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:
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show all references

References:
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[21]

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[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

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[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

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

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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]

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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

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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]

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