Compressive sampling and dynamic mode decomposition

TL;DR

This paper introduces compressive sampling techniques for dynamic mode decomposition (DMD) that enable accurate eigenvalue computation and mode reconstruction from heavily subsampled data, supported by theoretical guarantees and demonstrated on model systems.

Contribution

It develops new compressive sampling strategies for DMD, establishing theoretical connections and invariance properties, allowing efficient computation from sparse or projected data.

Findings

DMD eigenvalues are preserved under subsampling and projection.

Full-state DMD modes can be reconstructed using $\,l_1$-minimization or greedy algorithms.

The approach is validated on spatial signals and chaotic flow models.

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.