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Flow state estimation in the presence of discretization errors

Published online by Cambridge University Press:  11 March 2020

Andre F. C. da Silva Open the ORCID record for Andre F. C. da Silva [Opens in a new window]  and
Tim Colonius Open the ORCID record for Tim Colonius [Opens in a new window]
Andre F. C. da Silva*
Affiliation:
Department of Civil and Mechanical Engineering, California Institute of Technology, Pasadena, CA 91101, USA
Tim Colonius
Affiliation:
Department of Civil and Mechanical Engineering, California Institute of Technology, Pasadena, CA 91101, USA
*
†Email address for correspondence: [email protected]
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Abstract

Ensemble data assimilation methods integrate measurement data and computational flow models to estimate the state of fluid systems in a robust, scalable way. However, discretization errors in the dynamical and observation models lead to biased forecasts and poor estimator performance. We propose a low-rank representation for this bias, whose dynamics is modelled by data-informed, time-correlated processes. State and bias parameters are simultaneously corrected online with the ensemble Kalman filter. The proposed methodology is then applied to the problem of estimating the state of a two-dimensional flow at modest Reynolds number using an ensemble of coarse-mesh simulations and pressure measurements at the surface of an immersed body in a synthetic experiment framework. Using an ensemble size of 60, the bias-aware estimator is demonstrated to achieve at least 70 % error reduction when compared to its bias-blind counterpart. Strategies to determine the bias statistics and their impact on the estimator performance are discussed.

