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Nudged Particle Filters in Multiscale Chaotic Systems with Correlated Sensor Noise

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Modeling, Stochastic Control, Optimization, and Applications

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 164))

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Abstract

In this work we present recent and new results for the theory and algorithms of efficient estimation of the coarse-grain dynamics of a multiscale chaotic dynamical system, where observations may be limited both spatial and in time, and the observations are correlated with the slow states. The rigorous mathematical statement and convergence result with a rate of convergence to the reduced order filter problem is given for the case of correlated sensor noise. Based on this result, algorithms for efficient numerical solution of the filtering problem for the coarse-grain dynamics are provided. We then address a second issue, which presents itself in the case of chaotic systems and degrades particle filtering performance; the growth of small errors at an exponential rate. We solve this problem by introducing an optimal control problem for the solution of the proposal distribution and develop a numerical algorithm for it’s solution. The algorithms developed in this work are demonstrated on the widely used multiscale chaotic Lorenz 1996 model, that mimics mid-latitude convection.

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References

  1. Anderson, J.L.: An Ensemble Adjustment Kalman Filter for Data Assimilation. Monthly Weather Review 129, 2884–2903 (2001)

    Article  Google Scholar 

  2. Arulampalam, M.S., Maskell, S., Gordon, N., Clapp, T.: A tutorial on particle filters for online nonlinear/non-gaussian bayesian tracking. IEEE Transactions on Signal Processing 50(2), 174–188 (2002). https://doi.org/10.1109/78.978374

    Article  Google Scholar 

  3. Bain, A., Crisan, D.: Fundamentals of Stochastic Filtering. Springer (2009)

    Google Scholar 

  4. Beeson, R.: Reduced order nonlinear filters for multi-scale systems with correlated sensor noise. In: 21st International Conference on Information Fusion (FUSION) 2018 (FUSION 2018). Cambridge, United Kingdom (Great Britain) (2018)

    Google Scholar 

  5. Beeson, R., Namachchivaya, N.S., Perkowski, N.: Dimensional reduction in nonlinear filter- ing: Multi-scale systems and correlated sensor noise. In Preparation

    Google Scholar 

  6. Crisan, D.: Particle approximations for a class of stochastic partial differential equations. Ap- plied Mathematics and Optimization 54, 293–314 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Del Moral, P., Miclo, L.: Branching and Interacting Particle Systems Approximations of Feynman-Kac Formulae with Applications to Non-linear Filtering. In: Se´minaire de Prob- abilite´s XXXIV, Lecture Notes in Mathematics, vol. 1729, pp. 1–145. Springer-Verlag Berlin (2000)

    Google Scholar 

  8. Doucet, A.: On sequential simulation-based methods for Bayesian filtering. Tech. rep., Cam- bridge University (1998)

    Google Scholar 

  9. E, W., Liu, D., Vanden-Eijnden, E.: Analysis of multiscale methods for stochastic differential equations. Communication on Pure and Applied Mathematics 58, 1544–1585 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fatkullin, I., Vanden-Eijnden, E.: A computational strategy for multiscale systems. Journal of Computational Physics 200(2), 605–638 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fleming, W.H.: Exit probabilities and optimal stochastic12. control. Applied Mathematics and Optimization 4, 329–346 (1978)

    Google Scholar 

  12. Fleming, W.H.: PLogarithmic transformations and stochastic control, pp. 131–141. Springer Berlin Heidelberg, Berlin, Heidelberg (1982). https://doi.org/10.1007/bfb0004532. URL http://dx.doi.org/10.1007/BFb0004532

  13. Gordon, N.J., Salmond, D.J., Smith, A.F.M.: Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEEE Proceedings F 140(2), 107–113 (1993)

    Article  Google Scholar 

  14. Gustafsson, F., Saha, S.: Particle filtering with dependent noise. In: 2010 13th International Conference on Information Fusion, pp. 1–4 (2010). https://doi.org/10.1109/icif.2010.5712052

  15. Harlim, J., Kang, E.L.: Filtering partially observed multiscale systems with heterogeneous multiscale methods-based reduced climate models. Mon. Wea. Rev. 140(3), 860–873 (2012)

    Article  Google Scholar 

  16. Imkeller, P., Namachchivaya, N.S., Perkowski, N., Yeong, H.C.: Dimensional reduction in nonlinear filtering: A homogenization approach. Ann. Appl. Probab. 23(6), 2290–2326 (2013). https://doi.org/10.1214/12-aap901. URL http://dx.doi.org/10.1214/12-AAP901

    Article  MathSciNet  MATH  Google Scholar 

  17. Kang, E.L., Harlim, J.: Filtering partially observed multiscale systems with heterogeneous multiscale methods-based reduced climate models. Monthly Weather Review 140, 860–873 (2012)

    Article  Google Scholar 

  18. Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer-Verlag New York Inc. (1988)

    Google Scholar 

  19. Kushner, H.J.: Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems. Birkha¨user, Boston (1990)

    Book  MATH  Google Scholar 

  20. van Leeuwen, P.J.: Nonlinear data assimilation in geosciences: an extremely efficient particle filter. Quart. J. Royal Meteor. Soc. 136, 1991–1999 (2010)

