
A Stabilization of a Continuous Limit of the Ensemble Kalman Filter
The ensemble Kalman filter belongs to the class of iterative particle fi...
read it

On the Mathematical Theory of Ensemble (LinearGaussian) KalmanBucy Filtering
The purpose of this review is to present a comprehensive overview of the...
read it

Recent developments in controlled crowd dynamics
We survey recent results on controlled particle systems. The control asp...
read it

Continuous Limits for Constrained Ensemble Kalman Filter
The Ensemble Kalman Filter method can be used as an iterative particle n...
read it

Ensemble Kalman Sampling: meanfield limit and convergence analysis
Ensemble Kalman sampling (EKS) is a method to find i.i.d. samples from a...
read it

Note on Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler by GarbunoInigo, Hoffmann, Li and Stuart
An interacting system of Langevin dynamics driven particles has been pro...
read it

Gain function approximation in the Feedback Particle Filter
This paper is concerned with numerical algorithms for the problem of gai...
read it
An Optimal Transport Formulation of the Ensemble Kalman Filter
Controlled interacting particle systems such as the ensemble Kalman filter (EnKF) and the feedback particle filter (FPF) are numerical algorithms to approximate the solution of the nonlinear filtering problem in continuous time. The distinguishing feature of these algorithms is that the Bayesian update step is implemented using a feedback control law. It has been noted in the literature that the control law is not unique. This is the main problem addressed in this paper. To obtain a unique control law, the filtering problem is formulated here as an optimal transportation problem. An explicit formula for the (meanfield type) optimal control law is derived in the linear Gaussian setting. Comparisons are made with the control laws for different types of EnKF algorithms described in the literature. Via empirical approximation of the meanfield control law, a finiteN controlled interacting particle algorithm is obtained. For this algorithm, the equations for empirical mean and covariance are derived and shown to be identical to the Kalman filter. This allows strong conclusions on convergence and error properties based on the classical filter stability theory for the Kalman filter. It is shown that, under certain technical conditions, the mean squared error (m.s.e.) converges to zero even with a finite number of particles. A detailed propagation of chaos analysis is carried out for the finiteN algorithm. The analysis is used to prove weak convergence of the empirical distribution as N→∞. For a certain simplified filtering problem, analytical comparison of the m.s.e. with the importance samplingbased algorithms is described. The analysis helps explain the favorable scaling properties of the controlbased algorithms reported in several numerical studies in recent literature.
READ FULL TEXT
Comments
There are no comments yet.