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Basis Identification for Nonlinear Dynamical Systems Using Sparse Coding

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Abstract

Basis identification is a critical step in the construction of accurate reduced order models using Galerkin projection. This is particularly challenging in unsteady nonlinear flow fields due to the presence of multi-scale phenomena that cannot be ignored and are not well captured using the ubiquitous Proper Orthogonal Decomposition. This study focuses on this issue by exploring an approach known as sparse coding for the basis identification problem. Compared to Proper Orthogonal Decomposition, which seeks to truncate the basis spanning an observed data set into a small set of dominant modes, sparse coding is used to select a compact basis that best spans the entire data set. Thus, the resulting bases are inherently multi-scale, enabling improved reduced order modeling of unsteady flow fields. The approach is demonstrated for a canonical problem of an incompressible flow inside a 2-D lid-driven cavity. Results indicate that Galerkin reduction of the governing equations using sparse modes yields significantly improved fluid predictions.

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References

  1. Holmes, P., Lumley, J.L., Berkooz, G., Rowley, C.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn. Cambridge University Press, New York (2012)

    Book  MATH  Google Scholar 

  2. Ilak, M., Bagheri, S., Brandt, L., Rowley, C.W., Henningson, D.S.: Model reduction of the nonlinear complex Ginzburg-Landau equation. SIAM J. Appl. Dyn. Syst. 9(4), 1284–1302 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Kalashnikova, I., van Bloemen, W.B., Arunajatesan, S., Barone, M.: Stabilization of projection-based reduced order models for linear time-invariant systems via optimization-based eigenvalue reassignment. Comput. Methods Appl. Mech. Eng. 272, 251–270 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Rowley, C.: Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. Bifurcation Chaos 15(3), 997–1013 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Noack, B.R., Afanasiev, K., Morzynski, M., Tadmor, G., Thiele, F.: A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Noack, B.R., Eckelmann, H.: A low-dimensional Galerkin method for the three-dimensional flow around a circular cylinder. Phys. Fluids (1994–present) 6(1), 124–143 (1994)

    Google Scholar 

  7. Lucia, D.J., Beran, P.S., Silva, W.A.: Reduced-order modeling: new approaches for computational physics. Prog. Aerosp. Sci. 40(1), 51–117 (2004)

    Article  Google Scholar 

  8. Berkooz, G., Holmes, P., Lumley, J.L.: The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25(1), 539–575 (1993)

    Article  MathSciNet  Google Scholar 

  9. Chatterjee, A.: An introduction to the proper orthogonal decomposition. Curr. Sci. 78(7), 808–817 (2000)

    Google Scholar 

  10. Sirovich, L.: Turbulence and the dynamics of coherent structures. I-Coherent structures. II-symmetries and transformations. III-Dynamics and scaling. Q. Appl. Math. 45, 561–571 (1987)

    MathSciNet  MATH  Google Scholar 

  11. Balajewicz, M.J., Dowell, E.H., Noack, B.R.: Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation. J. Fluid Mech. 729, 285–308 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ilak, M., Rowley, C.W.: Modeling of transitional channel flow using balanced proper orthogonal decomposition. Phys. Fluids (1994–present) 20(3), 034103 (2008)

    Google Scholar 

  13. Amsallem, D., Farhat, C.: Stabilization of projection-based reduced-order models. Int. J. Numer. Methods Eng. 91(4), 358–377 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pope, S.B.: Turbulent Flows, 6th edn. Cambridge University Press, New York (2009)

    MATH  Google Scholar 

  15. Amsallem, D., Zahr, M.J., Farhat, C.: Nonlinear model order reduction based on local reduced-order bases. Int. J. Numer. Methods Eng. 92(10), 891–916 (2012)

    Article  MathSciNet  Google Scholar 

  16. Zhou, K., Doyle, J.C., Glover, K.: Robust and Optimal Control, 1st edn. Prentice Hall, Upper Saddle River (1996)

    MATH  Google Scholar 

  17. Ma, Z., Ahuja, S., Rowley, C.W.: Reduced-order models for control of fluids using the eigensystem realization algorithm. Theor. Comput. Fluid Dyn. 25(1–4), 233–247 (2010)

    MATH  Google Scholar 

  18. Olshausen, B., Field, D.: Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature 381(6583), 607–609 (1996)

    Article  Google Scholar 

  19. Kreutz-Delgado, K., Murray, J.F., Rao, B.D., Engan, K., Lee, T., Sejnowski, T.J.: Dictionary learning algorithms for sparse representation. Neural Comput. 15(2), 349–396 (2003)

    Article  MATH  Google Scholar 

  20. Yang, J., Yu, K., Huang, T.: Efficient highly over-complete sparse coding using a mixture model. In: Computer Vision—ECCV 2010, pp. 113–126. Springer, Berlin (2010)

    Google Scholar 

  21. Yang, J., Yu, K., Gong, Y., Huang, T.: Linear spatial pyramid matching using sparse coding for image classification. In: IEEE Computer Vision and Pattern Recognition, pp. 1794–1801 (2009)

    Google Scholar 

  22. Grosse, R., Raina, R., Kwong, H., Ng, A.: Shift-invariance sparse coding for audio classification (2012). arXiv preprint arXiv:1206.5241

