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Model order reduction using sparse coding exemplified for the lid-driven cavity

Published online by Cambridge University Press:  27 October 2016

Rohit Deshmukh ,
Jack J. McNamara ,
Zongxian Liang ,
J. Zico Kolter  and
Abhijit Gogulapati
Rohit Deshmukh
Affiliation:
Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA
Jack J. McNamara*
Affiliation:
Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA
Zongxian Liang
Affiliation:
Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA
J. Zico Kolter
Affiliation:
School of Computer Science, Carnegie Mellon University Pittsburgh, PA 15213, USA
Abhijit Gogulapati
Affiliation:
Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA
*
†Email address for correspondence: mcnamara.190@osu.edu
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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 flow fields due to the presence of multi-scale phenomena that cannot be ignored and may not be captured using a small set of modes extracted 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 with 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 identify a compact representation that spans all scales of the observed data. As such, the inherently multi-scale bases may improve reduced-order modelling of unsteady flow fields. The approach is examined for a canonical problem of an incompressible flow inside a two-dimensional lid-driven cavity. The results demonstrate that Galerkin reduction of the governing equations using sparse modes yields a significantly improved predictive model of the fluid dynamics.

JFM classification

Mathematical Foundations: Computational methods Nonlinear Dynamical Systems: Low-dimensional models Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
Papers
Information
Journal of Fluid Mechanics , Volume 808 , 10 December 2016 , pp. 189 - 223
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Copyright
© 2016 Cambridge University Press 

