Numerical approximation of the Frobenius-Perron operator using the finite volume method
For researchers in dynamical systems and Bayesian inference, this provides a new numerical method for approximating the Frobenius-Perron operator with guaranteed convergence, though the demonstration is limited to a low-dimensional example.
The authors develop a finite-dimensional approximation of the Frobenius-Perron operator using the finite volume method, ensuring Markov property via a CFL condition and convergence as mesh size tends to zero. They demonstrate the method on a sequential inference problem for a low-dimensional mechanical system with multi-modal observations.
We develop a finite-dimensional approximation of the Frobenius-Perron operator using the finite volume method applied to the continuity equation for the evolution of probability. A Courant-Friedrichs-Lewy condition ensures that the approximation satisfies the Markov property, while existing convergence theory for the finite volume method guarantees convergence of the discrete operator to the continuous operator as mesh size tends to zero. Properties of the approximation are demonstrated in a computed example of sequential inference for the state of a low-dimensional mechanical system when observations give rise to multi-modal distributions.