  The filtering problems are derived from a sequential minimization of a
quadratic function representing a compromise between model and data. In this
paper, we use the Perron-Frobenius operator in stochastic process to develop a
Perron-Frobenius operator filter. The proposed method belongs to Bayesian
filtering and works for non-Gaussian distributions for nonlinear stochastic
dynamical systems. The recursion of the filtering can be characterized by the
composition of Perron-Frobenius operator and likelihood operator. This gives a
significant connection between the Perron-Frobenius operator and Bayesian
filtering. We numerically fulfil the recursion through approximating the
Perron-Frobenius operator by Ulam's method. In this way, the posterior measure
is represented by a convex combination of the indicator functions in Ulam's
method. To get a low rank approximation for the Perron-Frobenius operator
filter, we take a spectral decomposition for the posterior measure by using the
eigenfunctions of the discretized Perron-Frobenius operator. A convergence
analysis is carried out and shows that the Perron-Frobenius operator filter
achieves a higher convergence rate than the particle filter, which uses Dirac
measures for the posterior. The proposed method is explored for the data
assimilation of the stochastic dynamical systems. A few numerical examples are
presented to illustrate the advantage of the Perron-Frobenius operator filter
over particle filter and extend Kalman filter.