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References

Abdi, H. & Williams, L. J. 2010 Principal component analysis. Wiley Interdisciplinary Rev.: Comput. Stat. 2 (4), 433–459.Google Scholar
Amsallem, D. & Farhat, C. 2012 Stabilization of projection-based reduced-order models. Intl J. Numer. Meth. Engng 91 (4), 358–377.Google Scholar
Amsallem, D., Zahr, M. J. & Farhat, C. 2012 Nonlinear model order reduction based on local reduced-order bases. Intl J. Numer. Meth. Engng 92 (10), 891–916.CrossRefGoogle Scholar
Balajewicz, M. J., Dowell, E. H. & Noack, B. R. 2013 Low-dimensional modelling of high-Reynolds-number shear flows incorporating constraints from the Navier–Stokes equation. J. Fluid Mech. 729, 285–308.Google Scholar
Bendat, J. S. & Piersol, A. G. 2010 Random Data: Analysis and Measurement Procedures, 4th edn. John Wiley & Sons.Google Scholar
Berkooz, G., Holmes, P. & Lumley, J. L. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25 (1), 539–575.Google Scholar
Chatterjee, A. 2000 An introduction to the proper orthogonal decomposition. Current Sci. 78 (7), 808–817.Google Scholar
Engan, K., Aase, S. O. & Hakon, H. J. 1999 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.Google Scholar
Friedman, J., Hastie, T. & Tibshirani, R. 2008 Sparse inverse covariance estimation with the graphical lasso. Biostatistics 9 (3), 432–441.CrossRefGoogle ScholarPubMed
Friedman, J., Hastie, T. & Tibshirani, R. 2010 Regularization paths for generalized linear models via coordinate descent. J. Stat. Software 33 (1), 1–22.Google Scholar
Grosse, R., Raina, R., Kwong, H. & Ng, A. Y.2012 Shift-invariance sparse coding for audio classification. arXiv:1206.5241.Google Scholar
Holmes, P., Lumley, J. L., Berkooz, G. & Rowley, C. W. 2012 Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn. Cambridge University Press.Google Scholar
Ilak, M., Bagheri, S., Brandt, L., Rowley, C. W. & Henningson, D. S. 2010 Model reduction of the nonlinear complex Ginzburg–Landau equation. SIAM J. Appl. Dyn. Syst. 9 (4), 1284–1302.Google Scholar
Ilak, M. & Rowley, C. W. 2008 Modeling of transitional channel flow using balanced proper orthogonal decomposition. Phys. Fluids 20 (3), 034103.Google Scholar
Jovanović, M. R., Schmid, P. J. & Nichols, J. W. 2014 Sparsity-promoting dynamic mode decomposition. Phys. Fluids 26 (2), 024103.CrossRefGoogle Scholar
Kalashnikova, I., van Bloemen, W. B., Arunajatesan, S. & Barone, M. 2014 Stabilization of projection-based reduced order models for linear time-invariant systems via optimization-based eigenvalue reassignment. Comput. Meth. Appl. Mech. Engng 272, 251–270.Google Scholar
Kolter, J. Z., Batra, S. & Ng, A. Y. 2010 Energy disaggregation via discriminative sparse coding. In Advances in Neural Information Processing Systems, NIPS, pp. 1153–1161.Google Scholar
Kreutz-Delgado, K., Murray, J. F., Rao, B. D., Engan, K., Lee, T. & Sejnowski, T. J. 2003 Dictionary learning algorithms for sparse representation. Neural Comput. 15 (2), 349–396.CrossRefGoogle ScholarPubMed
Leblond, C., Allery, C. & Inard, C. 2011 An optimal projection method for the reduced-order modeling of incompressible flows. Comput. Meth. Appl. Mech. Engng 200 (33), 2507–2527.Google Scholar
Lee, H., Battle, A., Raina, R. & Ng, A. Y. 2006 Efficient sparse coding algorithms. In Advances in Neural Information Processing Systems, NIPS, pp. 801–808.Google Scholar
Lucia, D. J., Beran, P. S. & Silva, W. A. 2004 Reduced-order modeling: new approaches for computational physics. Prog. Aerosp. Sci. 40 (1), 51–117.Google Scholar
Ma, Z., Ahuja, S. & Rowley, C. W. 2010 Reduced-order models for control of fluids using the eigensystem realization algorithm. Theor. Comput. Fluid Dyn. 25 (1–4), 233–247.CrossRefGoogle Scholar
Mittal, R., Dong, H., Bozkurttas, M., Najjar, F. M., Vargas, A. & von Loebbecke, A. 2008 A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries. J. Comput. Phys. 227 (10), 4825–4852.Google Scholar
Noack, B. R., Afanasiev, K., Morzynski, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.Google Scholar
Noack, B. R. & Eckelmann, H. 1994 A low-dimensional Galerkin method for the three-dimensional flow around a circular cylinder. Phys. Fluids 6 (1), 124–143.CrossRefGoogle Scholar
Noack, B. R., Papas, P. & Monkewitz, P. A. 2005 The need for a pressure-term representation in empirical Galerkin models of incompressible shear flows. J. Fluid Mech. 523, 339–365.CrossRefGoogle Scholar
Olshausen, B. A. & Field, D. J. 1996 Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature 381 (6583), 607–609.Google Scholar
Olshausen, B. A. & Field, D. J. 2004 Sparse coding of sensory inputs. Curr. Opin. Neurobiol. 14 (4), 481–487.CrossRefGoogle ScholarPubMed
Pope, S. B. 2009 Turbulent Flows, 6th edn. Cambridge University Press.Google Scholar
Rowley, C. W. 2005 Model reduction for fluids, using balanced proper orthogonal decomposition. Intl J. Bifurcation Chaos 15 (03), 997–1013.Google Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures part I: coherent structures. Q. Appl. Maths 45, 561–571.CrossRefGoogle Scholar
Terragni, F., Valero, E. & Vega, J. 2011 Local POD plus Galerkin projection in the unsteady lid-driven cavity problem. SIAM J. Sci. Comput. 33 (6), 3538–3561.Google Scholar
Tibshirani, R. 1996 Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B 267–288.Google Scholar
Welch, P. 1967 The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoust. 15 (2), 70–73.Google Scholar
Yang, J., Yu, K., Gong, Y. & Huang, T. 2009 Linear spatial pyramid matching using sparse coding for image classification. In Computer Vision and Pattern Recognition, pp. 1794–1801. IEEE.Google Scholar
Yang, J., Yu, K. & Huang, T. 2010 Efficient highly over-complete sparse coding using a mixture model. In Computer Vision–ECCV 2010, pp. 113–126. Springer.Google Scholar
Zhou, K., Doyle, J. C. & Glover, K. 1996 Robust and Optimal Control, 1st edn. Prentice Hall.Google Scholar
Zuo, W., Meng, D., Zhang, L., Feng, X. & Zhang, D. 2013 A generalized iterated shrinkage algorithm for non-convex sparse coding. In Proceedings of the IEEE International Conference on Computer Vision, pp. 217–224.Google Scholar