Nonlinear model reduction using group proper orthogonal decomposition

International Journal of Numerical Analysis and Modeling

Volumn 7, Issue 2, 2010, Pages 356-372

  Dickinson, Benjamin T ^a   Singler, John R ^b  

^a Oregon State University   (United States)

^b MISSOURI UNIVERSITY OF SCIENCE AND TECHNOLOGY   (United States)

Author keywords

Group finite element; Model reduction; Nonlinear; Proper orthogonal decomposition

Indexed keywords

EID: 70350315742     PISSN: 17055105     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (38)

1. L. Sirovich. Turbulence and the dynamics of coherent structures part 1: coherent structures. Quarterly of Applied Mathematics, 45(3):561 571, October 1987.
2. L. Sirovich. Turbulence and the dynamics of coherent structures part 2: symmetries and transformations. Quarterly of Applied Mathematics, 45(3):573 582, October 1987.
3. D. H. Chambers, R. J. Adrian, P. Moin, D. S. Stewart, and H. J. Sung. Karhunen-Loéve expansion of Burgers' model of turbulence. Phys. Fluids, 31:2573-2582, 1988.
4. P. Holmes, J. L. Lumley, and G.Berkooz. Turbulence, Coherent Structures, Dynamical Systems, and Symmetry. Cambridge University Press, 1996.
5. M. Fahl. Computation of POD basis functions for fluid flows with Lanczos methods. Mathematical and Computer Modeling, 31:91-97, 2001.
6. A. Iollo, S. Lanteri, and J Désidéri. Stability properties of POD-Galerkin approximations for the compressible Navier-Stokes equations. Theoretical and Computational Fluid Dynamics, 13:377-396, 2000.
7. K. Kunisch and W. Volkwein. Galerkin proper orthogonal decomposition methods for parabolic problems. Numerical Mathematics, 90:117 148, 2001.
8. K. Kunisch and S. Volkwein. Galerkin proper orthongonal decomposition methods for a general equation in fluid dynamics. SIAM Journal of Numerical Analysis, 40:492-515, 2002.
9. T. Henri and J. Yvon. Stability of the POD and convergence of the POD Galerkin method for parabolic problems. Technical report, Institut de Recherche Mathematiqe de Rennes, September 2002. Prepublication.
10. C. Rowley, T. Colonius, and R. Murray. Model reduction for compressible flows using POD and Galerkin projection. Physica D, 189:115-129, 2004.
11. R. C. Camphouse. Boundary feedback control using proper orthogonal decomposition models.Journal of Guidance, Control, and Dynamics, 28:931-938, 2005.
12. I. Christie, D. F. Griffiths, A. R. Mitchell, and J. M. Sanz-Serna. Product approximation for non-linear problems in the finite element method. IMA Journal of Numerical Analysis, 1:253-266, 1981.
13. C. A. J. Fletcher. The group finite element formulation. Computer Methods in Applied Mechanics and Engineering, 37:225-243, 1983.
14. C. A. J. Fletcher. Computational Techniques for Fluid Dynamics I. Springer-Verlag, 1991.
15. P. J. Roache. Verification and Validation in Computational Science and Engineering. Hermosa Publishers, Albuquerque NM, 1998.
16. W. L. Oberkampf, M. N. Sindar, and A. T. Conlisk. Guide for the verification and validation of computational fluid dynamics simulations. Technical report, American Institute of Aeronautics and Astronautics, 1998.
17. P. J. Roache. Code verification by the method of manufactured solutions. Transaction of the ASME, 124:4-10, 2002.
18. C. J. Roy, C. C. Nelson, T. M. Smith, and C. C. Ober. Verification of Euler/Navier Stokes codes using the method of manufactured solutions. International Journal for Numerical Methods in Fluids, 44:599-620, 2003.
19. P. Knupp and K. Salari. Verification of Computer Codes in Computational Science and Engineering. Chapman & Hall/CRC, 2003.
20. K. Pearson. On lines and planes of closest fit to systems of points in space. Edinburgh Dublin Philosophical Mag. J. Sci., 2:559-572, 1901.
21. H. Hotelling. Analysis of a complex of statistical variables into principal components. The Journal of Educational Psychology, 24(6):417-441, 1933.
22. H. Bjornsson and S. A. Venegas. A manual for EOF and SVD analyses of climatic data.Technical report, Department of Atmospheric and Oceanic Sciences and Centre for Climate and Global Change Research, McGill University, February 1997.
23. S. S. Ravindran. A reduced-order approach for optimal control of fluids using proper orthogonal decomposition. International Journal for Numerical Methods in Fluids, 34:425-448,2000.
24. J. A. Atwell, J. T. Borggaard, and B. B. King. Reduced order controllers for Burgers' equation with a nonlinear observer. Int. J. Appl. Math. Comput. Sci., 11(6):1311-1330, 2001.
25. J. A. Atwell and B. B. King. Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations. Math. Comput. Modelling, 33(1-3):1-19, 2001.
26. H. T. Banks, S. C. Beeler, G. M. Kepler, and H. T. Tran. Reduced order modeling and control of thin film growth in an HPCVD reactor. SIAM J. Appl. Math., 62(4):1251-1280, 2002.
27. J. A. Atwell and B. B. King. Reduced order controllers for spatially distributed systems via proper orthogonal decomposition. SIAM J. Sci. Comput., 26(1):128-151, 2004.
28. C. H. Lee and H. T. Tran. Reduced-order-based feedback control of the Kuramoto-Sivashinsky equation. J. Comput. Appl. Math., 173(1):1-19, 2005.
29. T. Henri and J. P. Yvon. Convergence estimates of POD-Galerkin methods for parabolic problems. In Proceedings of the 21st IFIP TC7 Conference of System Modeling and Optimization,2003.
30. L. Abia and J. M. Sanz-Serna. Interpolation of the coefficients in nonlinear elliptic Galerkin procedures. SIAM Journal on Numerical Analysis, 21(1):77-83, 1984.
31. C. M. Chen, S. Larsson, and N. Y. Zhang. Error estimates of optimal order for finite element methods with interpolated coefficients for the nonlinear heat equation. IMA Journal of Numerical Analysis, 9:507-524, 1989.
32. J. Douglas and T. Dupont. The effect of interpolating the coefficients in nonlinear parabolic Galerkin procedures. Mathematics of Computation, 29:360-389, 1975.
33. S. Larsson, V. Thomée, and N. Y. Zhang. Interpolation of coefficients and transformation of the dependent variable in the finite element method for non-linear heat equation. Mathematical Methods in the Applied Sciences, 11:105-124, 1989.
34. T. Murdoch. An error bound for the product approximation of nonlinear problems in the finite element method. Numerical Analysis Group Research Report 86/17, Oxford U. Computing Laboratory, 1986.
35. Y. Tourigny. Product approximation for the nonlinear Klein-Gordon equation. IMA Journal of Numerical Analysis, 9:449-462, 1990.
36. S. Lall, J. Marsden, and S. Glavaski. Empirical model reduction of controlled nonlinear systems. In Proceedings of the IFAC World Congress, 1999.
37. S. Lall, J. E. Marsden, and S. Glavaski. A subspace approach to balanced truncation for model reduction of nonlinear systems. Int. J. Robust Nonlinear Control, 12:519-535, 2002.
38. A. Chatterjee. An introduction to the proper orthogonal decomposition. Current Science, 78(7):808 817, April 2000.