June  2015, 2(2): 165-191. doi: 10.3934/jcd.2015002

Compressed sensing and dynamic mode decomposition

1. 

Dept. of Applied Mathematics, University of Washington, Seattle, WA 98195

2. 

Institute for Disease Modeling, Intellectual Ventures Laboratory, Bellevue, WA 98004, United States

3. 

Department of Electrical Engineering and Computer Science, University of California, Berkeley, Berkeley, CA 94720, United States

Received  December 2013 Revised  August 2015 Published  December 2016

This work develops compressed sensing strategies for computing the dynamic mode decomposition (DMD) from heavily subsampled or compressed data. The resulting DMD eigenvalues are equal to DMD eigenvalues from the full-state data. It is then possible to reconstruct full-state DMD eigenvectors using $\ell_1$-minimization or greedy algorithms. If full-state snapshots are available, it may be computationally beneficial to compress the data, compute DMD on the compressed data, and then reconstruct full-state modes by applying the compressed DMD transforms to full-state snapshots.
    These results rely on a number of theoretical advances. First, we establish connections between DMD on full-state and compressed data. Next, we demonstrate the invariance of the DMD algorithm to left and right unitary transformations. When data and modes are sparse in some transform basis, we show a similar invariance of DMD to measurement matrices that satisfy the restricted isometry property from compressed sensing. We demonstrate the success of this architecture on two model systems. In the first example, we construct a spatial signal from a sparse vector of Fourier coefficients with a linear dynamical system driving the coefficients. In the second example, we consider the double gyre flow field, which is a model for chaotic mixing in the ocean.

    A video abstract of this work may be found at: http://youtu.be/4tLSq_PEFms.
Citation: Steven L. Brunton, Joshua L. Proctor, Jonathan H. Tu, J. Nathan Kutz. Compressed sensing and dynamic mode decomposition. Journal of Computational Dynamics, 2015, 2 (2) : 165-191. doi: 10.3934/jcd.2015002
References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, volume 75 of Applied Mathematical Sciences,, Springer-Verlag, (1988). doi: 10.1007/978-1-4612-1029-0. Google Scholar

[2]

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[3]

S. Bagheri, Effects of small noise on the DMD/Koopman spectrum,, Bulletin Am. Phys. Soc., 58 (2013). Google Scholar

[4]

S. Bagheri, Koopman-mode decomposition of the cylinder wake,, Journal of Fluid Mechanics, 726 (2013), 596. doi: 10.1017/jfm.2013.249. Google Scholar

[5]

Z. Bai, T. Wimalajeewa, Z. Berger, G. Wang, M. Glauser and P. K. Varshney, Physics Based Compressive Sensing Approach Applied to Airfoil Data Collection and Analysis,, AIAA Paper 2013-0772, (2013), 2013. Google Scholar

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[7]

R. G. Baraniuk, Compressive sensing,, IEEE Signal Processing Magazine, 24 (2007), 118. Google Scholar

[8]

R. G. Baraniuk, V. Cevher, M. F. Duarte and C. Hegde, Model-based compressive sensing,, IEEE Transactions on Information Theory, 56 (2010), 1982. doi: 10.1109/TIT.2010.2040894. Google Scholar

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G. Berkooz, P. Holmes and J. L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows,, Annual Review of Fluid Mechanics, 23 (1993), 539. Google Scholar

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I. Bright, G. Lin and J. N. Kutz, Compressive sensing and machine learning strategies for characterizing the flow around a cylinder with limited pressure measurements,, Physics of Fluids, 25 (2013). doi: 10.1063/1.4836815. Google Scholar

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B. W. Brunton, S. L. Brunton, J. L. Proctor and J. N. Kutz, Optimal sensor placement and enhanced sparsity for classification,, arXiv preprint arXiv:1310.4217, (2013). Google Scholar

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B. W. Brunton, L. A. Johnson, J. G. Ojemann and J. N. Kutz, Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition,, Journal of Neuroscience Methods, 258 (2016), 1. doi: 10.1016/j.jneumeth.2015.10.010. Google Scholar

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S. L. Brunton, J. H. Tu, I. Bright and J. N. Kutz, Compressive sensing and low-rank libraries for classification of bifurcation regimes in nonlinear dynamical systems,, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1716. doi: 10.1137/130949282. Google Scholar

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show all references

References:
[1]

R. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, volume 75 of Applied Mathematical Sciences,, Springer-Verlag, (1988). doi: 10.1007/978-1-4612-1029-0. Google Scholar

[2]