JFM classification

Flow Control: Control theory Mathematical Foundations: Computational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 890 , 10 May 2020 , A10
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Ahuja, S. & Rowley, C. W. 2010 Feedback control of unstable steady states of flow past a flat plate using reduced-order estimators. J. Fluid Mech. 645, 447–478.CrossRefGoogle Scholar
Anderson, J. L. & Anderson, S. L. 1999 A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and forecasts. Mon. Weath. Rev. 127 (12), 2741–2758.2.0.CO;2>CrossRefGoogle Scholar
Bell, B. M. & Cathey, F. W. 1993 The iterated Kalman filter update as a Gauss–Newton method. IEEE Trans. Autom. Control 38 (2), 294–297.CrossRefGoogle Scholar
Bengtsson, T., Snyder, C. & Nychka, D. 2003 Toward a nonlinear ensemble filter for high-dimensional systems. J. Geophys. Res. 108 (D24), doi:10.1029/2002JD002900.CrossRefGoogle Scholar
Bishop, C. H., Etherton, B. J. & Majumdar, S. J. 2001 Adaptive sampling with the ensemble transform Kalman filter. Part I. Theoretical aspects. Mon. Weath. Rev. 129 (3), 420–436.2.0.CO;2>CrossRefGoogle Scholar
Burgers, G., Jan van Leeuwen, P. & Evensen, G. 1998 Analysis scheme in the ensemble Kalman filter. Mon. Weath. Rev. 126 (6), 1719–1724.2.0.CO;2>CrossRefGoogle Scholar
Chui, C. K. & Chen, G. 2009 Kalman Filtering with Real Time Applications. Springer.Google Scholar
Cohn, S. E. 1997 An introduction to estimation theory. J. Met. Soc. Japan II 75 (1B), 257–288.Google Scholar
Cohn, S. E. & Parrish, D. F. 1991 The behavior of forecast error covariances for a Kalman filter in two dimensions. Mon. Weath. Rev. 119 (8), 1757–1785.2.0.CO;2>CrossRefGoogle Scholar
Colburn, C. H., Cessna, J. B. & Bewley, T. R. 2011 State estimation in wall-bounded flow systems. Part 3. The ensemble Kalman filter. J. Fluid Mech. 682, 289–303.CrossRefGoogle Scholar
Colonius, T. & Taira, K. 2008 A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions. Comput. Meth. Appl. Mech. Engng 197 (25), 2131–2146.CrossRefGoogle Scholar
Darakananda, D., da Silva, A. F. C., Colonius, T. & Eldredge, J. D. 2018 Data-assimilated low-order vortex modeling of separated flows. Phys. Rev. Fluids 3, 124701.CrossRefGoogle Scholar
Dee, D. P. 1995 On-line estimation of error covariance parameters for atmospheric data assimilation. Mon. Weath. Rev. 123 (4), 1128–1145.2.0.CO;2>CrossRefGoogle Scholar
Dee, D. P. & Da Silva, A. M. 1998 Data assimilation in the presence of forecast bias. Q. J. R. Meteorol. Soc. 124 (545), 269–295.CrossRefGoogle Scholar
Drecourt, J.-P., Madsen, H. & Rosbjerg, D. 2006 Bias aware Kalman filters: comparison and improvements. Adv. Water Resour. 29 (5), 707–718.CrossRefGoogle Scholar
Evensen, G. 1994 Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. J. Geophys. Res. 99 (C5), 10143–10162.CrossRefGoogle Scholar
Evensen, G. 2003 The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn. 53 (4), 343–367.CrossRefGoogle Scholar
Evensen, G. 2004 Sampling strategies and square root analysis schemes for the EnKF. Ocean Dyn. 54 (6), 539–560.CrossRefGoogle Scholar
Evensen, G. 2009 Data Assimilation: The Ensemble Kalman Filter. Springer.CrossRefGoogle Scholar
Evensen, G. & van Leeuwen, P. J. 1996 Assimilation of geosat altimeter data for the Agulhas current using the ensemble Kalman filter with a quasigeostrophic model. Mon. Weath. Rev. 124 (1), 85–96.2.0.CO;2>CrossRefGoogle Scholar
Flinois, T. L. B. & Morgans, A. S. 2016 Feedback control of unstable flows: a direct modelling approach using the eigensystem realisation algorithm. J. Fluid Mech. 793, 41–78.CrossRefGoogle Scholar
Friedland, B. 1969 Treatment of bias in recursive filtering. IEEE Trans. Autom. Control 14 (4), 359–367.CrossRefGoogle Scholar
Gelb, A. 1974 Applied Optimal Estimation. MIT Press.Google Scholar
Giles, M. B. 2015 Multilevel Monte Carlo methods. Acta Numerica 24, 259–328.CrossRefGoogle Scholar
Gu, Y. & Oliver, D. S. 2007 An iterative ensemble Kalman filter for multiphase fluid flow data assimilation. SPE J. 12 (04), 438–446.CrossRefGoogle Scholar
