Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-13T04:42:07.442Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal snapshot locationfor computingPOD basis functions

Published online by Cambridge University Press:  04 February 2010

Karl Kunisch  and
Stefan Volkwein
Karl Kunisch
Affiliation:
University of Graz, Institute for Mathematics and Scientific Computing, Heinrichstrasse 36, 8010 Graz, Austria. karl.kunisch@uni-graz.at
Stefan Volkwein
Affiliation:
University of Constance, Department for Mathematics and Statistics, Universitätsstraße 10, 78464 Konstanz, Germany. stefan.volkwein@uni-konstanz.de
Get access

Abstract

The construction of reduced order models for dynamical systems usingproper orthogonal decomposition (POD) is based on the informationcontained in so-called snapshots. These provide the spatialdistribution of the dynamical system at discrete time instances.This work is devoted to optimizing the choice of these timeinstances in such a manner that the error between the POD-solutionand the trajectory of the dynamical system is minimized. First andsecond order optimality systems are given. Numerical examplesillustrate that the proposed criterion is sensitive with respect tothe choice of the time instances and further they demonstrate thefeasibility of the method in determining optimal snapshot locationsfor concrete diffusion equations.

Keywords

Proper orthogonal decompositionoptimal snapshotlocationsfirst and second order optimality conditions
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 3 , May 2010 , pp. 509 - 529
Copyright
© EDP Sciences, SMAI, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

G. Berkooz, P. Holmes and J.L. Lumley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry – Cambridge Monographes in Mechanics. Cambridge Universtity Press, UK (1996).
T. Bui-Thanh, Model-constrained optimization methods for reduction of parameterized systems. Ph.D. Thesis, MIT, USA (2007).
Bui-Thanh, T., Damodoran, M. and Willcox, K., Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition. AIAA Journal 42 (2004) 1505–1516. CrossRef
Bui-Thanh, T., Willcox, K., Ghattas, O. and van Bloemen Wanders, B., Goal-oriented, model-constrained optimization for reduction of large-scale systems. J. Comput Phys. 224 (2007) 880–896. CrossRef
Everson, R. and Sirovich, L., The Karhunen-Loeve procedure for gappy data. J. Opt. Soc. Am. 12 (1995) 1657–1664. CrossRef
K. Fukunaga, Introduction to Statistical Recognition. Academic Press, New York, USA (1990).
Grepl, M.A., Maday, Y., Nguyen, N.C. and Patera, A.T., Efficient reduced-basis treatment of affine and nonlinear partial differential equations. ESAIM: M2AN 41 (2007) 575–605. CrossRef
M. Heinkenschloss, Formulation and Analysis of a Sequential Quadratic Programming Method for the Optimal Dirichlet Boundary Control of Navier Stokes Flow – Optimal Control: Theory, Methods and Applications. Kluwer Academic Publisher, B.V. (1998) 178–203.
Hinze, M. and Kunisch, K., Second order methods for optimal control of time – Dependent fluid flow. SIAM J. Contr. Optim. 40 (2001) 925–946. CrossRef
Ito, K. and Ravindran, S.S., A reduced-order method for simulation and control of fluid flows. J. Comput. Phys. 143 (1998) 403–425. CrossRef
T. Kato, Perturbation Theory for Linear Operators. Springer Verlag, Germany (1980).
Kunisch, K. and Volkwein, S., Control of Burgers' equation by reduced order approach using proper orthogonal decomposition. J. Optim. Theory Appl. 102 (1999) 345–371. CrossRef
Kunisch, K. and Volkwein, S., Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117–148. CrossRef
S. Lall, J.E. Marsden and S. Glavaski, Empirical model reduction of controlled nonlinear systems, in Proceedings of the IFAC Congress, Vol. F (1999) 473–478.
Ly, H.V. and Tran, H.T., Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor. Quarterly Appl. Math. 60 (2002) 631–656. CrossRef
J. Nocedal and S.J. Wright, Numerical Optimization, Springer Series in Operation Research. Second Edition, Springer Verlag, New York, USA (2006).
Rathinam, M. and Petzold, L.R., A new look at proper orthogonal decomposition. SIAM J. Numer. Anal. 41 (2003) 1893–1925. CrossRef
Ravindran, S.S., Adaptive reduced-order controllers for a thermal flow system using proper orthogonal decomposition. SIAM J. Sci. Comput. 23 (2002) 1924–1942. CrossRef
Rowley, C.W., Model reduction for fluids using balanced proper orthogonal decomposition. Int. J. Bifur. Chaos 15 (2005) 997–1013. CrossRef
Rozza, G., Huynh, D.B.P. and Patera, A.T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations: Application to transport and continuum mechanics. Arch. Comput. Method. E. 15 (2008) 229–275. CrossRef
R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics. Second edition, Springer, Berlin, Germany (1997).
K. Willcox, O. Ghattas, B. von Bloemen Wanders and W. Bader, An optimization framework for goal-oriented, model-based reduction of large-scale systems, in 44th IEEE Conference on Decision and Control, Sevilla, Spain (2005).