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S. Bagheri, Effects of small noise on the DMD/Koopman spectrum,, Bulletin Am. Phys. Soc., 58 (2013). Google Scholar

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[6]

Z. Bai, T. Wimalajeewa, Z. Berger, G. Wang, M. Glauser and P. K. Varshney, Low-dimensional approach for reconstruction of airfoil data via compressive sensing,, AIAA Journal, 53 (2015), 920. doi: 10.2514/1.J053287. Google Scholar

[7]

R. G. Baraniuk, Compressive sensing,, IEEE Signal Processing Magazine, 24 (2007), 118. Google Scholar

[8]

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[10]

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[11]

B. W. Brunton, S. L. Brunton, J. L. Proctor and J. N. Kutz, Optimal sensor placement and enhanced sparsity for classification,, arXiv preprint arXiv:1310.4217, (2013). Google Scholar

[12]

B. W. Brunton, L. A. Johnson, J. G. Ojemann and J. N. Kutz, Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition,, Journal of Neuroscience Methods, 258 (2016), 1. doi: 10.1016/j.jneumeth.2015.10.010. Google Scholar

[13]

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[14]

S. L. Brunton, J. L. Proctor and J. N. Kutz, Discovering governing equations from data by sparse identification of nonlinear dynamical systems,, Proceedings of the National Academy of Sciences, 113 (2016), 3932. doi: 10.1073/pnas.1517384113. Google Scholar

[15]

S. L. Brunton, J. H. Tu, I. Bright and J. N. Kutz, Compressive sensing and low-rank libraries for classification of bifurcation regimes in nonlinear dynamical systems,, SIAM Journal on Applied Dynamical Systems, 13 (2014), 1716. doi: 10.1137/130949282. Google Scholar

[16]

M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism a),, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012). doi: 10.1063/1.4772195. Google Scholar

[17]

E. J. Candès, Compressive sensing,, Proceedings of the International Congress of Mathematics, 3 (2006), 1433. Google Scholar

[18]

E. J. Candès, J. Romberg and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,, IEEE Transactions on Information Theory, 52 (2006), 489. doi: 10.1109/TIT.2005.862083. Google Scholar

[19]

E. J. Candès, J. Romberg and T. Tao, Stable signal recovery from incomplete and inaccurate measurements,, Communications in Pure and Applied Mathematics, 59 (2016), 1207. doi: 10.1002/cpa.20124. Google Scholar

[20]

E. J. Candès and T. Tao, Near optimal signal recovery from random projections: Universal encoding strategies?,, IEEE Transactions on Information Theory, 52 (2006), 5406. doi: 10.1109/TIT.2006.885507. Google Scholar

[21]

E. J. Candès and M. B. Wakin, An introduction to compressive sampling,, IEEE Signal Processing Magazine, (2008), 21. Google Scholar

[22]

K. K. Chen, J. H. Tu and C. W. Rowley, Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses,, Journal of Nonlinear Science, 22 (2012), 887. doi: 10.1007/s00332-012-9130-9. Google Scholar

[23]

S. Dawson, M. Hemati, M. Williams and C. Rowley, Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition,, Experiments in Fluids, 57 (2016). doi: 10.1007/s00348-016-2127-7. Google Scholar

[24]

D. L. Donoho, Compressed sensing,, IEEE Transactions on Information Theory, 52 (2006), 1289. doi: 10.1109/TIT.2006.871582. Google Scholar

[25]

J. E. Fowler, Compressive-projection principal component analysis,, IEEE Transactions on Image Processing, 18 (2009), 2230. doi: 10.1109/TIP.2009.2025089. Google Scholar

[26]

M. Gavish and D. L. Donoho, The optimal hard threshold for singular values is $4/\sqrt{3}$,, IEEE Transactions on Information Theory, 60 (2014), 5040. doi: 10.1109/TIT.2014.2323359. Google Scholar

[27]

A. C. Gilbert and P. Indyk, Sparse recovery using sparse matrices,, Proceedings of the IEEE, 98 (2010), 937. doi: 10.1109/JPROC.2010.2045092. Google Scholar

[28]

A. C. Gilbert, J. Y. Park and M. B. Wakin, Sketched SVD: Recovering spectral features from compressive measurements., ArXiv e-prints, (2012). Google Scholar

[29]

J. Gosek and J. N. Kutz, Dynamic mode decomposition for real-time background/foreground separation in video,, arXiv preprint, (2014). Google Scholar

[30]

