Convergence analysis of a finite element approximation of minimum action methods
This work provides a theoretical foundation for numerical methods used to study rare transitions in dynamical systems, but the contribution is incremental as it extends existing convergence analysis to a specific discretization.
The authors prove convergence of a finite element approximation for the minimizer of the Freidlin-Wentzell action functional in non-gradient dynamical systems with small noise, using Γ-convergence. No concrete numerical results are provided.
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 $Γ$-convergence.