    Google Scholar 

  21. Lingala, N., Perkowski, N., Yeong, H.C., Namachchivaya, N.S., Rapti, Z.: Probabilistic Engi- neering Mechanics. Probabilistic Engineering Mechanics 37(C), 160–169 (2014)

    Article  Google Scholar 

  22. Lingala, N., Sri Namachchivaya, N., Perkowski, N., Yeong, H.C.: Particle filtering in high- dimensional chaotic systems. Chaos: An Interdisciplinary Journal of Nonlinear Science 22(4), 047,509 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lorenz, E.N.: Predictability: A problem partly solved. In: Proceedings of the ECMWF Semi- nar on Predictability, vol. 1, pp. 1–18. ECMWF (1996)

    Google Scholar 

  24. Majda, A., Timofeyev, I., Vanden-Eijnden, E.: A Mathematical Framework for Stochastic Cli- mate Models. Comm. Pure Appl. Math. 54, 891–974 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Majda, A.J., Timofeyev, I., Vanden-Eijnden, E.: Systematic Strategies for Stochastic Mode Reduction in Climate. J. of Atmospheric Sciences 60, 1705–1722 (2003)

    Article  MathSciNet  Google Scholar 

  26. Øksendal, B.: An introduction to Malliavin calculus with applications to economics (1997). Lecture notes

    Google Scholar 

  27. Ott, E., Hunt, B.R., Szunyogh, I., Zimin, A.V., Kostelich, E.J., Corazza, M., Kalnay, E., Patil, D.J., Yorke, J.A.: A local ensemble Kalman filter for atmospheric data assimilation. Tellus 56A, 415–428 (2004)

    Article  Google Scholar 

  28. Papanicolaou, G.C., Stroock, D., Varadhan, S.R.S.: Martingale approach to some limit the- orems. In: Papers from the Duke Turbulence Conference. Duke University, Durham, North Carolina (1976)

    Google Scholar 

  29. Pardoux, E.: Stochastic partial differential equations and filtering of diffusion pro- cesses. Stochastics 3(1-4), 127–167 (1980). https://doi.org/10.1080/17442507908833142. URL http://www.tandfonline.com/doi/abs/10.1080/17442507908833142

    Article  MathSciNet  Google Scholar 

  30. Pardoux, E., Peng, S.: Backward doubly stochastic differential equations and systems of quasi- linear SPDEs. Probability Theory and Related Fields (98), 209–227 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  31. Pardoux, E., Veretennikov, A.Y.: On Poisson Equation and Diffusion Approximation 2. The Annals of Probability 31(3), 1166–1192 (2003)

    Google Scholar 

  32. Park, J.H., Namachchivaya, N.S., Yeong, H.C.: Particle filters in a multiscale environment: Homogenized hybrid particle filter. Journal of Applied Mechanics 78 (2011)

    Article  Google Scholar 

  33. Rozovskii, B.L.: Stochastic Evolution System: Linear Theory and Aplications to Non-linear Filtering. Kluwer Academic Publishers, Dordrecht (1990)

    Google Scholar 

  34. Saha, S., Gustafsson, F.: Marginalized particle filter for dependent gaussian noise processes. In: 2012 IEEE Aerospace Conference, pp. 1–6 (2012). https://doi.org/10.1109/aero.2012.6187212

  35. Vanden-Eijnden, E.: Numerical techniques for multi-scale dynamical systems with stochastic effects. Communications in Mathematical Sciences 1(2), 385–391 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Veretennikov, A.Y.: On polynomial mixing bounds for stochastic differential equations. Stochastic Processes and their Applications 70, 115–127 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. Yang, T., Laugesen, R.S., Mehta, P.G., Meyn, S.P.: Multivariable feedback particle filter. Au- tomatica 71, 10–23 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. Yeong, H.C., Beeson, R., Namachchivaya, N.S., Perkowski, N.: Particle filters with nudging in multiscale chaotic system: with application to the lorenz-96 atmospheric model. In Sub- mission (2018)

    Google Scholar 

  39. Yeong, H.C., Park, J.H., Namachchivaya, N.S.: Particle filters in a multiscale en- vironment: with application to the lorenz-96 atmospheric model. Stochastics and Dynamics 11(02n03), 569–591 (2011). https://doi.org/10.1142/s0219493711003450. URL http://www.worldscientific.com/doi/abs/10.1142/S0219493711003450

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

    Ryne Beeson

  2. Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada

    N. Sri Namachchivaya

Authors
  1. Ryne Beeson
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  2. N. Sri Namachchivaya
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Corresponding author

Correspondence to Ryne Beeson .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI, USA

    George Yin

  2. Department of Mathematics, University of Georgia, Athens, GA, USA

    Qing Zhang

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Beeson, R., Sri Namachchivaya, N. (2019). Nudged Particle Filters in Multiscale Chaotic Systems with Correlated Sensor Noise. In: Yin, G., Zhang, Q. (eds) Modeling, Stochastic Control, Optimization, and Applications. The IMA Volumes in Mathematics and its Applications, vol 164. Springer, Cham. https://doi.org/10.1007/978-3-030-25498-8_2

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