Nonlinear model reduction using group proper orthogonal decomposition. (English) Zbl 1499.35732
Summary: We propose a new method to reduce the cost of computing nonlinear terms in projection based reduced order models with global basis functions. We develop this method by extending ideas from the group finite element (GFE) method to proper orthogonal decomposition (POD) and call it the group POD method. Here, a scalar two-dimensional Burgers’ equation is used as a model problem for the group POD method. Numerical results show that group POD models of Burgers’ equation are as accurate and are computationally more efficient than standard POD models of Burgers’ equation.
MSC:
| 35R35 | Free boundary problems for PDEs |
| 49J40 | Variational inequalities |
| 60G40 | Stopping times; optimal stopping problems; gambling theory |
References:
| [1] | L. Sirovich. Turbulence and the dynamics of coherent structures part 1: coherent structures. Quarterly of Applied Mathematics, 45(3):561 -571, October 1987. · Zbl 0676.76047 |
| [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 Sys-tems, and Symmetry. Cambridge University Press, 1996. · Zbl 0890.76001 |
| [5] | M. Fahl. Computation of POD basis functions for fluid flows with Lanczos methods. Mathe-matical 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. · Zbl 0987.76077 |
| [7] | K. Kunisch and W. Volkwein. Galerkin proper orthogonal decomposition methods for para-bolic problems. Numerical Mathematics, 90:117 -148, 2001. · Zbl 1005.65112 |
| [8] | K. Kunisch and S. Volkwein. Galerkin proper orthongonal decomposition methods for a gen-eral equation in fluid dynamics. SIAM Journal of Numerical Analysis, 40:492-515, 2002. · Zbl 1075.65118 |
| [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. · Zbl 1098.76602 |
| [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. · Zbl 0469.65072 |
| [13] | C. A. J. Fletcher. The group finite element formulation. Computer Methods in Applied Me-chanics and Engineering, 37:225-243, 1983. |
| [14] | C. A. J. Fletcher. Computational Techniques for Fluid Dynamics I. Springer-Verlag, 1991. · Zbl 0717.76001 |
| [15] | P. J. Roache. Verification and Validation in Computational Science and Engineering. Her-mosa Publishers, Albuquerque NM, 1998. · Zbl 0914.68133 |
| [16] | W. L. Oberkampf, M. N. Sindar, and A. T. Conlisk. Guide for the verification and valida-tion 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. · Zbl 1067.76580 |
| [19] | P. Knupp and K. Salari. Verification of Computer Codes in Computational Science and Engineering. Chapman & Hall/CRC, 2003. · Zbl 1025.68022 |
| [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. · JFM 32.0710.04 |
| [21] | H. Hotelling. Analysis of a complex of statistical variables into principal components. The Journal of Educational Psychology, 24(6):417-441, 1933. · JFM 59.1182.04 |
| [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 orthog-onal decomposition. International Journal for Numerical Methods in Fluids, 34:425-448, 2000. · Zbl 1005.76020 |
| [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. · Zbl 1051.93045 |
| [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. · Zbl 0964.93032 |
| [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. · Zbl 1005.93022 |
| [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. · Zbl 1075.65090 |
| [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. · Zbl 1107.93028 |
| [29] | T. Henri and J. P. Yvon. Convergence estimates of POD-Galerkin methods for parabolic prob-lems. In Proceedings of the 21st IFIP TC7 Conference of System Modeling and Optimization, 2003. · Zbl 1088.65576 |
| [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. · Zbl 0542.65042 |
| [31] | C. M. Chen, S. Larsson, and N. Y. Zhang. Error estimates of optimal order for finite ele-ment methods with interpolated coefficients for the nonlinear heat equation. IMA Journal of Numerical Analysis, 9:507-524, 1989. · Zbl 0686.65077 |
| [32] | J. Douglas and T. Dupont. The effect of interpolating the coefficients in nonlinear parabolic Galerkin procedures. Mathematics of Computation, 29:360-389, 1975. · Zbl 0311.65060 |
| [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. Mathemat-ical Methods in the Applied Sciences, 11:105-124, 1989. · Zbl 0663.65118 |
| [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. · Zbl 0707.65088 |
| [36] | S. Lall, J. Marsden, and S. Glavaski. Empirical model reduction of controlled nonlinear sys-tems. 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. · Zbl 1006.93010 |
| [38] | A. Chatterjee. An introduction to the proper orthogonal decomposition. Current Science, 78(7):808 -817, April 2000. |
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