# An Improved Adaptive Minimum Action Method for the Calculation of Transition Path in Non-gradient Systems

## TL;DR

This paper introduces an enhanced adaptive minimum action method that effectively addresses path-tangling and non-smoothness issues in calculating transition paths in non-gradient stochastic systems, improving accuracy and robustness.

## Contribution

The paper proposes a new monitor function and a generalized Euler-Lagrange scheme, combined with WENO interpolation, to improve the adaptive minimum action method for non-gradient systems.

## Key findings

- Reduces path-tangling near transition states.
- Handles non-smooth tangent directions at transition points.
- Demonstrates improved accuracy in numerical examples.

## Abstract

The minimum action method (MAM) is to calculate the most probable transition path in randomly perturbed stochastic dynamics, based on the idea of action minimization in the path space. The accuracy of the numerical path between different metastable states usually suffers from the "clustering problem" near fixed points. The adaptive minimum action method (aMAM) solves this problem by relocating image points equally along arc-length with the help of moving mesh strategy. However, when the time interval is large, the images on the path may still be locally trapped around the transition state in a tangle, due to the singularity of the relationship between arc-length and time at the transition state. Additionally, in most non-gradient dynamics, the tangent direction of the path is not continuous at the transition state so that a geometric corner forms, which brings extra challenges for the aMAM. In this note, we improve the aMAM by proposing a better monitor function that does not contain the numerical approximation of derivatives, and taking use of a generalized scheme of the Euler-Lagrange equation to solve the minimization problem, so that both the path-tangling problem and the non-smoothness in parametrizing the curve do not exist. To further improve the accuracy, we apply the Weighted Essentially non-oscillatory (WENO) method for the interpolation to achieve better performance. Numerical examples are presented to demonstrate the advantages of our new method.