An iterative implementation of the implicit nonlinear filter
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 3, p. 535-543

Implicit sampling is a sampling scheme for particle filters, designed to move particles one-by-one so that they remain in high-probability domains. We present a new derivation of implicit sampling, as well as a new iteration method for solving the resulting algebraic equations.

DOI : https://doi.org/10.1051/m2an/2011055
Classification:  60G35,  62M20,  86A05
Keywords: implicit sampling, filter, reference density, jacobian, iteration, particles
@article{M2AN_2012__46_3_535_0,
     author = {Chorin, Alexandre J. and Tu, Xuemin},
     title = {An iterative implementation of the implicit nonlinear filter},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {3},
     year = {2012},
     pages = {535-543},
     doi = {10.1051/m2an/2011055},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2012__46_3_535_0}
}
Chorin, Alexandre J.; Tu, Xuemin. An iterative implementation of the implicit nonlinear filter. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 3, pp. 535-543. doi : 10.1051/m2an/2011055. http://www.numdam.org/item/M2AN_2012__46_3_535_0/

[1] M. Arulampalam, S. Maskell, N. Gordon and T. Clapp, A tutorial on particle filters for online nonlinear/nongaussian Bayesia tracking. IEEE Trans. Signal Process. 50 (2002) 174-188.

[2] P. Bickel, B. Li and T. Bengtsson, Sharp failure rates for the bootstrap particle filter in high dimensions. IMS Collections : Pushing the Limits of Contemporary Statistics : Contributions in Honor of Jayanta K. Ghosh 3 (2008) 318-329. | MR 2459233

[3] S. Bozic, Digital and Kalman Filtering. Butterworth-Heinemann, Oxford (1994). | Zbl 0593.93057

[4] A.J. Chorin and P. Krause, Dimensional reduction for a Bayesian filter. Proc. Natl. Acad. Sci. USA 101 (2004) 15013-15017. | MR 2099822 | Zbl 1135.93377

[5] A.J. Chorin and X. Tu, Implicit sampling for particle filters. Proc. Natl. Acad. Sc. USA 106 (2009) 17249-17254.

[6] A.J. Chorin, M. Morzfeld and X. Tu, Implicit particle filters for data assimilation. Commun. Appl. Math. Comput. Sci. 5 (2010) 221-240. | MR 2765384 | Zbl 1229.60047

[7] A. Doucet and A. Johansen, Particle filtering and smoothing : Fifteen years later, in Handbook of Nonlinear Filtering, edited by D. Crisan and B. Rozovsky, to appear.

[8] A. Doucet, S. Godsill and C. Andrieu, On sequential Monte Carlo sampling methods for Bayesian filtering. Stat. Comput. 10 (2000) 197-208.

[9] A. Doucet, N. De Freitas and N. Gordon, Sequential Monte Carlo Methods in Practice. Springer, New York (2001). | MR 1847783 | Zbl 0967.00022

[10] M. Dowd, A sequential Monte Carlo approach for marine ecological prediction. Environmetrics 17 (2006) 435-455. | MR 2240936

[11] W. Gilks and C. Berzuini, Following a moving target-Monte Carlo inference for dynamic Bayesian models. J. Roy. Statist. Soc. B 63 (2001) 127-146. | MR 1811995 | Zbl 0976.62021

[12] J. Liu and C. Sabatti, Generalized Gibbs sampler and multigrid Monte Carlo for Bayesian computation. Biometrika 87 (2000) 353-369. | MR 1782484 | Zbl 0960.65015

[13] S. Maceachern, M. Clyde and J. Liu, Sequential importance sampling for nonparametric Bayes models : the next generation. Can. J. Stat. 27 (1999) 251-267. | MR 1704407 | Zbl 0957.62068

[14] M. Morzfeld, X. Tu, E. Atkins and A.J. Chorin, A random map implementation of implicit filters. Submitted to J. Comput. Phys. | Zbl 1242.65012

[15] C. Snyder, T. Bengtsson, P. Bickel and J. Anderson, Obstacles to high-dimensional particle filtering. Mon. Weather Rev. 136 (2008) 4629-4640.