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Performance assessment of the maximum likelihood ensemble filter and the ensemble Kalman filters for nonlinear problems

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Abstract

This study presents a thorough investigation of the performance comparison of three ensemble data assimilation (DA) methods, including the maximum likelihood ensemble filter (MLEF), the ensemble Kalman filter (EnKF), and the iterative EnKF (IEnKF), with respect to solution accuracy and computational efficiency for nonlinear problems. The convection–diffusion–reaction (CDR) problem is first tested, and then, the chaotic Lorenz 96 model is solved. Both linear and nonlinear observation operators are considered. The study demonstrates that MLEF consistently produces more accurate and efficient solution than the other two methods and provides more information on both states and their uncertainties. The IEnKF and MLEF are used to estimate model parameters and uncertainty in initial conditions using a nonlinear observation operator. The assimilation performance is assessed based on the quality metrics, such as the squared true error, the trace of the error covariance matrix, and the root-mean-square (RMS) error. Based on these DA performance assessments, MLEF demonstrates better convergence and higher accuracy. Results of the CDR problem show significant improvements in the estimate of model parameters and the solution accuracy by MLEF compared to the EnKF family. This study provides evidence supporting the choice of MLEF when solving large nonlinear problems.

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Data availability

Data sharing is not applicable to this article as no real-world datasets were generated or analyzed during the current study. The synthesized datasets generated during and/or analyzed during the present study are available from the corresponding author on reasonable request.

References

  1. Anderson, J.L.: An ensemble adjustment Kalman filter for data assimilation. Mon. Weather Rev. 129, 2884–2903 (2001)

    Article  Google Scholar 

  2. Bannister, R.N.: A review of operational methods of variational and ensemble-variational data assimilation. Q. J. R. Meteorol. Soc. 143, 607–633 (2017)

    Article  Google Scholar 

  3. Bell, B.M.: The iterated Kalman smoother as a Gauss–Newton method. SIAM J. Optim. 4, 626–636 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bishop, C., Etherton, J., Majmudar, S.J.: Adaptive sampling with the ensemble transform Kalman filter. Part I: theoretical aspect. Mon. Weather Rev. 129, 420–430 (2001)

    Article  Google Scholar 

  5. Evensen, G.: The ensemble Kalman filter: theoretical formulation and practical implementation. Ocean Dyn. 53, 343–367 (2003)

    Article  Google Scholar 

  6. Evensen, G.: Data Assimilation: The Ensemble Kalman Filter. Springer, Heidelberg (2007)

    MATH  Google Scholar 

  7. Gao, X., Liu, J.: Data assimilation for computational fluid dynamics modeling of combustion. In: 26th International Conference on Parallel Computational Fluid Dynamics (2014)

  8. Gao, X., Wang, Y., Overton, N., May, I., Tu, X.: Estimation of flame speed model parameter using ensemble Kalman filter algorithm. In: Fall Meeting of the Western States Section of the Combustion Institute (2015). WSSCI 2015-134IE-0013

  9. Gao, X., Wang, Y., Overton, N., May, I., Tu, X.: Data assimilated computational fluid dynamics algorithm for combustion. In: AIAA 2016-1810, 54th AIAA Aerospace Sciences Meeting (2016)

  10. Gao, X., Wang, Y., Overton, N., Zupanski, M., Tu, X.: Properties of a modified ensemble Kalman filter algorithm for combustion application. In: AIAA 2016-3484, 46th AIAA Fluid Dynamics Conference (2016)

  11. Gao, X., Wang, Y., Overton, N., Zupanski, M., Tu, X.: Data-assimilated computational fluid dynamics modeling of convection–diffusion–reaction problems. J. Comput. Sci. 21, 38–59 (2017)

    Article  MathSciNet  Google Scholar 

  12. Haug, E.J., Arora, J.S., Matsui, K.: A steepest-descent method for optimization of mechanical systems. J. Optim. Theory Appl. 19, 401–424 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  13. Houtekamer, P.L., Mitchell, H.L.: A sequential ensemble Kalman filter for atmospheric data assimilation. Am. Meteorol. Soc. 129, 123–137 (2001)

