Nonlinear filtering for observations on a random vector field along a random path. Application to atmospheric turbulent velocities
ESAIM: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 5, pp. 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 (1 N) asymptotic convergence. An application to nonlinear filtering for 3D atmospheric turbulent fluids will be described.

DOI : 10.1051/m2an/2010047
Classification : 82B31, 65C35, 65C05, 62M20, 60G57, 60J85
Mots clés : 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: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {921--945},
     publisher = {EDP-Sciences},
     volume = {44},
     number = {5},
     year = {2010},
     doi = {10.1051/m2an/2010047},
     mrnumber = {2731398},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2010047/}
}
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: Modélisation mathématique et analyse numérique
PY  - 2010
SP  - 921
EP  - 945
VL  - 44
IS  - 5
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2010047/
DO  - 10.1051/m2an/2010047
LA  - en
ID  - M2AN_2010__44_5_921_0
ER  - 
%0 Journal Article
%A Baehr, Christophe
%T Nonlinear filtering for observations on a random vector field along a random path. Application to atmospheric turbulent velocities
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2010
%P 921-945
%V 44
%N 5
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2010047/
%R 10.1051/m2an/2010047
%G en
%F 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: Modélisation mathématique et analyse numérique, Tome 44 (2010) no. 5, pp. 921-945. doi : 10.1051/m2an/2010047. http://www.numdam.org/articles/10.1051/m2an/2010047/

[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

[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

[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

[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

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

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

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

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

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

[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

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

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

[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

Cité par Sources :