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.

Overview of Compressive Sampling and Dynamic Mode Decomposition

The paper by Brunton et al. presents a sophisticated integration of compressive sampling techniques with Dynamic Mode Decomposition (DMD), aiming to enhance the computational efficiency and practical applicability of DMD in high-dimensional systems. The primary focus of the study is to demonstrate that DMD can be accurately computed from subsampled or output-projected data, thereby potentially reducing the volume of measurements required in practical scenarios such as fluid dynamics or oceanographic monitoring.

Dynamic Mode Decomposition is a robust tool for analyzing complex dynamical systems, offering insights into spatial-temporal coherent structures that oscillate at constant frequencies. It has gained prominence due to its data-driven approach, applicable equally to high-dimensional datasets from simulations or experimental setups. Given that DMD is closely related to the eigendecomposition of the infinite-dimensional Koopman operator, it provides a linear model for temporal evolution of modal coefficients without the need for explicit equations governing the underlying system.

The research introduces two key methodologies: Compressive Sampling DMD and Compressed DMD. The former involves reconstructing full-state DMD modes from limited and spatially subsampled measurements using compressive sampling. The latter, on the other hand, benefits from an initial compression of full-state snapshots before calculating the DMD and subsequently reconstructing the full-state modes, relying on projected DMD algorithm results.

Methodological Contributions

1. Theoretical Integration: The authors establish theoretical links between full-state and projected DMD, demonstrating invariance of DMD algorithms under unitary transformations. They extend this analysis by showing that DMD remains invariant when data and modes are sparse in some basis.

2. Practical Implementation: Leveraging the Restricted Isometry Principle (RIP), the paper outlines a methodology to apply DMD robustly despite significant data subsampling, making it viable for applications where complete data acquisition is challenging or expensive.

3. Applications and Validation: The studies provide examples illustrating the application of the proposed methodologies in fluid dynamics, notably examining a sparse linear system in the Fourier domain and the time-varying double gyre flow, a model for ocean mixing. These examples validate the efficacy and accuracy of Compressive Sampling DMD and Compressed DMD in capturing essential dynamical features with reduced data input.

Numerical Results and Implications

Strong numerical results corroborate the efficacy of the proposed approaches. For instance, using simulated data where spatial structures are driven by low-order dynamics, the authors demonstrate accurate recovery of DMD modes and eigenvalues even when data is heavily subsampled. This exemplifies potential practical applications such as reducing the burden of data collection in experiments or enhancing temporal resolution capacities in computational fluid dynamics settings.

Future Directions

The integration of compressive sampling with DMD opens avenues for expanding its applicability to real-world problems in fluid dynamics and environmental sciences, where data acquisition is constrained by logistic or economic factors. Future work could explore the effects of noise on DMD robustness further and extend the framework to combine spatial sampling strategies with innovative temporal sampling methodologies for comprehensive real-time system identification.

This paper represents a significant stride in computational methods for dynamical systems, reaffirming DMD’s potential in scenarios characterized by sparse or incomplete data without compromising analytical integrity or computational efficiency.