The Annals of Applied Probability

Dimensional reduction in nonlinear filtering: A homogenization approach

Peter Imkeller, N. Sri Namachchivaya, Nicolas Perkowski, and Hoong C. Yeong

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Abstract

We propose a homogenized filter for multiscale signals, which allows us to reduce the dimension of the system. We prove that the nonlinear filter converges to our homogenized filter with rate $\sqrt{\varepsilon}$. This is achieved by a suitable asymptotic expansion of the dual of the Zakai equation, and by probabilistically representing the correction terms with the help of BDSDEs.

Article information

Source
Ann. Appl. Probab., Volume 23, Number 6 (2013), 2290-2326.

Dates
First available in Project Euclid: 22 October 2013

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1382447689

Digital Object Identifier
doi:10.1214/12-AAP901

Mathematical Reviews number (MathSciNet)
MR3127936

Zentralblatt MATH identifier
1288.60049

Subjects
Primary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 60H15: Stochastic partial differential equations [See also 35R60] 60H35: Computational methods for stochastic equations [See also 65C30]

Keywords
Nonlinear filtering dimensional reduction homogenization particle filtering asymptotic expansion SPDE BDSDE

Citation

Imkeller, Peter; Namachchivaya, N. Sri; Perkowski, Nicolas; Yeong, Hoong C. Dimensional reduction in nonlinear filtering: A homogenization approach. Ann. Appl. Probab. 23 (2013), no. 6, 2290--2326. doi:10.1214/12-AAP901. https://projecteuclid.org/euclid.aoap/1382447689


