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 asymptotic convergence. An application to nonlinear filtering for 3D atmospheric turbulent fluids will be described.
@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},
volume = {44},
year = {2010},
pages = {921-945},
doi = {10.1051/m2an/2010047},
mrnumber = {2731398},
zbl = {pre05798938},
language = {en},
url = {http://dml.mathdoc.fr/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, Tome 44 (2010) pp. 921-945. doi : 10.1051/m2an/2010047. http://gdmltest.u-ga.fr/item/M2AN_2010__44_5_921_0/
[1] , 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] and , Some Mean-Field Processes Filtering using Particles System Approximations. In preparation.
[3] and , 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] , Flots et séries de Taylor stochastiques. Probab. Theor. Relat. Fields 81 (1989) 29-77. | Zbl 0639.60062
[5] and , 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] and 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] and , A stochastic Lagrangian representation of 3-dimensional incompressible Navier-Stokes equations. Commun. Pure Appl. Math. 61 (2008) 330-345. | Zbl 1156.60048
[8] and , A Lagrangian stochastic model for dispersion in stratified turbulence. Phys. Fluids 17 (2005) 025109. | Zbl 1187.76115
[9] , Feynman-Kac Formulae, Genealogical and Interacting Particle Systems with Applications. Springer-Verlag (2004). | Zbl 1130.60003
[10] , Turbulence. Cambridge University Press, Cambridge (1995). | Zbl 0832.76001
[11] and , A stochastic-Lagrangian particle system for the Navier-Stokes equations. Nonlinearity 21 (2008) 2537-2553. | Zbl 1158.60383
[12] and , Brownian Motion and Stochastic Calculus. Springer-Verlag (1988). | Zbl 0734.60060
[13] , 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] and , Stochastic Navier-Stokes Equations for turbulent flows. SIAM J. Math. Anal. 35 (2004) 1250-1310. | Zbl 1062.60061
[15] , Turbulent Flows. Cambridge University Press, Cambridge (2000). | Zbl 0966.76002
[16] , 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