An error analysis of the dynamic mode decomposition

Duke, Daniel J.; Soria, Julio; Honnery, Damon

doi:10.1007/s00348-011-1235-7

Cited by 143 publications

(89 citation statements)

References 18 publications

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“…The present investigation is based on 1000 snapshots in time and 75 snapshots in space. Criteria of [5] are applied where possible. Data is sampled in a cylindrical 3D region of the computational domain of the LES shown in Fig.…”

Section: Dynamic Mode Decomposition Of Les Resultsmentioning

confidence: 99%

Dynamic Mode Decomposition for Swirling Jet Flow Undergoing Vortex Breakdown

Luginsland

Kleiser

2012

Proc Appl Math and Mech

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Swirling jets undergoing vortex breakdown occur in many technical applications, e.g. vortex burners, turbines and jet engines. At the stage of vortex breakdown the flow is dominated by a conical shear layer and a large recirculation zone around the jet axis. We performed Large-Eddy Simulations (LES) of compressible swirling jet flows at Re=5000, Ma=0.6 in the high swirl number regime (S=1). A nozzle is included in our computational setup to account for more realistic inflow conditions. The obtained velocity fields are analyzed by means of temporal and spatial dynamic mode decomposition (DMD) to get further insight into the characteristic structures dominating the flow. We present eigenvalue spectra for the case under consideration and discuss the stability behaviour in time and space. LES of swirling jet flow undergoing vortex breakdownWe performed Large-Eddy Simulations (LES) of swirling jet flow undergoing vortex breakdown, see [1] for a review. We solve the compressible Navier-Stokes equations on a cylindrical grid using high-order temporal and spatial discretisation schemes, see [2] and the references therein for details. The Reynolds number is set to Re = ρ cl r in u z cl /µ cl = 5000 and the Mach number to Ma = 0.6 using the centerline values at the inflow plane of the computational domain. The flow medium is air. We capture the fundamental physics of the flow configuration under investigation, namely a large recirculation zone at the centerline of the jet, a strong bursting behind the nozzle lip and helical instabilities dominating the flow, see [3] and Fig. 1. To get further insight into the structures dominating the flow we apply the Dynamic Mode Decomposition method (DMD) to the simulation results. Dynamic Mode Decomposition of LES resultsThe Dynamic Mode Decomposition (DMD) is a relatively new method to get insight into coherent structures dominating the dynamics of a flow in time and space. It is based on snapshot sequences of either experimental or numerical time-and spaceresolved data (see [4] for a detailed description of the method and first applications). The present investigation is based on 1000 snapshots in time and 75 snapshots in space. Criteria of [5] are applied where possible. Data is sampled in a cylindrical 3D region of the computational domain of the LES shown in Fig. 1 (right). We apply DMD to all three velocity components u r , u θ , u z one after the other and restrict ourselves here to presenting results for the azimuthal velocity u θ only for the sake of brevity. Data is sampled after 100tu cl /r in dimensionless time units for a time interval of 25tu cl /r in . The maximum resolved Strouhal number in the snapshot basis is St = 20 and the maximum resolved streamwise wave number k/r in = 9.5. With the snapshot basis used in the present investigation the residual converged over 4 orders of magnitude.Figure 2 (left) shows the temporal spectrum of the flow with the unstable region shaded in grey. Since transitional processes are still going on during sampling of the flow most of th...

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Section: Dynamic Mode Decomposition Of Les Resultsmentioning

confidence: 99%

Dynamic Mode Decomposition for Swirling Jet Flow Undergoing Vortex Breakdown

Luginsland

Kleiser

2012

Proc Appl Math and Mech

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“…This flow was used by Duke et al (2012) to analyse the DMD algorithm. Algorithm performance is determined by the relative growth rate error statistic.…”

Section: Comparison With Dmd Using a Synthetic Data Ensemblementioning

confidence: 99%

“…At each covariance and temporal frequency pair (σ 2 , ω) ∈ [0.05 2 , 1] × [0.6, 1.6], 10 3 data ensembles were created and both DMD and algorithm 1 were applied to each simulation ensemble. Calculation of the DMD eigenvalues was performed using the method described in Duke et al (2012) with a rank-reduction ratio of 10 −1 . A rank reduction ratio of 10…”

Section: Comparison With Dmd Using a Synthetic Data Ensemblementioning

confidence: 99%

Optimal mode decomposition for unsteady flows

et al. 2013

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A new method, herein referred to as Optimal Mode Decomposition (OMD), of finding a linear model to describe the evolution of a fluid flow is presented. The method estimates the linear dynamics of a high-dimensional system which is first projected onto a subspace of a user-defined fixed rank. An iterative procedure is used to find the optimal combination of linear model and subspace that minimises the system residual error. The OMD method is shown to be a generalisation of Dynamic Mode Decomposition (DMD), in which the subspace is not optimised but rather fixed to be the Proper Orthogonal Decomposition (POD) modes. Furthermore, OMD is shown to provide an approximation to the Koopman modes and eigenvalues of the underlying system. A comparison between OMD and DMD is made using both a synthetic waveform and an experimental data set. The OMD technique is shown to have lower residual errors than DMD and is shown on a synthetic waveform to provide more accurate estimates of the system eigenvalues. This new method can be used with experimental and numerical data to calculate the "optimal" low-order model with a user-defined rank that best captures the system dynamics of unsteady and turbulent flows.

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“…Innovative measurement strategies utilizing recent developments in compressive sensing [111] allow for experimentalists to collect less data in time [57] and in space [63,112,113] while still reconstructing relevant dynamic characteristics. Other relevant innovations include memory-efficient algorithms for BPOD based on DMD [102], error and uncertainty analysis of growth rates [114], and more accurate low-order models via optimal mode decomposition [115]. DMD has also been extended to include inputs and control expanding the method to complex systems that allow for exogenous inputs and nonautonomous dynamical systems [61].…”

Section: The Dynamic Mode Decompositionmentioning

confidence: 99%

Including inputs and control within equation-free architectures for complex systems

Proctor

Brunton

Kutz

2016

Eur. Phys. J. Spec. Top.

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Abstract. The increasing ubiquity of complex systems that require control is a challenge for existing methodologies in characterization and controller design when the system is high-dimensional, nonlinear, and without physics-based governing equations. We review standard model reduction techniques such as Proper Orthogonal Decomposition (POD) with Galerkin projection and Balanced POD (BPOD). Further, we discuss the link between these equation-based methods and recently developed equation-free methods such as the Dynamic Mode Decomposition and Koopman operator theory. These data-driven methods can mitigate the challenge of not having a well-characterized set of governing equations. We illustrate that this equation-free approach that is being applied to measurement data from complex systems can be extended to include inputs and control. Three specific research examples are presented that extend current equation-free architectures toward the characterization and control of complex systems. These examples motivate a potentially revolutionary shift in the characterization of complex systems and subsequent design of objective-based controllers for data-driven models.

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An error analysis of the dynamic mode decomposition

Cited by 143 publications

References 18 publications

Dynamic Mode Decomposition for Swirling Jet Flow Undergoing Vortex Breakdown

Dynamic Mode Decomposition for Swirling Jet Flow Undergoing Vortex Breakdown

Optimal mode decomposition for unsteady flows

Including inputs and control within equation-free architectures for complex systems

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