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This paper is concerned with a duality-based approach to derive the linearfeedback particle filter (FPF). The FPF is a controlled interacting particlesystem where the control law is designed to provide an exact solution for thenonlinear filtering problem. For the linear Gaussian special case, certainsimplifications arise whereby the linear FPF is identical to the square-rootform of the ensemble Kalman filter. For this and for the more general nonlinearnon-Gaussian case, it has been an open problem to derive/interpret the FPFcontrol law as a solution of an optimal control problem. In this paper, certainduality-based arguments are employed to transform the filtering problem into anoptimal control problem. Its solution is shown to yield the deterministic formof the linear FPF. An extension is described to incorporate stochastic effectsdue to noise leading to a novel homotopy of exact ensemble Kalman filters. Allthe derivations are based on duality formalisms.

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