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The filtering problems are derived from a sequential minimization of aquadratic function representing a compromise between model and data. In thispaper, we use the Perron-Frobenius operator in stochastic process to develop aPerron-Frobenius operator filter. The proposed method belongs to Bayesianfiltering and works for non-Gaussian distributions for nonlinear stochasticdynamical systems. The recursion of the filtering can be characterized by thecomposition of Perron-Frobenius operator and likelihood operator. This gives asignificant connection between the Perron-Frobenius operator and Bayesianfiltering. We numerically fulfil the recursion through approximating thePerron-Frobenius operator by Ulam's method. In this way, the posterior measureis represented by a convex combination of the indicator functions in Ulam'smethod. To get a low rank approximation for the Perron-Frobenius operatorfilter, we take a spectral decomposition for the posterior measure by using theeigenfunctions of the discretized Perron-Frobenius operator. A convergenceanalysis is carried out and shows that the Perron-Frobenius operator filterachieves a higher convergence rate than the particle filter, which uses Diracmeasures for the posterior. The proposed method is explored for the dataassimilation of the stochastic dynamical systems. A few numerical examples arepresented to illustrate the advantage of the Perron-Frobenius operator filterover particle filter and extend Kalman filter.

Perron-Frobenius operator filter for stochastic dynamical systems