Henderson, H. V. & Searle, S. R. 1981 On deriving the inverse of a sum of matrices. SIAM Rev. 23 (1), 53–60.CrossRefGoogle Scholar
Hoel, H., Law, K. J. H. & Tempone, R. 2016 Multilevel ensemble Kalman filtering. SIAM J. Numer. Anal. 54 (3), 1813–1839.CrossRefGoogle Scholar
Holmes, P., Lumley, J. L., Berkooz, G. & Rowley, C. W. 2012 Turbulence, Coherent Structures, Dynamical Systems and Symmetry, 2nd edn. Cambridge University Press.CrossRefGoogle Scholar
Houtekamer, P. L. & Mitchell, H. L. 2001 A sequential ensemble Kalman filter for atmospheric data assimilation. Mon. Weath. Rev. 129 (1), 123–137.2.0.CO;2>CrossRefGoogle Scholar
Houtekamer, P. L. & Zhang, F. 2016 Review of the ensemble Kalman filter for atmospheric data assimilation. Mon. Weath. Rev. 144 (12), 4489–4532.CrossRefGoogle Scholar
Jazwinski, A. H. 1970 Stochastic Processes and Filtering Theory. Courier Corporation.Google Scholar
Julier, S. J. & Uhlmann, J. K. 2004 Unscented filtering and nonlinear estimation. Proc. IEEE 92 (3), 401–422.CrossRefGoogle Scholar
Kato, H., Yoshizawa, A., Ueno, G. & Obayashi, S. 2015 A data assimilation methodology for reconstructing turbulent flows around aircraft. J. Comput. Phys. 283 (C), 559–581.CrossRefGoogle Scholar
Kikuchi, R., Misaka, T. & Obayashi, S. 2015 Assessment of probability density function based on POD reduced-order model for ensemble-based data assimilation. Fluid Dyn. Res. 47 (5), 051403.CrossRefGoogle Scholar
Law, K., Stuart, A. & Zygalakis, K. 2015 Data Assimilation: A Mathemathical Introduction. Springer.CrossRefGoogle Scholar
Lawson, W. G. & Hansen, J. A. 2004 Implications of stochastic and deterministic filters as ensemble-based data assimilation methods in varying regimes of error growth. Mon. Weath. Rev. 132 (8), 1966–1981.2.0.CO;2>CrossRefGoogle Scholar
van Leeuwen, P. J. 1999 Comment on ‘Data assimilation using an ensemble Kalman filter technique’. Mon. Weath. Rev. 127 (6), 1374–1377.2.0.CO;2>CrossRefGoogle Scholar
Liska, S. & Colonius, T. 2014 A parallel fast multipole method for elliptic difference equations. J. Comput. Phys. 278, 76–91.CrossRefGoogle Scholar
Liska, S. & Colonius, T. 2017 A fast immersed boundary method for external incompressible viscous flows using lattice Green’s functions. J. Comput. Phys. 331, 257–279.CrossRefGoogle Scholar
Mons, V., Chassaing, J.-C., Gomez, T. & Sagaut, P. 2016 Reconstruction of unsteady viscous flows using data assimilation schemes. J. Comput. Phys. 316, 255–280.CrossRefGoogle Scholar
Papadakis, N., Mémin, É., Cuzol, A. & Gengembre, N. 2010 Data assimilation with the weighted ensemble Kalman filter. Tellus A 62 (5), 673–697.CrossRefGoogle Scholar
Peherstorfer, B., Willcox, K. & Gunzburger, M. 2018 Survey of multifidelity methods in uncertainty propagation, inference, and optimization. SIAM Rev. 60 (3), 550–591.CrossRefGoogle Scholar
Rabier, F. 2005 Overview of global data assimilation developments in numerical weather-prediction centres. Q. J. R. Meteorol. Soc. 131 (613), 3215–3233.CrossRefGoogle Scholar
Sakov, P., Evensen, G. & Bertino, L. 2009 Asynchronous data assimilation with the EnKF. Tellus A 62 (1), 24–29.CrossRefGoogle Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.CrossRefGoogle Scholar
da Silva, A. F. C.2019 An EnKF-based flow state estimator for aerodynamic problems. PhD thesis, California Institute of Technology.Google Scholar
da Silva, A. F. C. & Colonius, T. 2018 Ensemble-based state estimator for aerodynamic flows. AIAA J. 56, 2568–2578.CrossRefGoogle Scholar
Taira, K. & Colonius, T. 2007 The immersed boundary method: a projection approach. J. Comput. Phys. 225 (2), 2118–2137.CrossRefGoogle Scholar
Trémolet, Y. 2006 Accounting for an imperfect model in 4D-Var. Q. J. R. Meteorol. Soc. 132 (621), 2483–2504.CrossRefGoogle Scholar
Whitaker, J. S. & Hamill, T. M. 2012 Evaluating methods to account for system errors in ensemble data assimilation. Mon. Weath. Rev. 140 (9), 3078–3089.CrossRefGoogle Scholar
Zupanski, M. 2005 Maximum likelihood ensemble filter: theoretical aspects. Mon. Weath. Rev. 133 (6), 1710–1726.CrossRefGoogle Scholar