Derivation and Extensions of the Linear Feedback Particle Filter based on Duality Formalisms

TL;DR

This paper uses duality formalisms to derive and extend the linear feedback particle filter, connecting it to optimal control and ensemble Kalman filters, and addressing nonlinear filtering challenges.

Contribution

It introduces a duality-based derivation of the linear FPF and extends it to include stochastic effects, linking it to ensemble Kalman filters.

Findings

Linear FPF is identical to the square-root ensemble Kalman filter in the Gaussian case

Duality-based approach transforms filtering into an optimal control problem

Extension incorporates stochastic effects, creating a homotopy of ensemble Kalman filters

Abstract

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.