Experiments in Fluids

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

An error analysis of the dynamic mode decomposition

  • Daniel DukeEmail author
  • Julio Soria
  • Damon Honnery
Research Article


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.


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


Initial amplitude of synthetic instability


Velocity of convective instability


Synthetic instability function


DMD mode number


Spatial wavenumber


Periodic part of synthetic instability


Temporal co-ordinate


Spatial co-ordinate


Linear mapping between V 1 and V 2


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


Dynamic mode eigenvector of mode i


Number of samples in dimension n


Number of periods of f in the dimension n


Orthogonal and upper triangular matrices in QR decomposition


Number of samples per period of f in dim. n


Low-dimensional representative system matrix


Signal to noise ratio


Left singular vectors in singular value decomposition (SVD)


DMD input matrix


Column n of V; one sample in DMD dimension


Right singular vectors in SVD


Complex eigenvector of mode i


Constant space/time step between each v n


Exponential growth or decay rate per wavelength;

DMD as measured by DMD,

true as the exact value from f(xt)


Fractional error in γ


Dynamic mode eigenvalue of mode i


Standard deviation


Matrix of singular values


Ritz Eigenvalue of mode i


Temporal frequency



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.


  1. Aubry N, Holmes P, Lumley J, Stone E (1988) The dynamics of coherent structures in the wall region of a turbulent boundary layer. J Fluid Mech 192:115–173MathSciNetzbMATHCrossRefGoogle Scholar
  2. Basley J, Pastur L, Lusseyran F, Faure T, Delprat N (2010) Experimental investigation of global structures in an incompressible cavity flow using time-resolved PIV. Exp Fluids 50(4):905–918CrossRefGoogle Scholar
  3. Berkooz G, Holmes P (1993) The proper orthogonal decomposition in the analysis of turbulent flows. Annu Rev Fluid Mech 25:539–575MathSciNetCrossRefGoogle Scholar
  4. Chatterjee A (2000) An introduction to the proper orthogonal decomposition. Curr Sci 78(7):808–817Google Scholar
  5. Davies P, Yule A (1975) Coherent structures in turbulence. J Fluid Mech 69(3):513–537zbMATHCrossRefGoogle Scholar
  6. delÁlamo J, Jiménez J (2009) Estimation of turbulent convection velocities and corrections to Taylor’s approximation. J Fluid Mech 640:5–26MathSciNetCrossRefGoogle Scholar
  7. Duke D, Honnery D, Soria J (2010) A cross-correlation velocimetry technique for breakup of an annular liquid sheet. Exp Fluids 49:435–445CrossRefGoogle Scholar
  8. Duke D, Honnery D, Soria J (2011) A comparison of subpixel edge detection and correlation algorithms for the measurement of sprays. Int J Spray Comb Dyn 3(2):93–110CrossRefGoogle Scholar
  9. Hussain A (1983) Coherent structures-reality and myth. Phys Fluids 26:2816zbMATHCrossRefGoogle Scholar
  10. Hussain A (1986) Coherent structures and turbulence. J Fluid Mech 173:303–356CrossRefGoogle Scholar
  11. Lloyd N, Bau D (1997) Numerical linear algebra. Society for industrial mathematicsGoogle Scholar
  12. Lozano A, García-Olivares A, Dopazo C (1998) The instability growth leading to a liquid sheet breakup. Phys Fluids 10(9):2188–2197MathSciNetzbMATHCrossRefGoogle Scholar
  13. Moin P (2009) Revisiting Taylor’s hypothesis. J Fluid Mech 640:1–4MathSciNetzbMATHCrossRefGoogle Scholar
  14. Nash J (1990) Compact numerical methods for computers: linear algebra and function minimisation 2. Taylor & Francis, New YorkGoogle Scholar
  15. Priestley M (1996) Wavelets and time-dependent spectral analysis. J Time Ser Anal 17(1):85–103Google Scholar
  16. Rowley C, Mezić I, Bagheri S (2009) Spectral analysis of nonlinear flows. J Fluid Mech 641:115–127MathSciNetzbMATHCrossRefGoogle Scholar
  17. Schmid P (2010) Dynamic mode decomposition of numerical and experimental data. J Fluid Mech 656:5–28MathSciNetzbMATHCrossRefGoogle Scholar
  18. Schmid P (2011) Application of the dynamic mode decomposition to experimental data. Exp Fluids 50(4):1123–1130CrossRefGoogle Scholar
  19. Schmid P, Henningson D (2001) Stability and transition in shear flows. Springer, New YorkzbMATHCrossRefGoogle Scholar
  20. Schneider K, Vasilyev O (2010) Wavelet methods in computational fluid dynamics. Annu Rev Fluid Mech 42:473–503MathSciNetCrossRefGoogle Scholar
  21. Simens M, Jiménez J, Hoyas S, Mizuno Y (2009) A high-resolution code for turbulent boundary layers. J Comput Phys 228(11):4218–4231zbMATHCrossRefGoogle Scholar

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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