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Journal of Computational Dynamics

Open Access Articles

Novel computational approaches and their applications
Bernd Krauskopf and Carlo R. Laing
2020, 7(2): ⅰ-ⅰ doi: 10.3934/jcd.2020007 +[Abstract](778) +[HTML](283) +[PDF](77.34KB)
Preface Special issue in honor of Reinout Quispel
Elena Celledoni and Robert I. McLachlan
2019, 6(2): ⅰ-ⅴ doi: 10.3934/jcd.2019007 +[Abstract](1312) +[HTML](378) +[PDF](194.73KB)
Compressed sensing and dynamic mode decomposition
Steven L. Brunton, Joshua L. Proctor, Jonathan H. Tu and J. Nathan Kutz
2015, 2(2): 165-191 doi: 10.3934/jcd.2015002 +[Abstract](7735) +[PDF](9556.1KB)
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:
Preface: Special issue on the occasion of the 4th International Workshop on Set-Oriented Numerics (SON 13, Dresden, 2013)
Gary Froyland, Oliver Junge and Kathrin Padberg-Gehle
2015, 2(1): i-ii doi: 10.3934/jcd.2015.2.1i +[Abstract](2343) +[PDF](86.8KB)
This issue comprises manuscripts collected on the occasion of the 4th International Workshop on Set-Oriented Numerics which took place at the Technische Universität Dresden in September 2013. The contributions cover a broad spectrum of different subjects in computational dynamics ranging from purely discrete problems on graphs to computer assisted proofs of bifurcations in dissipative PDEs. In many cases, ideas related to set-oriented paradigms turn out to be useful in the computations, for example by quantizing the state space, or by using interval arithmetic to perform rigorous computations.

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