\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Analysis of the feedback particle filter with diffusion map based approximation of the gain

  • * Corresponding author: Sahani Pathiraja

    * Corresponding author: Sahani Pathiraja 
This research has been partially funded by the Deutsche Forschungsgemeinschaft (DFG)-Project-ID 318763901 - SFB1294.
Abstract Full Text(HTML) Related Papers Cited by
  • Control-type particle filters have been receiving increasing attention over the last decade as a means of obtaining sample based approximations to the sequential Bayesian filtering problem in the nonlinear setting. Here we analyse one such type, namely the feedback particle filter and a recently proposed approximation of the associated gain function based on diffusion maps. The key purpose is to provide analytic insights on the form of the approximate gain, which are of interest in their own right. These are then used to establish a roadmap to obtaining well-posedness and convergence of the finite $ N $ system to its mean field limit. A number of possible future research directions are also discussed.

    Mathematics Subject Classification: 60H10, 35Q93, 35J05, 60J60, 60J27, 62M05.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] K. Bergemann and S. Reich, An ensemble Kalman-Bucy filter for continuous data assimilation, Meteorologische Zeitschrift, 21 (2012), 213-219.  doi: 10.1127/0941-2948/2012/0307.
    [2] K. Berntorp and P. Grover, Data-driven gain computation in the feedback particle filter, Proceedings of the American Control Conference, (2016), 2711–2716. doi: 10.1109/ACC.2016.7525328.
    [3] T. Berry and J. Harlim, Variable bandwidth diffusion kernels, Applied and Computational Harmonic Analysis, 40 (2016), 68-96.  doi: 10.1016/j.acha.2015.01.001.
    [4] A. N. BishopP. del Moral and S. D. Pathiraja, Perturbations and projections of Kalman-Bucy semigroups, Stochastic Processes and their Applications, 128 (2018), 2857-2904.  doi: 10.1016/j.spa.2017.10.006.
    [5] F. BolleyA. Guillin and F. Malrieu, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, ESAIM: Mathematical Modelling and Numerical Analysis, 44 (2010), 867-884.  doi: 10.1051/m2an/2010045.
    [6] H. J. Brascamp and E. H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, Journal of Functional Analysis, 22 (1976), 366-389.  doi: 10.1016/0022-1236(76)90004-5.
    [7] E. A. CarlenD. Cordero-erausquin and E. H. Lieb, Asymmetric covariance estimates of Brascamp-Lieb type and related inequalities for log-concave measures, Ann. Inst. Henri Poincaré Probab. Stat., 49 (2013), 1-12.  doi: 10.1214/11-AIHP462.
    [8] P. Cattiaux and A. Guillin, On the Poincaré constant of log-concave measures, in Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2017-2019 Volume I (eds. B. Klartag), Springer International Publishing, (2020), 171–217. doi: 10.1007/978-3-030-36020-7_9.
    [9] R. R. Coifman and S. Lafon, Diffusion maps, Applied and Computational Harmonic Analysis, 21 (2006), 5-30.  doi: 10.1016/j.acha.2006.04.006.
    [10] D. Crisan and J. Xiong, Approximate McKean-Vlasov representations for a class of SPDEs, Stochastics, 82 (2010), 53-68.  doi: 10.1080/17442500902723575.
    [11] J. de WiljesS. Reich and W. Stannat, Long-time stability and accuracy of the ensemble Kalman-Bucy filter for fully observed processes and small measurement noise, SIAM J. Applied Dynamical Systems, 17 (2018), 1152-1181.  doi: 10.1137/17M1119056.
    [12] J. de Wiljes and X. Tong, Analysis of a localised nonlinear Ensemble Kalman Bucy Filter with complete and accurate observations, Nonlinearity, 33 (2020), 4752-4782.  doi: 10.1088/1361-6544/ab8d14.
    [13] P. del MoralA. Kurtzmann and J. Tugaut, On the stability and the uniform propagation of chaos of a class of extended ensemble Kalman–Bucy filters, SIAM Journal on Control and Optimization, 55 (2017), 119-155.  doi: 10.1137/16M1087497.