    Google Scholar 

  14. Hurst, C., Gao, X.: Increasing the finite limit of predictability in turbulence by data assimilation. Technical report, AIAA Science & Technology 2020 Forum (2020)

  15. Jazwinski, A.H.: Stochastic Processes and Filtering Theory. Academic Press, New York (1970)

    MATH  Google Scholar 

  16. Kalnay, E.: Atmospheric Modeling, Data Assimilation and Predictabilty, 1st edn. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  17. Kleist, D.T., Ide, K.: An OSSE-based evaluation of hybrid variational-ensemble data assimilation for the NCEP GFS. Part I: system description and 3d-hybrid results. Mon. Weather Rev. 143, 433–451 (2015)

    Article  Google Scholar 

  18. Labahn, J., Wu, H., Coriton, B., Frnak, J., Ihme, M.: Data assimilation using high-speed measurement and LES to examine local extinction events in turbulent flames. Proc. Combust. Inst. 37, 2259–2266 (2019)

    Article  Google Scholar 

  19. Le Dimet, F.-X., Talagrand, O.: Variational algorithms for analysis and assimilation of meteorological observations: theoretical aspects. Tellus A 38A, 97–110 (1986)

    Article  Google Scholar 

  20. Li, Z., Navon, I.M.: Optimality of variational data assimilation and its relationship with the Kalman filter and smoother. Q. J. R. Meteorol. Soc. 127, 661–683 (2001)

    Article  Google Scholar 

  21. Lorenz, A.C.: Analysis methods for numerical weather prediction. Q. J. R. Meteorol. Soc. 112, 1177–1194 (1986)

    Article  Google Scholar 

  22. Lorentzen, R.J., Naevdal, G.: An iterative ensemble Kalman filter. IEEE Trans. Autom. Control 56(8), 1990–1995 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Luenberger, D.L.: Linear and Non-linear Programming, 2nd edn. Addison-Wesley, Boston (1984)

    MATH  Google Scholar 

  24. Navon, I.M., Wright, S.J.: Conjugate gradient methods for large-scale minimization in meteorology. Mon. Weather Rev. 115, 1479–1502 (1987)

    Article  Google Scholar 

  25. Neave, H.R.: On using the box-muller transformation with multiplicative congruential pseudo-random number generators. J. Roy. Stat. Soc. Ser. C (Appl. Stat.) 21(1), 92–97 (1973)

    Google Scholar 

  26. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, Berlin (1999)

    Book  MATH  Google Scholar 

  27. Parrish, D.F., Derber, J.C.: The national meteorological center’s spectral statistical interpolation analysis system. Mon. Weather Rev. 120, 1747–1763 (1992)

    Article  Google Scholar 

  28. Rabier, F., Jarvinen, H., Klinker, E., Simmons, A.J.: The ECMWF operational implementation of four-dimensional variational assimilation. I: experimental results with simplified physics. Q. J. R. Meteorol. Soc. 126, 1143–1170 (2007)

    Article  Google Scholar 

  29. Sakov, P., Oliver, D.S., Bertino, L.: An iterative EnKF for strongly nonlinear systems. Mon. Weather Rev. 140, 1988–2004 (2012)

    Article  Google Scholar 

  30. Suzuki, K., Zupanski, M.: Uncertainty in solid precipitation and snow depth prediction for Siberia using the Noah and Noah-MP land surface models. Front. Earth Sci. (2018). https://doi.org/10.1007/s11707-018-0691-2

    Article  Google Scholar 

  31. Talagrand, O.: Assimilation of observations-an introduction. J. Meteorol. Soc. Jpn 75, 191–209 (1997)

    Article  Google Scholar 

  32. Talagrand, O., Courtier, P.: Variational assimilation of meteorological observations with the adjoint vorticity equation. I: Theory. J. Meteorol. Soc. 113, 1311–1328 (1987)