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References

  • Arulampalam, M. S., Maskell, S., Gordon, N. and Clapp, T. (2002). A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Trans. Signal Process. 50 174–188.
  • Bain, A. and Crisan, D. (2009). Fundamentals of Stochastic Filtering. Stochastic Modelling and Applied Probability 60. Springer, New York.
    Mathematical Reviews (MathSciNet): MR2454694
  • Bensoussan, A. and Blankenship, G. L. (1986). Nonlinear filtering with homogenization. Stochastics 17 67–90.
    Mathematical Reviews (MathSciNet): MR878554
    Digital Object Identifier: doi:10.1080/17442508608833383
  • Bensoussan, A., Lions, J.-L. and Papanicolaou, G. (1978). Asymptotic Analysis for Periodic Structures. Studies in Mathematics and Its Applications 5. North-Holland, Amsterdam.
    Mathematical Reviews (MathSciNet): MR503330
  • Crisan, D. and Rozovskiĭ, B. (2011). The Oxford Handbook of Nonlinear Filtering. Oxford Univ. Press, Oxford.
    Mathematical Reviews (MathSciNet): MR2882749
  • Doucet, A., de Freitas, N. and Gordon, N. (2001). Sequential Monte Carlo Methods in Practice. Springer, New York.
    Mathematical Reviews (MathSciNet): MR1847783
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
    Mathematical Reviews (MathSciNet): MR838085
  • Fujisaki, M., Kallianpur, G. and Kunita, H. (1972). Stochastic differential equations for the non linear filtering problem. Osaka J. Math. 9 19–40.
    Mathematical Reviews (MathSciNet): MR336801
    Project Euclid: euclid.ojm/1200693535
  • Harlim, J. and Kang, E. L. (2012). Filtering partially observed multiscale systems with heterogeneous multiscale methods based reduced climate models. Mon. Wea. Rev. 140 860–873.
  • Ichihara, N. (2004). Homogenization problem for stochastic partial differential equations of Zakai type. Stoch. Stoch. Rep. 76 243–266.
    Mathematical Reviews (MathSciNet): MR2072382
    Digital Object Identifier: doi:10.1080/10451120410001714107
  • Kallianpur, G. (1980). Stochastic Filtering Theory. Applications of Mathematics 13. Springer, New York.
    Mathematical Reviews (MathSciNet): MR583435
  • Karandikar, R. L. (1983). Interchanging the order of stochastic integration and ordinary differentiation. Sankhyā Ser. A 45 120–124.
    Mathematical Reviews (MathSciNet): MR749360
  • Kleptsina, M. L., Liptser, R. S. and Serebrovski, A. P. (1997). Nonlinear filtering problem with contamination. Ann. Appl. Probab. 7 917–934.
    Mathematical Reviews (MathSciNet): MR1484791
    Digital Object Identifier: doi:10.1214/aoap/1043862418
    Project Euclid: euclid.aoap/1043862418
  • Kushner, H. J. (1990). Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems. Systems & Control: Foundations & Applications 3. Birkhäuser, Boston, MA.
    Mathematical Reviews (MathSciNet): MR1102242
  • Lingala, N., Namachchivaya, N. S., Perkowski, N. and Yeong, H. C. (2012). Particle filtering in high-dimensional chaotic systems. Chaos 22 047509.
  • Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of Random Processes. II, expanded ed. Applications of Mathematics (New York) 6. Springer, Berlin.
    Mathematical Reviews (MathSciNet): MR1800858
  • Papanicolaou, G. C. and Kohler, W. (1975). Asymptotic analysis of deterministic and stochastic equations with rapidly varying components. Comm. Math. Phys. 45 217–232.
    Mathematical Reviews (MathSciNet): MR413265
    Digital Object Identifier: doi:10.1007/BF01608329
    Project Euclid: euclid.cmp/1103899493
  • Papanicolaou, G. C., Stroock, D. and Varadhan, S. R. S. (1977). Martingale approach to some limit theorems. In Papers from the Duke Turbulence Conference (Duke Univ., Durham, NC, 1976), Paper No. 6. Duke Univ. Math. Ser. III ii+120. Duke Univ., Durham, NC.
    Mathematical Reviews (MathSciNet): MR461684
  • Pardoux, E. (1979). Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3 127–167.
    Mathematical Reviews (MathSciNet): MR553909
    Digital Object Identifier: doi:10.1080/17442507908833142
  • Pardoux, É. and Peng, S. G. (1994). Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab. Theory Related Fields 98 209–227.
    Mathematical Reviews (MathSciNet): MR1258986
    Digital Object Identifier: doi:10.1007/BF01192514
  • Pardoux, E. and Veretennikov, A. Y. (2001). On the Poisson equation and diffusion approximation. Ann. Probab. 29 1061–1085.
    Mathematical Reviews (MathSciNet): MR1872736
    Digital Object Identifier: doi:10.1214/aop/1015345596
    Project Euclid: euclid.aop/1015345596
  • Pardoux, È. and Veretennikov, A. Y. (2003). On Poisson equation and diffusion approximation. II. Ann. Probab. 31 1166–1192.
    Mathematical Reviews (MathSciNet): MR1988467
    Digital Object Identifier: doi:10.1214/aop/1055425774
    Project Euclid: euclid.aop/1055425774
  • Park, J. H., Namachchivaya, N. S. and Yeong, H. C. (2011). Particle filters in a multiscale environment: Homogenized hybrid particle filter. J. Appl. Mech. 78 1–10.
  • Park, J. H., Sowers, R. B. and Sri Namachchivaya, N. (2010). Dimensional reduction in nonlinear filtering. Nonlinearity 23 305–324.
    Mathematical Reviews (MathSciNet): MR2578480
    Digital Object Identifier: doi:10.1088/0951-7715/23/2/005
  • Rozovskiĭ, B. L. (1990). Stochastic Evolution Systems: Linear Theory and Applications to Nonlinear Filtering. Mathematics and Its Applications (Soviet Series) 35. Kluwer Academic, Dordrecht.
    Mathematical Reviews (MathSciNet): MR1135324
  • Snyder, C., Bengtsson, T., Bickel, P. and Anderson, J. (2008). Obstacles to high-dimensional particle filtering. Mon. Weather Rev. 136 4629–4640.
  • Stroock, D. W. (2008). Partial Differential Equations for Probabilists. Cambridge Studies in Advanced Mathematics 112. Cambridge Univ. Press, Cambridge.
    Mathematical Reviews (MathSciNet): MR2410225
  • Veretennikov, A. Y. (1997). On polynomial mixing bounds for stochastic differential equations. Stochastic Process. Appl. 70 115–127.
    Mathematical Reviews (MathSciNet): MR1472961
    Digital Object Identifier: doi:10.1016/S0304-4149(97)00056-2
  • Zakai, M. (1969). On the optimal filtering of diffusion processes. Z. Wahrsch. Verw. Gebiete 11 230–243.
    Mathematical Reviews (MathSciNet): MR242552
    Digital Object Identifier: doi:10.1007/BF00536382

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