Optimal snapshot location for computing POD basis functions

Karl Kunisch; Stefan Volkwein

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 3, page 509-529
  • ISSN: 0764-583X

Abstract

top
The construction of reduced order models for dynamical systems using proper orthogonal decomposition (POD) is based on the information contained in so-called snapshots. These provide the spatial distribution of the dynamical system at discrete time instances. This work is devoted to optimizing the choice of these time instances in such a manner that the error between the POD-solution and the trajectory of the dynamical system is minimized. First and second order optimality systems are given. Numerical examples illustrate that the proposed criterion is sensitive with respect to the choice of the time instances and further they demonstrate the feasibility of the method in determining optimal snapshot locations for concrete diffusion equations.

How to cite

top

Kunisch, Karl, and Volkwein, Stefan. "Optimal snapshot location for computing POD basis functions." ESAIM: Mathematical Modelling and Numerical Analysis 44.3 (2010): 509-529. <http://eudml.org/doc/250703>.

@article{Kunisch2010,
abstract = { The construction of reduced order models for dynamical systems using proper orthogonal decomposition (POD) is based on the information contained in so-called snapshots. These provide the spatial distribution of the dynamical system at discrete time instances. This work is devoted to optimizing the choice of these time instances in such a manner that the error between the POD-solution and the trajectory of the dynamical system is minimized. First and second order optimality systems are given. Numerical examples illustrate that the proposed criterion is sensitive with respect to the choice of the time instances and further they demonstrate the feasibility of the method in determining optimal snapshot locations for concrete diffusion equations. },
author = {Kunisch, Karl, Volkwein, Stefan},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Proper orthogonal decomposition; optimal snapshot locations; first and second order optimality conditions; proper orthogonal decomposition (POD); optimal time snapshot locations; optimality conditions},
language = {eng},
month = {4},
number = {3},
pages = {509-529},
publisher = {EDP Sciences},
title = {Optimal snapshot location for computing POD basis functions},
url = {http://eudml.org/doc/250703},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Kunisch, Karl
AU - Volkwein, Stefan
TI - Optimal snapshot location for computing POD basis functions
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/4//
PB - EDP Sciences
VL - 44
IS - 3
SP - 509
EP - 529
AB - The construction of reduced order models for dynamical systems using proper orthogonal decomposition (POD) is based on the information contained in so-called snapshots. These provide the spatial distribution of the dynamical system at discrete time instances. This work is devoted to optimizing the choice of these time instances in such a manner that the error between the POD-solution and the trajectory of the dynamical system is minimized. First and second order optimality systems are given. Numerical examples illustrate that the proposed criterion is sensitive with respect to the choice of the time instances and further they demonstrate the feasibility of the method in determining optimal snapshot locations for concrete diffusion equations.
LA - eng
KW - Proper orthogonal decomposition; optimal snapshot locations; first and second order optimality conditions; proper orthogonal decomposition (POD); optimal time snapshot locations; optimality conditions
UR - http://eudml.org/doc/250703
ER -

References

top
  1. G. Berkooz, P. Holmes and J.L. Lumley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry – Cambridge Monographes in Mechanics. Cambridge Universtity Press, UK (1996).  
  2. T. Bui-Thanh, Model-constrained optimization methods for reduction of parameterized systems. Ph.D. Thesis, MIT, USA (2007).  
  3. T. Bui-Thanh, M. Damodoran and K. Willcox, Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition. AIAA Journal42 (2004) 1505–1516.  
  4. T. Bui-Thanh, K. Willcox, O. Ghattas and B. van Bloemen Wanders, Goal-oriented, model-constrained optimization for reduction of large-scale systems. J. Comput Phys.224 (2007) 880–896.  
  5. R. Everson and L. Sirovich, The Karhunen-Loeve procedure for gappy data. J. Opt. Soc. Am.12 (1995) 1657–1664.  
  6. K. Fukunaga, Introduction to Statistical Recognition. Academic Press, New York, USA (1990).  
  7. M.A. Grepl, Y. Maday, N.C. Nguyen and A.T. Patera, Efficient reduced-basis treatment of affine and nonlinear partial differential equations. ESAIM: M2AN41 (2007) 575–605.  
  8. 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.  
  9. M. Hinze and K. Kunisch, Second order methods for optimal control of time – Dependent fluid flow. SIAM J. Contr. Optim.40 (2001) 925–946.  
  10. K. Ito and S.S. Ravindran, A reduced-order method for simulation and control of fluid flows. J. Comput. Phys.143 (1998) 403–425.  
  11. T. Kato, Perturbation Theory for Linear Operators. Springer Verlag, Germany (1980).  
  12. K. Kunisch and S. Volkwein, Control of Burgers' equation by reduced order approach using proper orthogonal decomposition. J. Optim. Theory Appl.102 (1999) 345–371.  
  13. K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math.90 (2001) 117–148.  
  14. 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.  
  15. H.V. Ly and H.T. Tran, Proper orthogonal decomposition for flow calculations and optimal control in a horizontal CVD reactor. Quarterly Appl. Math.60 (2002) 631–656.  
  16. J. Nocedal and S.J. Wright, Numerical Optimization, Springer Series in Operation Research. Second Edition, Springer Verlag, New York, USA (2006).  
  17. M. Rathinam and L.R. Petzold, A new look at proper orthogonal decomposition. SIAM J. Numer. Anal.41 (2003) 1893–1925.  
  18. S.S. Ravindran, Adaptive reduced-order controllers for a thermal flow system using proper orthogonal decomposition. SIAM J. Sci. Comput.23 (2002) 1924–1942.  
  19. C.W. Rowley, Model reduction for fluids using balanced proper orthogonal decomposition. Int. J. Bifur. Chaos15 (2005) 997–1013.  
  20. G. Rozza, D.B.P. Huynh and A.T. Patera, 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.  
  21. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics. Second edition, Springer, Berlin, Germany (1997).  
  22. 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).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.