    Article  Google Scholar 

  33. VanLeeuwen, P.J.: Particle filtering in geophysical system. Mon. Weather Rev. 2137, 4089–4114 (2009)

    Article  Google Scholar 

  34. Wang, X.: Application of the WRF hybrid ETKF-3DVAR data assimilation system for hurricane track forecasts. Am. Meteorol. Soc. 26, 868–884 (2011)

    Google Scholar 

  35. Wang, Y., Gao, X.: Estimation of multiple model parameters using a data-assimilated CFD algorithm. In: Fall Meeting of the Western States Section of the Combustion Institute (2017). WSSCI 2017-29IT-0076

  36. Wang, Y., Gao, X.: Covariance correction to the ensemble Kalman filter in a data-assimilated CFD algorithm for the convection–diffusion–reaction equation. In: AIAA 2017-3630, 23rd AIAA Computational Fluid Dynamics Conference (2017). https://doi.org/10.2514/6.2017-3630

  37. Wang, X., Barker, D.M., Snyder, C., Hamill, T.M.: A hybrid ETKF-3DVAR data assimilation scheme for the WRF model. Part I: observation system simulation experiment. Mon. Weather Rev. 136, 5116–5131 (2008)

    Article  Google Scholar 

  38. Wang, Y., Walters, S., Guzik, S., Gao, X.: CFD modeling of bluff-body stabilized premixed flames with data assimilation. In: AIAA 2020-0352, 2020 AIAA Science and Technology Forum (January 2020)

  39. Wang, Y., Guzik, S., Zupanski, M., Gao, X.: The maximum likelihood ensemble filter for computational flame and fluid dynamics. IMA J. Appl. Math. 86(4), 631–661 (2021). https://doi.org/10.1093/imamat/hxab010

    Article  MathSciNet  MATH  Google Scholar 

  40. Whitaker, J.S., Hamill, T.M.: Ensemble data assimilation without perturbed observations. Mon. Weather Rev. 130, 1913–1924 (2002)

    Article  Google Scholar 

  41. Xiao, H., Wu, J.-L., Wang, J.-X., Sun, R., Roy, C.J.: Quantifying and reducing model-form uncertainties in Reynolds-averaged Navier–Stokes simulations: a data-driven, physics-informed Bayesian approach. J. Comput. Phys. 324, 115–136 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  42. Yang, S., Kalnay, E., Hunt, B.: Handling nonlinearity in an ensemble Kalman filter: experiments with the three-variable Lorenz model. Mon. Weather Rev. 140, 2628–2646 (2012)

    Article  Google Scholar 

  43. Zupanski, M.: Regional four-dimensional variational data assimilation in a quasi-operational forecasting environment. Mon. Weather Rev. 121, 2396–2408 (1993)

    Article  Google Scholar 

  44. Zupanski, M.: Maximum likelihood ensemble filter: theoretical aspects. Mon. Weather Rev. 133, 1710–1726 (2005)

    Article  Google Scholar 

  45. Zupanski, M., Navon, I.M., Zupanski, D.: The maximum likelihood ensemble filter as a non-differentiable minimization algorithm. Q. J. R. Meteorol. Soc. 113(D20110) (2008)

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Funding

National Science Foundation (1723191 and 1723066).

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Authors and Affiliations

  1. Computational Fluid Dynamics and Propulsion Laboratory, Colorado State University, Fort Collins, 80523, CO, USA

    Yijun Wang & Xinfeng Gao

  2. Cooperative Institution for Research in the Atmosphere, Colorado State University, Fort Collins, 80523, CO, USA

    Milija Zupanski

  3. Department of Mathematics, University of Kansas, Lawrence, 66045, KS, USA

    Xuemin Tu

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  1. Yijun Wang
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  2. Milija Zupanski
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Correspondence to Yijun Wang.

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Milija Zupanski, Xuemin Tu and Xinfeng Gao these authors contributed equally to this work.

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Wang, Y., Zupanski, M., Tu, X. et al. Performance assessment of the maximum likelihood ensemble filter and the ensemble Kalman filters for nonlinear problems. Res Math Sci 9, 62 (2022). https://doi.org/10.1007/s40687-022-00359-7

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