    Google Scholar 

  23. Lee, H., Battle, A., Raina, R., Ng, A.: Efficient sparse coding algorithms. In: Advances in Neural Information Processing Systems, pp. 801–808 (2006)

    Google Scholar 

  24. Olshausen, B., Field, D.: Sparse coding of sensory inputs. Curr. Opin. Neurobiol. 14(4), 481–487 (2004)

    Article  Google Scholar 

  25. Kolter, J., Batra, S., Ng, A.: Energy disaggregation via discriminative sparse coding. In: Advances in Neural Information Processing Systems, pp. 1153–1161 (2010)

    Google Scholar 

  26. Deshmukh, R., Liang, Z., Gogulapati, A., McNamara, J.J., Kolter, J.Z.: Basis identification for reduced order modeling of unsteady flows using sparse coding. In: Proceedings of the 56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (SciTech 2015), No. AIAA 2015-0688, Kissimmee, FL, 5–9 Jan 2015

    Google Scholar 

  27. Friedman, J., Hastie, T., Tibshirani, R.: Regularization paths for generalized linear models via coordinate descent. J. Stat. Softw. 33(1), 1–22 (2010)

    Article  Google Scholar 

  28. Mittal, R., Dong, H., Bozkurttas, M., Najjar, F., Vargas, A., von Loebbecke, A.: A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries. J. Comput. Phys. 227(10), 4825–4852 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Abdi, H., Williams, L.: Principal component analysis. Wiley Interdiscip. Rev. Comput. Stat. 2(4), 433–459 (2010)

    Article  Google Scholar 

  30. Zuo, W., Meng, D., Zhang, L., Feng, X., Zhang, D.: A generalized iterated shrinkage algorithm for non-convex sparse coding. In: Proceedings of the IEEE International Conference on Computer Vision, pp. 217–224 (2013)

    Google Scholar 

  31. Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B Methodol. 58, 267–288 (1996)

    MathSciNet  MATH  Google Scholar 

  32. Friedman, J., Hastie, T., Tibshirani, R.: Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9(3), 432–441 (2008)

    Article  MATH  Google Scholar 

  33. Engan, K., Aase, S.O., Hakon Husoy, J.: Method of optimal directions for frame design. In: Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 5, pp. 2443–2446. IEEE, New York (1999)

    Google Scholar 

  34. Olshausen, B.A., Field, D.J.: Sparse coding with an overcomplete basis set: a strategy employed by V1? Vis. Res. 37(23), 3311–3325 (1997)

    Article  Google Scholar 

  35. Xiang, Z.J., Xu, H., Ramadge, P.J.: Learning sparse representations of high dimensional data on large scale dictionaries. In: Advances in Neural Information Processing Systems, pp. 900–908 (2011)

    Google Scholar 

  36. Ullmann, S., Lang, J.: A POD-Galerkin reduced model with updated coefficients for Smagorinsky LES. In: Proceeding of V. European Conference on Computational Fluid Dynamics, ECCOMAS CFD (2010)

    Google Scholar 

  37. Graham, W., Peraire, J., Tang, K.: Optimal control of vortex shedding using low-order models. Part 1 – open-loop model development. Int. J. Numer. Methods Eng. 44, (7), 945–972 (1999)

    Google Scholar 

  38. Bergmann, M., Cordier, L., Brancher, J.: Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced-order model. Phys. Fluids (1994–present) 17(9), 097–101 (2005)

    Google Scholar 

  39. Monin, A.S., Lumley, J.L., Yaglom, A.M.: Statistical Fluid Mechanics: Mechanics of Turbulence, vol. 1. MIT Press, New York (1971)

    Google Scholar 

  40. Noack, B.R., Papas, P., Monkewitz, P.A.: The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523, 339–365 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  41. Leblond, C., Allery, C., Inard, C.: An optimal projection method for the reduced-order modeling of incompressible flows. Comput. Methods Appl. Mech. Eng. 200(33), 2507–2527 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  42. Terragni, F., Valero, E., Vega, J.: Local POD plus Galerkin projection in the unsteady lid-driven cavity problem. SIAM J. Sci. Comput. 33(6), 3538–3561 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors gratefully acknowledge the support of ONR grant N00014-14-1-0018, under the direction of Dr. Judah Milgram, an HPCMPO Frontier PETTT Project Grant, under the direction of David Bartoe, and an allocation of computing time from the Ohio Supercomputer Center. The authors thank Dr. Lionel Agostini, Mr. Kalyan Goparaju, Mr. S. Unnikrishnan, and Dr. Datta Gaitonde, Mechanical & Aerospace Engineering, The Ohio State University for providing technical guidance in the study of local flow dynamics.

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  1. Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH, 43210, USA

    Rohit Deshmukh, Zongxian Liang & Jack J. McNamara

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  1. Rohit Deshmukh
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  2. Zongxian Liang
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  3. Jack J. McNamara
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Correspondence to Jack J. McNamara .

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  1. Liege, Belgium

    Gaetan Kerschen

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Deshmukh, R., Liang, Z., McNamara, J.J. (2016). Basis Identification for Nonlinear Dynamical Systems Using Sparse Coding. In: Kerschen, G. (eds) Nonlinear Dynamics, Volume 1. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-29739-2_26

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  • DOI: https://doi.org/10.1007/978-3-319-29739-2_26

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