    [14] G. Evensen, Data Assimilation. The Ensemble Kalman Filter, Springer-Verlag, New York, 2009. doi: 10.1007/978-3-642-03711-5.
    [15] G. Evensen, The ensemble Kalman filter: Theoretical formulation and practical implementation, Ocean Dynamics, 53 (2003), 343-367.  doi: 10.1007/s10236-003-0036-9.
    [16] G. Evensen and P. J. van Leeuwen, An ensemble Kalman smoother for nonlinear dynamics, Monthly Weather Review, 128 (2000), 1852-1867.  doi: 10.1175/1520-0493(2000)128<1852:AEKSFN>2.0.CO;2.
    [17] A. M. KulikIntroduction to Ergodic Rates for Markov Chains and Processes, with Applications to Limit Theorems, Potsdam University Press, Germany, 2015. 
    [18] T. Lange and W. Stannat, Mean field limit of ensemble square root filters - discrete and continuous time, Foundations of Data Science, (2021). doi: 10.3934/fods.2021003.
    [19] T. Lange and W. Stannat, On the continuous time limit of the ensemble Kalman filter, Mathematics of Computation, 90 (2021), 233-265.  doi: 10.1090/mcom/3588.
    [20] R. S. LaugesenP. G. MehtaS. P. Meyn and M. Raginsky, Poissons equation in nonlinear filtering, SIAM Journal on Control and Optimization, 53 (2015), 501-525.  doi: 10.1137/13094743X.
    [21] F. Le Gland, V. Monbet and V.-D. Tran, Large sample asymptotics for the ensemble Kalman filter, The Oxford Handbook of Nonlinear Filtering, Oxford Univ. Press, Oxford, 2011, 598-631.
    [22] A. J. Majda and X. T. Tong, Robustness and accuracy of finite ensemble Kalman filters in large dimensions, Comm. Pure Appl. Math., 71 (2018), 892–937, arXiv: 1606.0932. doi: 10.1002/cpa.21722.
    [23] S. Y. Olmez, A. Taghvaei and P. G. Mehta, Deep FPF : Gain function approximation in high-dimensional setting, arXiv: 2010.01183v1.
    [24] S. Pathiraja, L2 convergence of smooth approximations of stochastic differential equations with unbounded coefficients, preprint, arXiv: 2011.13009.
    [25] S. Pathiraja, S. Reich and W. Stannat, McKean-Vlasov SDEs in nonlinear filtering, SIAM Journal on Control and Optimization, (accepted), (2021)
    [26] A. Radhakrishnan, A. Devraj and S. P. Meyn, Learning techniques for feedback particle filter design, Proceedings of the IEEE 55th Conference on Decision and Control, (2016), 5453–5459. doi: 10.1109/CDC.2016.7799106.
    [27] S. Reich, A dynamical systems framework for intermittent data assimilation, BIT Numerical Mathematics, 51 (2010), 235-249.  doi: 10.1007/s10543-010-0302-4.
    [28] C. Schillings and A. M. Stuart, Analysis of the ensemble Kalman filter for inverse problems, SIAM Journal on Numerical Analysis, 55 (2016), 1264-1290.  doi: 10.1137/16M105959X.
    [29] A. TaghvaeiP. G. Mehta and S. P. Meyn, Diffusion map-based algorithm for gain function approximation in the feedback particle filter, SIAM-ASA Journal on Uncertainty Quantification, 8 (2020), 1090-1117.  doi: 10.1137/19M124513X.
    [30] X. TongA. J. Majda and D. Kelly, Nonlinear stability and ergodicity of ensemble based Kalman filters, Nonlinearity, 29 (2016), 657-691.  doi: 10.1088/0951-7715/29/2/657.
    [31] J. Touboul, Propagation of chaos in neural fields, Annals of Applied Probability, 24 (2014), 1298-1328.  doi: 10.1214/13-AAP950.
    [32] C. L. Wormell and S. Reich, Spectral convergence of diffusion maps: Improved error bounds and an alternative normalization, SIAM Journal on Numerical Analysis, 59 (2021), 1687-1734.  doi: 10.1137/20M1344093.
    [33] T. YangP. G. Mehta and S. P. Meyn, Feedback particle filter, IEEE Transactions on Automatic Control, 58 (2013), 2465-2480.  doi: 10.1109/TAC.2013.2258825.
    [34] T. Yang, P. G. Mehta and S. P. Meyn, A mean-field control-oriented approach to particle filtering, Proceedings of the American Control Conference, (2011), 2037–2043. doi: 10.1109/ACC.2011.5991422.
  • 加载中
Open Access Under a Creative Commons license
SHARE

Article Metrics

HTML views(677) PDF downloads(139) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return