The Annals of Applied Probability

Dimensional reduction in nonlinear filtering: A homogenization approach

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

Full-text: Open access

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


Export citation

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.
  • Bensoussan, A. and Blankenship, G. L. (1986). Nonlinear filtering with homogenization. Stochastics 17 67–90.
  • Bensoussan, A., Lions, J.-L. and Papanicolaou, G. (1978). Asymptotic Analysis for Periodic Structures. Studies in Mathematics and Its Applications 5. North-Holland, Amsterdam.
  • Crisan, D. and Rozovskiĭ, B. (2011). The Oxford Handbook of Nonlinear Filtering. Oxford Univ. Press, Oxford.
  • Doucet, A., de Freitas, N. and Gordon, N. (2001). Sequential Monte Carlo Methods in Practice. Springer, New York.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • Fujisaki, M., Kallianpur, G. and Kunita, H. (1972). Stochastic differential equations for the non linear filtering problem. Osaka J. Math. 9 19–40.
  • 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.
  • Kallianpur, G. (1980). Stochastic Filtering Theory. Applications of Mathematics 13. Springer, New York.
  • Karandikar, R. L. (1983). Interchanging the order of stochastic integration and ordinary differentiation. Sankhyā Ser. A 45 120–124.
  • Kleptsina, M. L., Liptser, R. S. and Serebrovski, A. P. (1997). Nonlinear filtering problem with contamination. Ann. Appl. Probab. 7 917–934.
  • Kushner, H. J. (1990). Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems. Systems & Control: Foundations & Applications 3. Birkhäuser, Boston, MA.
  • 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.
  • Papanicolaou, G. C. and Kohler, W. (1975). Asymptotic analysis of deterministic and stochastic equations with rapidly varying components. Comm. Math. Phys. 45 217–232.
  • 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.
  • Pardoux, E. (1979). Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3 127–167.
  • Pardoux, É. and Peng, S. G. (1994). Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probab. Theory Related Fields 98 209–227.
  • Pardoux, E. and Veretennikov, A. Y. (2001). On the Poisson equation and diffusion approximation. Ann. Probab. 29 1061–1085.
  • Pardoux, È. and Veretennikov, A. Y. (2003). On Poisson equation and diffusion approximation. II. Ann. Probab. 31 1166–1192.
  • 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.
  • Rozovskiĭ, B. L. (1990). Stochastic Evolution Systems: Linear Theory and Applications to Nonlinear Filtering. Mathematics and Its Applications (Soviet Series) 35. Kluwer Academic, Dordrecht.
  • 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.
  • Veretennikov, A. Y. (1997). On polynomial mixing bounds for stochastic differential equations. Stochastic Process. Appl. 70 115–127.
  • Zakai, M. (1969). On the optimal filtering of diffusion processes. Z. Wahrsch. Verw. Gebiete 11 230–243.