  In recent work it is shown that importance sampling can be avoided in the
particle filter through an innovation structure inspired by traditional
nonlinear filtering combined with Mean-Field Game formalisms. The resulting
feedback particle filter (FPF) offers significant variance improvements; in
particular, the algorithm can be applied to systems that are not stable. The
filter comes with an up-front computational cost to obtain the filter gain.
This paper describes new representations and algorithms to compute the gain in
the general multivariable setting. The main contributions are, (i) Theory
surrounding the FPF is improved: Consistency is established in the multivariate
setting, as well as well-posedness of the associated PDE to obtain the filter
gain. (ii) The gain can be expressed as the gradient of a function, which is
precisely the solution to Poisson's equation for a related MCMC diffusion (the
Smoluchowski equation). This provides a bridge to MCMC as well as to
approximate optimal filtering approaches such as TD-learning, which can in turn
be used to approximate the gain. (iii) Motivated by a weak formulation of
Poisson's equation, a Galerkin finite-element algorithm is proposed for
approximation of the gain. Its performance is illustrated in numerical
experiments.