M. Grilli, P. J. Schmid, S. Hickel and N. A. Adams, Analysis of unsteady behaviour in shockwave turbulent boundary layer interaction,, Journal of Fluid Mechanics, 700 (2012), 16. Google Scholar

[31]

F. Gueniat, L. Mathelin and L. Pastur, A dynamic mode decomposition approach for large and arbitrarily sampled systems,, Physics of Fluids, 27 (2015). doi: 10.1063/1.4908073. Google Scholar

[32]

Maziar S Hemati and Clarence W Rowley, De-biasing the dynamic mode decomposition for applied Koopman spectral analysis,, arXiv preprint, (2015). Google Scholar

[33]

B. L. Ho and R. E. Kalman, Effective construction of linear state-variable models from input/output data,, In Proceedings of the 3rd Annual Allerton Conference on Circuit and System Theory, (1965), 449. Google Scholar

[34]

P. J. Holmes, J. L. Lumley, G. Berkooz and C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry,, Cambridge Monographs in Mechanics. Cambridge University Press, (2012). doi: 10.1017/CBO9780511919701. Google Scholar

[35]

W. B. Johnson and J. Lindenstrauss, Extensions of Lipschitz mappings into a Hilbert space,, Contemporary mathematics, 26 (1984), 189. doi: 10.1090/conm/026/737400. Google Scholar

[36]

M. R. Jovanović, P. J. Schmid and J. W. Nichols, Low-rank and sparse dynamic mode decomposition,, Center for Turbulence Research, (2012). Google Scholar

[37]

J. N. Juang and R. S. Pappa, An eigensystem realization algorithm for modal parameter identification and model reduction,, Journal of Guidance, 8 (1985), 620. doi: 10.2514/3.20031. Google Scholar

[38]

I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg and C. Theodoropoulos, Equation-free, coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level analysis,, Communications in Mathematical Science, 1 (2003), 715. doi: 10.4310/CMS.2003.v1.n4.a5. Google Scholar

[39]

B. O. Koopman, Hamiltonian systems and transformation in Hilbert space,, Proceedings of the National Academy of Sciences, 17 (1931), 315. doi: 10.1073/pnas.17.5.315. Google Scholar

[40]

J. N. Kutz, Data-Driven Modeling & Scientific Computation: Methods for Complex Systems & Big Data,, Oxford University Press, (2013). Google Scholar

[41]

J. N. Kutz, X. Fu and S. L. Brunton, Multi-resolution dynamic mode decomposition,, SIAM Journal on Applied Dynamical Systems, 15 (2016), 713. doi: 10.1137/15M1023543. Google Scholar

[42]

J. L. Lumley, Stochastic Tools in Turbulence,, Academic Press, (1970). Google Scholar

[43]

Z. Ma, S. Ahuja and C. W. Rowley, Reduced order models for control of fluids using the eigensystem realization algorithm,, Theoretical and Computational Fluid Dynamics, 25 (2011), 233. doi: 10.1007/s00162-010-0184-8. Google Scholar

[44]

A. Mackey, H. Schaeffer and S. Osher, On the compressive spectral method,, Multiscale Modeling & Simulation, 12 (2014), 1800. doi: 10.1137/140965909. Google Scholar

[45]

I. Mezić, Analysis of fluid flows via spectral properties of the Koopman operator,, Annual Review of Fluid Mechanics, 45 (2013), 357. doi: 10.1146/annurev-fluid-011212-140652. Google Scholar

[46]

I. Mezić, Spectral properties of dynamical systems, model reduction and decompositions,, Nonlinear Dynamics, 41 (2005), 309. doi: 10.1007/s11071-005-2824-x. Google Scholar

[47]

D. Needell and J. A. Tropp, CoSaMP: iterative signal recovery from incomplete and inaccurate samples,, Appl. Comput. Harmon. Anal., 26 (2009), 301. doi: 10.1016/j.acha.2008.07.002. Google Scholar

[48]

B. R. Noack, K. Afanasiev, M. Morzynski, G. Tadmor and F. Thiele, A hierarchy of low-dimensional models for the transient and post-transient cylinder wake,, Journal of Fluid Mechanics, 497 (2003), 335. doi: 10.1017/S0022112003006694. Google Scholar

[49]

H. Nyquist, Certain topics in telegraph transmission theory,, Transactions of the A. I. E. E., (1928), 617. Google Scholar

[50]

V. M. Patel and R. Chellappa, Sparse Representations and Compressive Sensing for Imaging and Vision,, Briefs in Electrical and Computer Engineering. Springer, (2013). doi: 10.1007/978-1-4614-6381-8. Google Scholar

[51]

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