Perron-Frobenius operator filter for stochastic dynamical systems

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.