Nonlinear filtering for observations on a random vector field along a random path. Application to atmospheric turbulent velocities
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 5, p. 921-945

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 $𝒪$ $\left(\frac{1}{\sqrt{N}}\right)$ asymptotic convergence. An application to nonlinear filtering for 3D atmospheric turbulent fluids will be described.

DOI : https://doi.org/10.1051/m2an/2010047
Classification:  82B31,  65C35,  65C05,  62M20,  60G57,  60J85
Keywords: nonlinear filtering, Feynman-Kac, stochastic model, turbulence
@article{M2AN_2010__44_5_921_0,
author = {Baehr, Christophe},
title = {Nonlinear filtering for observations on a random vector field along a random path. Application to atmospheric turbulent velocities},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {44},
number = {5},
year = {2010},
pages = {921-945},
doi = {10.1051/m2an/2010047},
zbl = {pre05798938},
mrnumber = {2731398},
language = {en},
url = {http://www.numdam.org/item/M2AN_2010__44_5_921_0}
}

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 - Modélisation Mathématique et Analyse Numérique, Volume 44 (2010) no. 5, pp. 921-945. doi : 10.1051/m2an/2010047. http://www.numdam.org/item/M2AN_2010__44_5_921_0/

[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. Fields 81 (1989) 29-77. | Zbl 0639.60062

[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. | Zbl 0860.60038

[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. | Zbl 1075.76019

[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. | Zbl 1156.60048

[8] S. Das and P. Durbin, A Lagrangian stochastic model for dispersion in stratified turbulence. Phys. Fluids 17 (2005) 025109. | Zbl 1187.76115

[9] P. Del Moral, Feynman-Kac Formulae, Genealogical and Interacting Particle Systems with Applications. Springer-Verlag (2004). | Zbl 1130.60003

[10] U. Frisch, Turbulence. Cambridge University Press, Cambridge (1995). | Zbl 0832.76001

[11] G. Iyer and J. Mattingly, A stochastic-Lagrangian particle system for the Navier-Stokes equations. Nonlinearity 21 (2008) 2537-2553. | Zbl 1158.60383

[12] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus. Springer-Verlag (1988). | Zbl 0734.60060

[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). | Zbl 0864.60077

[14] R. Mikulevicius and B. Rozovskii, Stochastic Navier-Stokes Equations for turbulent flows. SIAM J. Math. Anal. 35 (2004) 1250-1310. | Zbl 1062.60061

[15] S.B. Pope, Turbulent Flows. Cambridge University Press, Cambridge (2000). | Zbl 0966.76002

[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). | Zbl 0732.60114