De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets

By Maziar S. Hemati, Clarence W. Rowley, Eric A. Deem, Louis N. Cattafesta. Theoretical and Computational Fluid Dynamics, 2015

373 citations recorded by the source corpus.

The dynamic mode decomposition (DMD)—a popular method for performing data-driven Koopman spectral analysis—has gained increased popularity for extracting dynamically meaningful spatiotemporal descriptions of fluid flows from snapshot measurements. Often times, DMD descriptions can be used for predictive purposes as well, which enables informed decision-making based on DMD model forecasts. Despite its widespread use and utility, DMD can fail to yield accurate dynamical descriptions when the measured snapshot data are imprecise due to, e.g., sensor noise. Here, we express DMD as a two-stage algorithm in order to isolate a source of systematic error. We show that DMD’s first stage, a subspace projection step, systematically introduces bias errors by processing snapshots asymmetrically. To remove this systematic error, we propose utilizing an augmented snapshot matrix in a subspace projection step, as in problems of total least-squares, in order to account for the error present in all snapshots. The resulting unbiased and noise-aware total DMD (TDMD) formulation reduces to standard DMD in the absence of snapshot errors, while the two-stage perspective generalizes the de-biasing framework to other related methods as well. TDMD’s performance is demonstrated in numerical and experimental fluids examples. In particular, in the analysis of time-resolved particle image velocimetry data for a separated flow, TDMD outperforms standard DMD by providing dynamical interpretations that are consistent with alternative analysis techniques. Further, TDMD extracts modes that reveal detailed spatial structures missed by standard DMD.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7a0f963cc2f6e1ba06c41ccba832c8428f85133a

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Citations in the corpus

Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition (2014, methodology): “An alternative and more focused exposition of tlsDMD is given in Hemati et al. (2015).”
Data-driven discovery of Koopman eigenfunctions for control (2017, methodology): “The DMD algorithm has also been extended to include sparsity and compressed sensing [23, 40, 46, 55, 113], actuation and control [92], multi-resolution analysis [69], de-noising [33, 50], and streaming variants [51].”
Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control (2015, positive): “Much of the success of DMD rests on it foundations in linear algebra, as the procedure is easy to implement [28], the results are highly interpretable, and the method lends itself naturally to extensions [17], such as compressed sensing [27, 29–31], disambiguating the effect of actuation [13], multi”
Adaptive separation control of a laminar boundary layer using online dynamic mode decomposition (2020, methodology): “Thus, this algorithm is well suited to the surface pressure array measurements with m = 13 unsteady pressure sensors in the present case, in contrast to streaming DMD which are beneficial for highdimensional data (Hemati et al. 2014, 2017).”
Model Reduction for Flow Analysis and Control (2017, background): “Although it has been observed empirically that DMD can be sensitive to noisy data (Duke et al. 2012b, Pan et al. 2015), recent work has explained this sensitivity and presented modifications to the algorithm to make its outputs more robust to noisy data (Hemati et al. 2015, Dawson et al. 2016).”