## Derivation and Extensions of the Linear Feedback Particle Filter based on Duality Formalisms. (arXiv:1804.04199v1 [math.OC])

This paper is concerned with a duality-based approach to derive the linear
feedback particle filter (FPF). The FPF is a controlled interacting particle
system where the control law is designed to provide an exact solution for the
nonlinear filtering problem. For the linear Gaussian special case, certain
simplifications arise whereby the linear FPF is identical to the square-root
form of the ensemble Kalman filter. For this and for the more general nonlinear
non-Gaussian case, it has been an open problem to derive/interpret the FPF
control law as a solution of an optimal control problem. In this paper, certain
duality-based arguments are employed to transform the filtering problem into an
optimal control problem. Its solution is shown to yield the deterministic form
of the linear FPF. An extension is described to incorporate stochastic effects
due to noise leading to a novel homotopy of exact ensemble Kalman filters. All
the derivations are based on duality formalisms.查看全文