Convergence of particle filtering method for nonlinear estimation of vortex dynamics

In this paper we formulate a numerical approximation method for the nonlinear filtering of vortex dynamics subject to noise using particle filter method. We prove the convergence of this scheme allowing the observation vector to be unbounded. Nonlinear estimation of turbulence and vortical structures has many applica- tions in engineering and in geophysical sciences. In (18) and (41), mathematical foundation of nonlinear filtering methods was developed for viscous flow and for reacting and diffusing systems. The current work is in part an effort to contribute towards concrete computational methods to solve the nonlinear filtering equations derived in the above papers. We will however focus our attention on much sim- pler fluid dynamic models in terms of point vortices, which nevertheless contain significant physical attributes of fluid mechanics. The particle filter method is a generalization of the traditional Monte-Carlo method and is often called the sequential Monte-Carlo method. The difference with Monte-Carlo method is the presence of an additional correction procedure applied at regular time intervals to the system of particles. At the correction time, each particle is replaced by a random number of particles. This amounts to particles branching into a random number of offsprings. The general principle is that particles with small weights have no offspring, and particles with large weights are replaced by several offsprings. As a numerical method for nonlinear filtering problem, particle filter can be used to approximate general stochastic differential equations. In recent years, dif- ferent variations of it have been studied, such as particle filter with occasional sampling (9), particle filter with variance reduction (10), branching particle filter (11), (12) and regularized particle filter (13), (27), most of which are applicable in discrete time setting and have been implemented computationally. In this paper, we will work in the continuous time setting and study the continuous time particle filter.