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
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show all references

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[2]

Siam, 1999. Google Scholar

[3]

Bulletin Am. Phys. Soc., 58 (2013), p230. Google Scholar

[4]

Journal of Fluid Mechanics, 726 (2013), 596-623. doi: 10.1017/jfm.2013.249.  Google Scholar

[5]

AIAA Paper 2013-0772, 51st Aerospace Sciences Meeting, January 2013. Google Scholar

[6]

AIAA Journal, 53 (2015), 920-933. doi: 10.2514/1.J053287.  Google Scholar

[7]

IEEE Signal Processing Magazine, 24 (2007), 118-120.  Google Scholar

[8]

IEEE Transactions on Information Theory, 56 (2010), 1982-2001. doi: 10.1109/TIT.2010.2040894.  Google Scholar

[9]

Annual Review of Fluid Mechanics, 23 (1993), 539-575.  Google Scholar

[10]

Physics of Fluids, 25 (2013), 127102. doi: 10.1063/1.4836815.  Google Scholar

[11]

arXiv preprint arXiv:1310.4217, 2013. Google Scholar

[12]

Journal of Neuroscience Methods, 258 (2016), 1-15. doi: 10.1016/j.jneumeth.2015.10.010.  Google Scholar

[13]

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[14]

Proceedings of the National Academy of Sciences, 113 (2016), 3932-3937. doi: 10.1073/pnas.1517384113.  Google Scholar

[15]

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[16]

Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012), 047510, 33pp. doi: 10.1063/1.4772195.  Google Scholar

[17]

Proceedings of the International Congress of Mathematics, 3 (2006), 1433-1452.  Google Scholar

[18]

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[19]

Communications in Pure and Applied Mathematics, 59 (2016), 1207-1223. doi: 10.1002/cpa.20124.  Google Scholar

[20]

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[25]

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[26]

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[36]

Center for Turbulence Research, 2012. Google Scholar

[37]

Journal of Guidance, Control, and Dynamics, 8 (1985), 620-627. doi: 10.2514/3.20031.  Google Scholar

[38]

Communications in Mathematical Science, 1 (2003), 715-762. doi: 10.4310/CMS.2003.v1.n4.a5.  Google Scholar

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Proceedings of the National Academy of Sciences, 17 (1931), 315-318. doi: 10.1073/pnas.17.5.315.  Google Scholar

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SIAM Journal on Applied Dynamical Systems, 15 (2016), 713-735, Available: arXiv:1506.00564. doi: 10.1137/15M1023543.  Google Scholar

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Annual Review of Fluid Mechanics, 45 (2013), 357-378. doi: 10.1146/annurev-fluid-011212-140652.  Google Scholar

[46]

Nonlinear Dynamics, 41 (2005), 309-325. doi: 10.1007/s11071-005-2824-x.  Google Scholar

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Appl. Comput. Harmon. Anal., 26 (2009), 301-321. doi: 10.1016/j.acha.2008.07.002.  Google Scholar

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Journal of Fluid Mechanics, 497 (2003), 335-363. doi: 10.1017/S0022112003006694.  Google Scholar

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Transactions of the A. I. E. E., (1928), 617-644. Google Scholar

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Briefs in Electrical and Computer Engineering. Springer, 2013. doi: 10.1007/978-1-4614-6381-8.  Google Scholar

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Mon. Weather Rev., 117 (1989), 2165-2185. Google Scholar

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J. Climate, 6 (1993), 1067-1076. Google Scholar

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SIAM Journal on Applied Dynamical Systems, 15 (2016), 142-161. doi: 10.1137/15M1013857.  Google Scholar

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IEEE International Conference on Image Processing, (2012), 937-940. doi: 10.1109/ICIP.2012.6467015.  Google Scholar

[55]

Journal of Fluid Mechanics, 641 (2009), 115-127. doi: 10.1017/S0022112009992059.  Google Scholar

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In Computer Vision-ECCV, (2010), 129-142. Google Scholar

[57]

Physical Review E, 92 (2015), 033304. doi: 10.1103/PhysRevE.92.033304.  Google Scholar

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Proceedings of the National Academy of Sciences USA, 110 (2013), 6634-6639. doi: 10.1073/pnas.1302752110.  Google Scholar

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Journal of Fluid Mechanics, 656 (2010), 5-28. doi: 10.1017/S0022112010001217.  Google Scholar

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Experiments in Fluids, 50 (2011), 1123-1130. doi: 10.1007/s00348-010-0911-3.  Google Scholar

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[64]

BMC Neuroscience, 13 (2012), p183. doi: 10.1186/1471-2202-13-S1-P183.  Google Scholar

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[66]

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[67]

Comptes Rendus Mécanique, 342 (2014), 410-416. doi: 10.1016/j.crme.2013.12.011.  Google Scholar

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IEEE Transactions on Information Theory, 50 (2004), 2231-2242. doi: 10.1109/TIT.2004.834793.  Google Scholar

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IEEE Transactions on Information Theory, 56 (2010), 520-544. doi: 10.1109/TIT.2009.2034811.  Google Scholar

[70]

Journal of Computational Dynamics, 1 (2014), 391-421. doi: 10.3934/jcd.2014.1.391.  Google Scholar

[71]

49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition, (2011), p2864. doi: 10.2514/6.2011-38.  Google Scholar

[72]

Experiments in Fluids, 55 (2014), p1805. doi: 10.1007/s00348-014-1805-6.  Google Scholar

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