Nonlinear filtering for observations on a random vector field along a random path. Application to atmospheric turbulent velocities

Christophe Baehr

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 5, page 921-945
  • ISSN: 0764-583X

Abstract

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To filter perturbed local measurements on a random medium, a dynamic model jointly with an observation transfer equation are needed. Some media given by PDE could have a local probabilistic representation by a Lagrangian stochastic process with mean-field interactions. In this case, we define the acquisition process of locally homogeneous medium along a random path by a Lagrangian Markov process conditioned to be in a domain following the path and conditioned to the observations. The nonlinear filtering for the mobile signal is therefore those of an acquisition process contaminated by random errors. This will provide a Feynman-Kac distribution flow for the conditional laws and an N particle approximation with a 𝒪 ( 1 N ) asymptotic convergence. An application to nonlinear filtering for 3D atmospheric turbulent fluids will be described.

How to cite

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Baehr, Christophe. "Nonlinear filtering for observations on a random vector field along a random path. Application to atmospheric turbulent velocities." ESAIM: Mathematical Modelling and Numerical Analysis 44.5 (2010): 921-945. <http://eudml.org/doc/250811>.

@article{Baehr2010,
abstract = { To filter perturbed local measurements on a random medium, a dynamic model jointly with an observation transfer equation are needed. Some media given by PDE could have a local probabilistic representation by a Lagrangian stochastic process with mean-field interactions. In this case, we define the acquisition process of locally homogeneous medium along a random path by a Lagrangian Markov process conditioned to be in a domain following the path and conditioned to the observations. The nonlinear filtering for the mobile signal is therefore those of an acquisition process contaminated by random errors. This will provide a Feynman-Kac distribution flow for the conditional laws and an N particle approximation with a $\mathcal\{O\}$$(\frac\{1\}\{\sqrt\{N\}\})$ asymptotic convergence. An application to nonlinear filtering for 3D atmospheric turbulent fluids will be described. },
author = {Baehr, Christophe},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Nonlinear filtering; Feynman-Kac; stochastic model; turbulence; nonlinear filtering},
language = {eng},
month = {8},
number = {5},
pages = {921-945},
publisher = {EDP Sciences},
title = {Nonlinear filtering for observations on a random vector field along a random path. Application to atmospheric turbulent velocities},
url = {http://eudml.org/doc/250811},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Baehr, Christophe
TI - Nonlinear filtering for observations on a random vector field along a random path. Application to atmospheric turbulent velocities
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/8//
PB - EDP Sciences
VL - 44
IS - 5
SP - 921
EP - 945
AB - To filter perturbed local measurements on a random medium, a dynamic model jointly with an observation transfer equation are needed. Some media given by PDE could have a local probabilistic representation by a Lagrangian stochastic process with mean-field interactions. In this case, we define the acquisition process of locally homogeneous medium along a random path by a Lagrangian Markov process conditioned to be in a domain following the path and conditioned to the observations. The nonlinear filtering for the mobile signal is therefore those of an acquisition process contaminated by random errors. This will provide a Feynman-Kac distribution flow for the conditional laws and an N particle approximation with a $\mathcal{O}$$(\frac{1}{\sqrt{N}})$ asymptotic convergence. An application to nonlinear filtering for 3D atmospheric turbulent fluids will be described.
LA - eng
KW - Nonlinear filtering; Feynman-Kac; stochastic model; turbulence; nonlinear filtering
UR - http://eudml.org/doc/250811
ER -

References

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  1. C. Baehr, Modélisation probabiliste des écoulements atmosphériques turbulents afin d'en filtrer la mesure par approche particulaire. Ph.D. Thesis University of Toulouse III – Paul Sabatier, Toulouse Mathematics Institute, France (2008).  
  2. C. Baehr and F. Legland, Some Mean-Field Processes Filtering using Particles System Approximations. In preparation.  
  3. C. Baehr and O. Pannekoucke, Some issues and results on the EnKF and particle filters for meteorological models, in Chaotic Systems: Theory and Applications, C.H. Skiadas and I. Dimotikalis Eds., World Scientific (2010).  
  4. G. BenArous, Flots et séries de Taylor stochastiques. Probab. Theor. Relat. Fields81 (1989) 29–77.  
  5. M. Bossy and D. Talay, Convergence rate for the approximation of the limit law of weakly interacting particles. 2: Application to the Burgers equation. Ann. Appl. Prob.6 (1996) 818–861.  Zbl0860.60038
  6. B. Busnello and F. Flandoli and M. Romito, A probabilistic representation for the vorticity of a 3d viscous fluid and for general systems of parabolic equations. Proc. Edinb. Math. Soc.48 (2005) 295–336.  Zbl1075.76019
  7. P. Constantin and G. Iyer, A stochastic Lagrangian representation of 3-dimensional incompressible Navier-Stokes equations. Commun. Pure Appl. Math.61 (2008) 330–345.  Zbl1156.60048
  8. S. Das and P. Durbin, A Lagrangian stochastic model for dispersion in stratified turbulence. Phys. Fluids17 (2005) 025109.  Zbl1187.76115
  9. P. Del Moral, Feynman-Kac Formulae, Genealogical and Interacting Particle Systems with Applications. Springer-Verlag (2004).  
  10. U. Frisch, Turbulence. Cambridge University Press, Cambridge (1995).  
  11. G. Iyer and J. Mattingly, A stochastic-Lagrangian particle system for the Navier-Stokes equations. Nonlinearity21 (2008) 2537–2553.  Zbl1158.60383
  12. I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag (1988).  Zbl0638.60065
  13. S. Méléard, Asymptotic behaviour of some particle systems: McKean Vlasov and Boltzmann models, in Probabilistic Models for Nonlinear Partial Differential Equations, Lecture Notes in Math.1627, Springer-Verlag (1996).  Zbl0864.60077
  14. R. Mikulevicius and B. Rozovskii, Stochastic Navier-Stokes Equations for turbulent flows. SIAM J. Math. Anal.35 (2004) 1250–1310.  Zbl1062.60061
  15. S.B. Pope, Turbulent Flows. Cambridge University Press, Cambridge (2000).  Zbl0966.76002
  16. A.S. Sznitman, Topics in propagation of chaos, in École d'Eté de Probabilités de Saint-Flour XIX-1989, Lecture Notes in Math.1464, Springer-Verlag (1991).  

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