Perron–Frobenius Operator Filter for Stochastic Dynamical Systems

SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 1, Page 182-211, March 2024.
Abstract.Filtering problems are derived from a sequential minimization of a quadratic function representing a compromise between the model and data. In this paper, we use the Perron–Frobenius operator in a 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 the Perron–Frobenius operator and likelihood operator. This gives a significant connection between the Perron–Frobenius operator and Bayesian filtering. We numerically fulfill the recursion by 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. The Perron–Frobenius operator filter employs data instead of flow equations to model the evolution of underlying stochastic dynamical systems. In contrast, standard particle filters require explicit equations or transition probability density for sampling. A few numerical examples are presented to illustrate the advantage of the Perron–Frobenius operator filter over the particle filter and extended Kalman filter.