# ADAPTIVE CVT-BASED REDUCED-ORDER MODELING OF BURGERS EQUATION

• Piao, Guang-Ri (DEPARTMENT OF MATHEMATICS, YANBIAN UNIVERSITY) ;
• Du, Qiang (DEPARTMENT OF MATHEMATICS, PENNSYLVANIA STATE UNIV., UNIVERSITY PARK) ;
• Lee, Hyung-Chun (DEPARTMENT OF MATHEMATICS, AJOU UNIVERSITY)
• Received : 2009.05.12
• Accepted : 2009.05.30
• Published : 2009.06.25

#### Abstract

In this article, we consider a weighted CVT-based reduced-order modelling for Burgers equation. Brief review of the CVT (centroidal Voronoi tessellation) approaches to reduced-order bases are provided. In CVT-reduced order modelling, we start with a snapshot set just as is done in a POD (Proper Orthogonal Decomposition)-based setting. So far, the CVT was researched with uniform density ($\rho$(y) = 1) to determine the basis elements for the approximatin subspaces. Here, we shall investigate the technique of CVT with nonuniform density as a procedure to determine the basis elements for the approximating subspaces. Some numerical experiments including comparison of two CVT (CVT-uniform and CVT-nonuniform)-based algorithm with numerical results obtained from FEM(finite element method) and POD-based algorithm are reported.

