Constrained Dynamic Mode Decomposition

Krake, Tim; Klötzl, Daniel; Eberhardt, Bernhard; Weiskopf, Daniel

doi:10.1109/tvcg.2022.3209437

Cited by 6 publications

(3 citation statements)

References 52 publications

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“…Symmetric DMD (Cohen et al 2020) mandates the dynamics matrix to be symmetric. Constrained DMD (Krake et al 2022) ensures the presence of specific frequencies by incorporating constraints into DMD.…”

Section: Further Methodsmentioning

confidence: 99%

Residual dynamic mode decomposition: robust and verified Koopmanism

2023

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Dynamic mode decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators. However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a significant challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes and dealing with continuous spectra, which both occur regularly in turbulent flows. Residual dynamic mode decomposition (ResDMD), introduced by Colbrook & Townsend (Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. 2021. arXiv:2111.14889), overcomes such challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems. ResDMD thus provides robust and verified Koopmanism. We implement ResDMD and demonstrate its application in various fluid dynamic situations at varying Reynolds numbers from both numerical and experimental data. Examples include vortex shedding behind a cylinder, hot-wire data acquired in a turbulent boundary layer, particle image velocimetry data focusing on a wall-jet flow and laser-induced plasma acoustic pressure signals. We present some advantages of ResDMD: the ability to resolve nonlinear and transient modes verifiably; the verification of learnt dictionaries; the verification of Koopman mode decompositions; and spectral calculations with reduced broadening effects. We also discuss how a new ordering of modes via residuals enables greater accuracy than the traditional modulus ordering (e.g. when forecasting) with a smaller dictionary. This result paves the way for more significant dynamic compression of large datasets without sacrificing accuracy.

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Section: Further Methodsmentioning

confidence: 99%

Residual dynamic mode decomposition: robust and verified Koopmanism

2023

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“…Concrete publications that deal with time series decomposition and visualization use decomposition techniques such as dynamic mode decomposition [18], singular spectrum analysis [19], and Karhunen-Loève decomposition [20]. However, none of them take uncertainty into account.…”

Section: Related Workmentioning

confidence: 99%

Uncertainty-Aware Seasonal-Trend Decomposition Based on Loess

Krake,

Klötzl,

Hägele

et al. 2024

IEEE Trans. Visual. Comput. Graphics

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Seasonal-trend decomposition based on loess (STL) is a powerful tool to explore time series data visually. In this paper, we present an extension of STL to uncertain data, named uncertaintyaware STL (UASTL). Our method propagates multivariate Gaussian distributions mathematically exactly through the entire analysis and visualization pipeline. Thereby, stochastic quantities shared between the components of the decomposition are preserved. Moreover, we present application scenarios with uncertainty modeling based on Gaussian processes, e.g., data with uncertain areas or missing values. Besides these mathematical results and modeling aspects, we introduce visualization techniques that address the challenges of uncertainty visualization and the problem of visualizing highly correlated components of a decomposition. The global uncertainty propagation enables the time series visualization with STL-consistent samples, the exploration of correlation between and within decomposition's components, and the analysis of the impact of varying uncertainty. Finally, we show the usefulness of UASTL and the importance of uncertainty visualization with several examples. Thereby, a comparison with conventional STL is performed.

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“…Dynamic Mode Decomposition (DMD) operates by decomposing time-resolved data to identify coherent spatio-temporal patterns, their growth rates, and their frequencies. Since its introduction by Schmid (2010), many different variations of the algorithm have been proposed (Belson et al, 2014;Vega and Le Clainche, 2017;Krake et al, 2019). As compared to POD, it provides additional information about the temporal behavior of the decomposed data considering the best-fitting linear operator A to approximate the dynamics of a system:…”

Section: Dynamic Mode Decompositionmentioning

confidence: 99%

Modal analysis reveals imprint of snowflake shape on wake flow structures

Tagliavini,

Holzner,

Corso

2024

Preprint

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This study investigates the complex interplay of wake flow structures, particle shape, and falling behavior of snowflakes through advanced flow analysis. We employ Proper Orthogonal Decomposition and Dynamic Mode Decomposition to analyze the wake flow patterns of three distinct snowflake geometries at Reynolds number of 1500: a dendrite crystal, a columnar crystal, and a rosette-like particle. Proper Orthogonal Decomposition reveals that spatial resolution significantly impacts the capture of flow structures, particularly for particles with with more intricate wake flow structure, corresponding to unstable falling motion. Dynamic Mode Decomposition demonstrates high sensitivity to temporal resolution, with data of the forces exerted on the snowflake incorporated in the matrix prior to the decomposition mitigating information loss at lower sampling rates. We establish a linear relationship between snowflake shape porosity and minimum and maximum Dynamic Mode Decomposition eigenfrequencies, absolute decay or growth rates, and wavenumbers of the most energetic mode, linking particle geometry to wake flow characteristics. Higher porosity corresponds to more stable, small-scale flow structures and steady falling motion, while lower porosity promotes larger, unstable structures and falling trajectories with random particle orientations. These findings reveal the interdependence of snowflake geometry, wake flow configuration, and falling behavior and highlight the importance of considering both spatial and temporal resolutions when dealing with modal analysis. This research contributes to improved predictions of snowflake falling behavior, with potential applications in meteorology and climate science.

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Constrained Dynamic Mode Decomposition

Cited by 6 publications

References 52 publications

Residual dynamic mode decomposition: robust and verified Koopmanism

Residual dynamic mode decomposition: robust and verified Koopmanism

Uncertainty-Aware Seasonal-Trend Decomposition Based on Loess

Modal analysis reveals imprint of snowflake shape on wake flow structures

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