Dynamic mode decomposition for non-uniformly sampled data

We propose an original approach to estimate dynamic mode decomposition (DMD) modes from non-uniformly sampled data. The proposed strategy processes a time-resolved sequence of flow snapshots in three steps. First, a reduced-order modeling of the non-missing data is made by proper orthogonal decomposition to obtain a low-order description of the state space. Second, the missing data are determined with maximum likelihood by coupling a linear dynamical state-space model with the Expectation-Maximization algorithm. Third, the DMD modes are finally estimated on the reconstructed data with a multiple linear regression method called orthonormalized partial least squares regression. This methodology is assessed for the flow past a NACA0012 airfoil at 20° of angle of attack and a Reynolds number of 103. The flow measurements are obtained with time-resolved particle image velocimetry and artificially subsampled at different ratios of missing data. The results show that the proposed method can reproduce the dominant DMD modes and the main structures of the flow fields for 50 and 75 % of missing data.

[1]  H. Wold.  Path Models with Latent Variables: The NIPALS Approach , 1975 .

[2]  H. Vinod.  Canonical ridge and econometrics of joint production , 1976 .

[3]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[4]  B. Moore.  Principal component analysis in linear systems: Controllability, observability, and model reduction , 1981 .

[5]  R. Manne.  Analysis of two partial-least-squares algorithms for multivariate calibration , 1987 .

[6]  L. Sirovich.  Turbulence and the dynamics of coherent structures. I. Coherent structures , 1987 .

[7]  Andrew Harvey,et al.  Forecasting, Structural Time Series Models and the Kalman Filter , 1990 .

[8]  J. R. Rohlicek,et al.  ML estimation of a stochastic linear system with the EM algorithm and its application to speech recognition , 1993, IEEE Trans. Speech Audio Process..

[9]  J. Friedman,et al.  A Statistical View of Some Chemometrics Regression Tools , 1993 .

[10]  P. Holmes,et al.  The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows , 1993 .

[11]  B. Silverman,et al.  Canonical correlation analysis when the data are curves. , 1993 .

[12]  Geoffrey E. Hinton,et al.  Parameter estimation for linear dynamical systems , 1996 .

[13]  Michael I. Jordan,et al.  On Convergence Properties of the EM Algorithm for Gaussian Mixtures , 1996, Neural Computation.

[14]  H. Knutsson,et al.  A Unified Approach to PCA, PLS, MLR and CCA , 1997 .

[15]  Sijmen de Jong,et al.  Multiway calibration in 3D QSAR , 1997 .

[16]  Karl J. Friston,et al.  Characterizing the Response of PET and fMRI Data Using Multivariate Linear Models , 1997, NeuroImage.

[17]  Jeff A. Bilmes,et al.  A gentle tutorial of the em algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov models , 1998 .

[18]  John Hulland,et al.  Use of partial least squares (PLS) in strategic management research: a review of four recent studies , 1999 .

[19]  Rong Fung Huang,et al.  Surface flow and vortex shedding of an impulsively started wing , 2001, Journal of Fluid Mechanics.

[20]  A. McIntosh,et al.  Spatiotemporal analysis of experimental differences in event-related potential data with partial least squares. , 2001, Psychophysiology.

[21]  Danh V. Nguyen,et al.  Tumor classification by partial least squares using microarray gene expression data , 2002, Bioinform..

[22]  Michael I. Jordan,et al.  Kernel independent component analysis , 2003 .

[23]  John Shawe-Taylor,et al.  Canonical Correlation Analysis: An Overview with Application to Learning Methods , 2004, Neural Computation.

[24]  Roman Rosipal,et al.  Overview and Recent Advances in Partial Least Squares , 2005, SLSFS.

[25]  Laurent Cordier,et al.  Control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[26]  I. Mezić.  Spectral Properties of Dynamical Systems, Model Reduction and Decompositions , 2005 .

[27]  I. Helland.  Partial Least Squares Regression , 2006 .

[28]  S. Geer,et al.  Regularization in statistics , 2006 .

[29]  Gustavo Camps-Valls,et al.  Efficient Kernel Orthonormalized PLS for Remote Sensing Applications , 2008, IEEE Transactions on Geoscience and Remote Sensing.

[30]  Jieping Ye,et al.  Finite Domain Constraint Solver Learning , 2009, IJCAI.

[31]  P. Meliga,et al.  Dynamics and Control of Global Instabilities in Open-Flows: A Linearized Approach , 2010 .

[32]  P. Schmid,et al.  Dynamic mode decomposition of numerical and experimental data , 2008, Journal of Fluid Mechanics.

[33]  H. Sung,et al.  Dynamic mode decomposition of turbulent cavity flows for self-sustained oscillations , 2011 .

[34]  Tahir Mehmood,et al.  A review of variable selection methods in Partial Least Squares Regression , 2012 .

[35]  Peter J. Schmid,et al.  Sparsity-promoting dynamic mode decomposition , 2012, 1309.4165.

[36]  Damon Honnery,et al.  An error analysis of the dynamic mode decomposition , 2011, Experiments in Fluids.

[37]  Peter J. Schmid,et al.  Decomposition of time-resolved tomographic PIV , 2012 .

[38]  Onofrio Semeraro,et al.  Analysis of time-resolved PIV measurements of a confined turbulent jet using POD and Koopman modes , 2012 .

[39]  P. Goulart,et al.  Optimal mode decomposition for unsteady flows , 2013, Journal of Fluid Mechanics.

[40]  Shervin Bagheri,et al.  Koopman-mode decomposition of the cylinder wake , 2013, Journal of Fluid Mechanics.

[41]  Jean-Michel Loubes,et al.  PLS: a new statistical insight through the prism of orthogonal polynomials , 2014, 1405.5900.

[42]  S. Bagheri.  Effects of weak noise on oscillating flows: Linking quality factor, Floquet modes, and Koopman spectrum , 2014 .

[43]  Clarence W. Rowley,et al.  Spectral analysis of fluid flows using sub-Nyquist-rate PIV data , 2014, Experiments in Fluids.

[44]  Réduction de modèle et contrôle d'écoulements , 2014 .

[45]  Clarence W. Rowley,et al.  Dynamic mode decomposition for large and streaming datasets , 2014, 1406.7187.

[46]  Steven L. Brunton,et al.  On dynamic mode decomposition: Theory and applications , 2013, 1312.0041.

[47]  Mingming Zhang,et al.  Experimental study of flow separation control on a low-Re airfoil using leading-edge protuberance method , 2014 .

[48]  F. Guéniat,et al.  A dynamic mode decomposition approach for large and arbitrarily sampled systems , 2015 .