Compressive sampling and dynamic mode decomposition

Abstract

This work develops compressive sampling strategies for computing the dynamic mode decomposition (DMD) from heavily subsampled or output-projected 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 a compressed DMD, and then reconstruct full-state modes by applying the projected DMD transforms to full-state snapshots. These results rely on a number of theoretical advances. First, we establish connections between the full-state and projected DMD. 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 so-called restricted isometry principle from compressive sampling. 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.

12 pages, 12 figures

References in corpus (4)

• Toward compressed DMD: spectral analysis of fluid flows using sub-Nyquist-rate PIV data
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• Interplay of Sensor Quantity, Placement and System Dimensionality on Energy Sparse Reconstruction of Fluid Flows
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• Multi-Resolution Dynamic Mode Decomposition
• Dynamic mode decomposition for interconnected control systems
• Cluster-based control of nonlinear dynamics
• Projection assisted Dynamic Mode Decomposition of large scale data
• Spectral identification of networks using sparse measurements
• On the Physical Interpretation of Proper Orthogonal Decomposition and Dynamic Mode Decomposition for Liquid Injection
• Predicting sparse circle maps from their dynamics