Dynamic Mode Decomposition

Kutz, J. Nathan; Brunton, Steven L.; Brunton, Bingni W.; Proctor, Joshua L.

doi:10.1137/1.9781611974508

Cited by 826 publications

(374 citation statements)

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Section: Discussionmentioning

confidence: 99%

“…Its popularity is also due to the fact that it does not make any assumptions about the underlying system. See (Kutz et al 2016) for a comprehensive overview of the algorithm and its connections to the Koopman-operator analysis, initiated in (Koopman 1931), along with examples in computational fluid dynamics.In the last years many variants arose, such as multiresolution DMD, compressed DMD, forward backward DMD, and higher order DMD among others, in order to deal with noisy data, big dataset, or spurius data for example.In the PyDMD package ("PyDMD: Python Dynamic Mode Decomposition. Available at: https://github.com/mathLab/PyDMD" n.d.) we implemented in Python the majority of the variants mentioned above with a user friendly interface.…”

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confidence: 99%

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PyDMD: Python Dynamic Mode Decomposition

Demo¹,

Tezzele²,

Rozza³

2018

JOSS

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SummaryDynamic mode decomposition (DMD) is a model reduction algorithm developed by Schmid (Schmid 2010). Since then has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. It is used for a data-driven model simplification based on spatiotemporal coherent structures. DMD relies only on the high-fidelity measurements, like experimental data and numerical simulations, so it is an equation-free algorithm. Its popularity is also due to the fact that it does not make any assumptions about the underlying system. See (Kutz et al. 2016) for a comprehensive overview of the algorithm and its connections to the Koopman-operator analysis, initiated in (Koopman 1931), along with examples in computational fluid dynamics.In the last years many variants arose, such as multiresolution DMD, compressed DMD, forward backward DMD, and higher order DMD among others, in order to deal with noisy data, big dataset, or spurius data for example.In the PyDMD package ("PyDMD: Python Dynamic Mode Decomposition. Available at: https://github.com/mathLab/PyDMD" n.d.) we implemented in Python the majority of the variants mentioned above with a user friendly interface. We also provide many tutorials that show all the characteristics of the software, ranging from the basic use case to the most sofisticated one allowed by the package.The research in the field is growing both in computational fluid dynamic and in structural mechanics, due to the equation-free nature of the model.As an exmaple, we show below few snapshots collected from a toy system with some noise. The DMD is able to reconstruct the entire system evolution, filtering the noise. It is also possible to predict the evolution of the system in the future with respect to the available data.Here we have the reconstruction of the dynamical system. You can observe the sensible reduction of the noise.

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Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

PyDMD: Python Dynamic Mode Decomposition

Demo¹,

Tezzele²,

Rozza³

2018

JOSS

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“…Reduced-order models provide a powerful framework to obtain surrogates for expensiveto-evaluate models. In the case of nonlinear systems, reduced-order models can be obtained via reduced-basis methods [19], dynamic mode decomposition [24], proper orthogonal decomposition [5], and many others; for a survey, see [4]. Here, we compute reduced-order models f (i) for our multifidelity approach via Proper Orthogonal Decomposition and the Discrete Empirical Interpolation Method (DEIM) for an efficient evaluation of the nonlinear term.…”

Section: Discretization and Reduced-order Modelsmentioning

confidence: 99%

Multifidelity probability estimation via fusion of estimators

Krämer

Marques

Peherstorfer

et al. 2019

Journal of Computational Physics

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This paper develops a multifidelity method that enables estimation of failure probabilities for expensive-to-evaluate models via information fusion and importance sampling. The presented general fusion method combines multiple probability estimators with the goal of variance reduction. We use low-fidelity models to derive biasing densities for importance sampling and then fuse the importance sampling estimators such that the fused multifidelity estimator is unbiased and has mean-squared error lower than or equal to that of any of the importance sampling estimators alone. By fusing all available estimators, the method circumvents the challenging problem of selecting the best biasing density and using only that density for sampling. A rigorous analysis shows that the fused estimator is optimal in the sense that it has minimal variance amongst all possible combinations of the estimators. The asymptotic behavior of the proposed method is demonstrated on a convection-diffusion-reaction partial differential equation model for which 10 5 samples can be afforded. To illustrate the proposed method at scale, we consider a model of a free plane jet and quantify how uncertainties at the flow inlet propagate to a quantity of interest related to turbulent mixing. Compared to an importance sampling estimator that uses the high-fidelity model alone, our multifidelity estimator reduces the required CPU time by 65% while achieving a similar coefficient of variation.

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“…In fact, the eigenvalues of the Koopman operator on Burger's equation (without process noise, i.e., σ p = 0) can be analytically obtained via the Cole-Hopf transformation and they correspond to the decaying modes of the solution [4,36]. When process noise is present (σ p > 0), the solution of Burger's equation becomes "rough," but its global appearance remains similar to the case of no process noise, as shown in Figure 2a.…”

Section: B Noisy Damping Modesmentioning

confidence: 99%

Subspace dynamic mode decomposition for stochastic Koopman analysis

2017

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The analysis of nonlinear dynamical systems based on the Koopman operator is attracting attention in various applications. Dynamic mode decomposition (DMD) is a data-driven algorithm for Koopman spectral analysis, and several variants with a wide range of applications have been proposed. However, popular implementations of DMD suffer from observation noise on random dynamical systems and generate inaccurate estimation of the spectra of the stochastic Koopman operator. In this paper, we propose subspace DMD as an algorithm for the Koopman analysis of random dynamical systems with observation noise. Subspace DMD first computes the orthogonal projection of future snapshots to the space of past snapshots and then estimates the spectra of a linear model, and its output converges to the spectra of the stochastic Koopman operator under standard assumptions. We investigate the empirical performance of subspace DMD with several dynamical systems and show its utility for the Koopman analysis of random dynamical systems.

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Dynamic Mode Decomposition

Cited by 826 publications

References 0 publications

PyDMD: Python Dynamic Mode Decomposition

PyDMD: Python Dynamic Mode Decomposition

Multifidelity probability estimation via fusion of estimators

Subspace dynamic mode decomposition for stochastic Koopman analysis

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