# American Institute of Mathematical Sciences

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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