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Experiments in Fluids

, Volume 52, Issue 2, pp 529–542 | Cite as

An error analysis of the dynamic mode decomposition

  • Daniel Duke
  • Julio Soria
  • Damon Honnery
Research Article

Abstract

Dynamic mode decomposition (DMD) is a new diagnostic technique in fluid mechanics which is growing in popularity. A powerful analysis tool, it has great potential for measuring the spatial and temporal dynamics of coherent structures in experimental fluid flows. To aid interpretation of experimental data, error-bars on the measured growth rates are needed. In this article, we undertake a massively parallel error analysis of the DMD algorithm using synthetic waveforms that are shown to be representative of the canonical instabilities observed in shear flows. We show that the waveform of the instability has a marked impact on the error of the measured growth rate. Sawtooth and square waves may have an order of magnitude larger error than sine waves under the same conditions. We also show that the effects of data quantity and quality are of critical importance in determining the error in the growth or decay rate, and that the effect of the key parametric variables are modulated by the growth rate itself. We further demonstrate methods by which ensemble and orthogonal data may be introduced to improve the noise response. With regard for the important variables, precise measurement of the growth rates of instabilities may be supplemented with an accurately estimated uncertainty. This opens many new possibilities for the measurement of coherent structure in shear flows.

Keywords

Linear Instability Data Quantity Decomposition Dimension Dynamic Mode Decomposition Rank Reduction 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

List of symbols

a0

Initial amplitude of synthetic instability

c

Velocity of convective instability

f(xt)

Synthetic instability function

i

DMD mode number

k

Spatial wavenumber

p(xt)

Periodic part of synthetic instability

t

Temporal co-ordinate

x

Spatial co-ordinate

A

Linear mapping between V 1 and V 2

AR

Aspect Ratio; ratio of product of data points in all non-DMD dimensions to data points in DMD dimension

DMi

Dynamic mode eigenvector of mode i

Nn

Number of samples in dimension n

NWn

Number of periods of f in the dimension n

QR

Orthogonal and upper triangular matrices in QR decomposition

Rn

Number of samples per period of f in dim. n

S

Low-dimensional representative system matrix

SNR

Signal to noise ratio

U

Left singular vectors in singular value decomposition (SVD)

V

DMD input matrix

vn

Column n of V; one sample in DMD dimension

W

Right singular vectors in SVD

Xi

Complex eigenvector of mode i

δ

Constant space/time step between each v n

γ

Exponential growth or decay rate per wavelength;

DMD as measured by DMD,

true as the exact value from f(xt)

\(\varepsilon\)

Fractional error in γ

λi

Dynamic mode eigenvalue of mode i

σ

Standard deviation

\(\Upsigma\)

Matrix of singular values

\(\varphi_{i}\)

Ritz Eigenvalue of mode i

ω

Temporal frequency

Notes

Acknowledgments

The authors wish to thank Mr. Daniel Mitchell for providing the data of Fig. 1a, and Dr. Yoshinori Mizuno for providing the data of Fig. 1b. The authors wish to acknowledge the support of the ARC and the National Computational Infrastructure (NCI) National Facility for providing the cluster computing resources required to undertake the research. The first author would also like to thank Mr. David Singleton and Dr. Margaret Khan of the NCI National Facility for their technical support. The first author was supported by an Australian Postgraduate Award whilst undertaking this research.

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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Laboratory for Turbulence Research in Aerospace and Combustion, Department of Mechanical and Aerospace Engineering Monash UniversityMelbourneAustralia

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