Convergence analysis of a finite element approximation of minimum action methods

TL;DR

This paper proves the convergence of a finite element method for approximating the minimizer of the Freidlin-Wentzell action functional in non-gradient stochastic systems, aiding the analysis of small-noise transitions.

Contribution

It introduces a finite element discretization of the F-W action functional and establishes its convergence via $ ext{Gamma}$-convergence, advancing numerical methods for large deviation analysis.

Findings

Finite element discretization of the F-W action functional.

Convergence of the approximation established via $ ext{Gamma}$-convergence.

Applicable to non-gradient stochastic dynamical systems.

Abstract

In this work, we address the convergence of a finite element approximation of the minimizer of the Freidlin-Wentzell (F-W) action functional for non-gradient dynamical systems perturbed by small noise. The F-W theory of large deviations is a rigorous mathematical tool to study small-noise-induced transitions in a dynamical system. The central task in the application of F-W theory of large deviations is to seek the minimizer and minimum of the F-W action functional. We discretize the F-W action functional using linear finite elements, and establish the convergence of {the approximation} through $\Gamma$-convergence.