Gain function approximation in the feedback particle filter

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.