#### References

1. B. O. Almroth, Automatic choice of global shape functions in structural analysis, AIAA J., 16 (1979), 525-528.
2. J. Atwell and B. King, Reduced order controllers for spatially distributed systems via proper orthogonal decomposition, SIAM J. Sci. Comput. 26 (2004), 128-151. https://doi.org/10.1137/S1064827599360091
3. A. K. Bangia, P. F. Batcho, I. G. Kevrekidis, and G. E. Karniadakis, Unsteady two-dimensional flows in complex geometries: comparative bifurcation studies with global eigen-function expansions, SIAM J. on Sci. Comput., 18 (1997), 775-805. https://doi.org/10.1137/S1064827595282246
4. S. C. Brenner and L. R. Scott, The mathematical theroy of finite element methods, Springer-Verlag, New York, 1994.
5. J. Borggaard, A. Hay, and D. Pelletier. Interval-based reduced-order models for unsteady uid ow. International Jounal of Numerical Analysis and Modeling, 4 (2007), 353367.
6. J. M. Burgers, Mathematical examples illustrating relations occuring in the theory of turbulent fluid motion, Trans. Roy. Neth. Acad. Sci. 17 (1939), Amsterdam, 1-53
7. J. M. Burgers, A mathematical model illustrating the theory of turbulence, Adv. in Appl. Mech, 1 (1948), 171-199.
8. J. M. Burgers, Statistical problems connected with asymptotic solution of one-dimensional nonlinear diffusion equation, in M. Rosenblatt and C. van Atta (eds.), Statistical Models and Turbulence, Springer, Berlin (1972), 41.
9. J. Burkardt, Q. Du, M. Gunzburger and H.-C. Lee, Reduced Order Modeling of Complex Systems, in Proceeding of the 20th Biennial Conference on Numerical Analysis, Ed. by D F Griffiths & G AWatson, University of Dundee, June, 2003, 29-38.
10. J. A. Burns and S. Kang, A control problem for Burgers equation with bounded input/oqtput, ICASE Report 90-45, 1990, NASA Langley research Center, Hampton, VA; Nonlinear Dynamics, 2 (1991), 235-262.
11. C. T. Chen, Linear System Theory and Design, Holt, Rinehart and Winston, New York, NY, 1984.
12. Q. Du, M. Emelianenko, and L. Ju, "Convergenece of the Lolyd algorithm for computing centroidal Voronoi tessellations, SIAM J. Numer. Anal., 44 (2006),. 102-119. https://doi.org/10.1137/040617364
13. Q. Du, V. Faber, and M. Gunzburger, Centroidal Voronoi Teesellations: applications and algorithms, SIAM Review 41 (1999), 637-676. https://doi.org/10.1137/S0036144599352836
14. Q. Du and M. Gunzburger, Model reduction by proper orthogonal decomposition coupled with centroidal Voronoi tessellation, Proc. Fluids Engineering Division Summer Meeting, FEDSM2002-31051, ASME, 2002
15. Q. Du and X.Wang, Tessellation and Clustering by Mixture Models and Their Parallel Implementations, in Proceeding of the fourth SIAM international conference on Data Mining, L ake Buena Vista, FL, 2004, SIAM, 257-268.
16. J. S. Gibson, The riccati integral equations for optimal control problems on Hilbert spces, SIAM Jounal on the Control and Optimazation, bf 17 (1979), 537-565. https://doi.org/10.1137/0317039
17. J. S. Gibson, and I. G. Rosen, Shifting the closed-loop spectrum in teh optimal linear quadratic regulator problem for hereditary system, Institute for Computer Applications for Science and Engineering, ICASE Report 86-16, 1986, NASA Langley Reserch Center, Hampton, VA.
18. L. Ju, Q. Du and M. Gunzburger, Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations, Parallel Computing, 28 (2002), 1477-1500. https://doi.org/10.1016/S0167-8191(02)00151-5
19. K. Kunisch and S. Volkwein, Control of burgers equation by a reduced order approach using proper orthogonal decomposition. J. Optim. Theory Appl. 102 (1999), 345-371. https://doi.org/10.1023/A:1021732508059
20. I. Lasiecka, and R. Triggiani, Dirichlet boundary control problem for parabolic equation with quadratic cost: analticity and Riccati's feedbacksynthesis, SIAM J. Control and Optimization 21 (1983), 41-67. https://doi.org/10.1137/0321003
21. H.-C. Lee, J. Burkardt, and M. Gunzburger, Centroidal Voronoi tessellation-based reduce-order modeling of complex systems, SIAM J. Sci. Comput. 28 (2006), 459-484.
22. H.-C. Lee, J. Burkardt, and M. Gunzburger, POD and CVT-based Reduced-order modeling of Navier-Stokes flows, Comput. Methods Appl. Mech. Engrg. 196 (2006), 337-355. https://doi.org/10.1016/j.cma.2006.04.004
23. S. Lloyd, Least squares quantization in PCM, IEEE Trans. Infor. Theory, 28 (1982), 129-137. https://doi.org/10.1109/TIT.1982.1056489
24. J. MacQueen, Some methods for classification and analysis of multivariate observations, Proc. Fifth Berkeley Symposium on Mathematical Statistics and Probability, 1 (1967), University of California, 281-297.
25. H. Marrekchi, Dynamic compensators for a nonlinear conservation law, Ph.D. Dissertation, Virginia Polytechnic Institute and State University, 1993.
26. D. A. Nagy, Modal representation of geometrically nonlinear behavior by the finite element method, Computers and Structures, 10 (1979), 683-688. https://doi.org/10.1016/0045-7949(79)90012-9
27. A. K. Noor, Recent advances in reduction methods for nonlinear problems, Computers and Structures, 13 (1981), 31-44. https://doi.org/10.1016/0045-7949(81)90106-1
28. A. K. Noor, C. M. Andersen, and J. M. Peters, Reduced basis technique for collapse analysis of shells, AIAA J., 19, 393-397.
29. A. K. Noor and J. M. Peters, Reduced basis technique for nonlinear analysis of structures, AIAA J., 18, 455-462.
30. J. S. Peterson, The reduced-basis method for incompressible viscous flow calculations, SIAM J. Scientific and Statistical computing, 10 (1989), 777-786. https://doi.org/10.1137/0910047
31. R. Triggiani, and R. Bulrisch, Boundary feedback stabilizability parabolic equations, Appl. Math. Optim. 6 (1980), 201-220. https://doi.org/10.1007/BF01442895