Gain function approximation in the feedback particle filter

Amirhossein Taghvaei, Prashant G. Mehta

Abstract: This paper is concerned with numerical algorithms for gain function approximation in the feedback particle filter. The exact gain function is the solution of a Poisson equation involving a probability-weighted Laplacian. The problem is to approximate this solution using only particles sampled from the probability distribution. Two algorithms are presented: a Galerkin algorithm and a kernel-based algorithm. Both the algorithms are adapted to the samples and do not require approximation of the probability distribution as an intermediate step. The paper contains a preliminary error analysis for the algorithms as well as some comparative numerical results for a non-Gaussian distribution. These algorithms are also applied and illustrated for a simple nonlinear filtering example.