Dynamic mode decomposition for large and streaming datasets

Abstract

We formulate a low-storage method for performing dynamic mode decomposition that can be updated inexpensively as new data become available; this formulation allows dynamical information to be extracted from large datasets and data streams. We present two algorithms: the first is mathematically equivalent to a standard "batch-processed" formulation; the second introduces a compression step that maintains computational efficiency, while enhancing the ability to isolate pertinent dynamical information from noisy measurements. Both algorithms reliably capture dominant fluid dynamic behaviors, as demonstrated on cylinder wake data collected from both direct numerical simulations and particle image velocimetry experiments.Dynamic mode decomposition (DMD) is a data-driven computational technique capable of extracting dynamical information from flowfields measured in physical experiments or generated by direct numerical simulations. 1 Since its introduction in 2008, 2 DMD has been used in the analysis of numerous fluid mechanical systems (e.g., bluff body flows, 3 jet flows, 4,5 and viscoelastic fluid flows 6 ) and has gained increasing popularity owing to its ability to reveal and quantify the dynamics of a flow, even when those dynamics are nonlinear. 4,7 DMD operates on snapshots of the flowfield (e.g., velocity, vorticity, pressure) and their time-shifted counterparts-obtained either from experiments or numerical simulations-to compute the eigenvalues ("DMD eigenvalues") and eigenvectors ("DMD modes") of a linear operator that best fits the associated dynamics in a least-squares sense. The DMD modes represent spatial fields that often highlight coherent structures in the flow, while the associated DMD eigenvalues dictate the decay/growth rates and oscillation frequencies of these modes. As such, access to DMD modes and eigenvalues enables a reconstruction of the dynamics associated with a given flowfield. Other modal decomposition techniques, such as the commonly employed proper orthogonal decomposition (POD), only compute spatial modes associated with the flow. 8 Although spatial modes can offer valuable information regarding coherent structures and other flow qualities (e.g., in the case of POD, they determine the most energetic modes), characterizing the underlying dynamics relies upon projecting these spatial modes onto an assumed dynamical form. DMD offers an advantage over these other modal decomposition techniques in that it computes both spatial modes and their associated temporal behaviors, thus removing any guesswork associated with realizing a dynamical representation of the system.To date, researchers have viewed DMD as a post-processing tool; that is, a method that requires the entire experimental or computational dataset to be available prior to commencing

On the accuracy of dynamic mode decomposition in estimating instability of wave packet

“…However, in practical applications, the choice of reduce rank relies on both the characteristics of the finest structures and the background noise level. Recent works by Jovanović et al (2014) and Hemati et al (2014) proposed promising strategies to balance the lowerorder estimation efficiency and the dynamic bias caused by truncation. The accuracy performance of these new DMD schemes should be evaluated in the future.…”

“…In the incrementally updated DMD algorithm, Hemati et al (2014) have observed a smooth effect on the mode shape by POD basis compression and attributed it to the truncation stage which filters out the contribution from noise. Such low-rank truncation procedure also appears in sparsity-promoting DMD (Jovanović et al 2014) and optimal mode decomposition (Wynn et al 2013), whose primary aims are both approximating the original dynamic system with minimal number of degrees of freedom.…”

“…Based on this new perspective, Tu et al (2014a) proposed a method of computing temporally oscillating spatial modes from subNyquist-rate data via DMD and compressed sensing. And Hemati et al (2014) formulated an incrementally updated DMD scheme to cope with the low-storage requirement or an online analysis of real-time measurement. Recently, an de-biasing noise-robust method based on solving the total least-squares problem was proposed by Hemati and Rowley (2015), which provides the unbiased dynamical approximations in the noise contamination circumstance.…”

Enstrophy-based proper orthogonal decomposition for reduced-order modeling of flow past a cylinder

“…dQ dt (15) For flow below the critical Reynolds number (Recr), o> is negative and if /,. > 0, any small perturbation introduced in the flow is eventually damped.…”

“…This linear mapping is equiv alent to the dynamic mode decomposition (DMD) technique developed [14][15][16] for applications. Koopman analysis results in modes that are orthogonal in time (single frequency modes), while POD gives spatially orthogonal, multi-time-periodic modes (as explained before), which capture most of the enstrophy of the flow.…”