer, Berlin, 2006)
F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Graduate Studies in Mathematics, vol. 112 (American Mathematical Society, Providence, 2010)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Akman, T., Karasözen, B. (2015). Reduced Order Optimal Control Using Proper Orthogonal Decomposition Sensitivities. In: Abdulle, A., Deparis, S., Kressner, D., Nobile, F., Picasso, M. (eds) Numerical Mathematics and Advanced Applications - ENUMATH 2013. Lecture Notes in Computational Science and Engineering, vol 103. Springer, Cham. https://doi.org/10.1007/978-3-319-10705-9_40
Download citation
DOI: https://doi.org/10.1007/978-3-319-10705-9_40
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10704-2
Online ISBN: 978-3-319-10705-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)