Gain function approximation in the Feedback Particle Filter
This paper is concerned with numerical algorithms for the problem of 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 numerical problem is to approximate this solution using only particles sampled from the probability distribution. A diffusion-map based algorithm is presented for this problem. The algorithm does not require approximation of the probability distribution as an intermediate step. A procedure for carrying out error analysis of the approximation is introduced and certain asymptotic estimates for bias and variance are derived. The paper contains some comparative numerical results for a problem with non-Gaussian distribution. The algorithm is also applied and illustrated for a numerical filtering example.
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