Numerical approximation of the Frobenius-Perron operator using the finite volume method

TL;DR

This paper introduces a finite volume method-based numerical approximation for the Frobenius-Perron operator, ensuring convergence and Markov property preservation, demonstrated through a sequential inference example.

Contribution

It presents a novel finite volume approach to approximate the Frobenius-Perron operator with proven convergence and property preservation.

Findings

The approximation satisfies the Markov property under CFL condition.

Convergence to the true operator is guaranteed as mesh size decreases.

Effective in sequential inference for multi-modal distributions.

Abstract

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.