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

Published online by Cambridge University Press:  26 August 2010

Christophe Baehr
Christophe Baehr*
Affiliation:
Météo-France-CNRS, CNRM-GAME URA1357, 42 avenue Coriolis, 31057 Toulouse Cedex 1, France. christophe.baehr@meteo.fr Associated member of the Laboratory of Statistics and Probability of the Toulouse Mathematics Institute (UMR 5219), 118 route de Narbonne, 31062 Toulouse Cedex 9, France.
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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.

Keywords

Nonlinear filteringFeynman-Kacstochastic modelturbulence
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 5: Special Issue on Probabilistic methods and their applications , September 2010 , pp. 921 - 945
Copyright
© EDP Sciences, SMAI, 2010

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References

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).
C. Baehr and F. Legland, Some Mean-Field Processes Filtering using Particles System Approximations. In preparation.
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).
BenArous, G., Flots et séries de Taylor stochastiques. Probab. Theor. Relat. Fields 81 (1989) 29–77.
Bossy, M. and Talay, D., 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.
Busnello, B. and Flandoli, F. and Romito, M., 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. CrossRef
Constantin, P. and Iyer, G., A stochastic Lagrangian representation of 3-dimensional incompressible Navier-Stokes equations. Commun. Pure Appl. Math. 61 (2008) 330–345. CrossRef
Das, S. and Durbin, P., Lagrangian, A stochastic model for dispersion in stratified turbulence. Phys. Fluids 17 (2005) 025109. CrossRef
P. Del Moral, Feynman-Kac Formulae, Genealogical and Interacting Particle Systems with Applications. Springer-Verlag (2004).
U. Frisch, Turbulence. Cambridge University Press, Cambridge (1995).
Iyer, G. and Mattingly, J., A stochastic-Lagrangian particle system for the Navier-Stokes equations. Nonlinearity 21 (2008) 2537–2553. CrossRef
I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag (1988).
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).
Mikulevicius, R. and Rozovskii, B., Stochastic Navier-Stokes Equations for turbulent flows. SIAM J. Math. Anal. 35 (2004) 1250–1310. CrossRef
S.B. Pope, Turbulent Flows. Cambridge University Press, Cambridge (2000).
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).