An Improved Adaptive Minimum Action Method for the Calculation of Transition Path in Non-Gradient Systems (Communications in Computational Physics, 2018)
The minimum action method (MAM) is to calculate the most probable transitionpath in randomly perturbed stochastic dynamics, based on the idea of actionminimization in the path space. The accuracy of the numerical path between differentmetastable states usually suffers from the "clustering problem" near fixed points.The adaptive minimum action method (aMAM) solves this problem by relocating imagepoints 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 trappedaround the transition state in a tangle, due to the singularity of the relationship betweenarc-length and time at the transition state. Additionally, in most non-gradientdynamics, the tangent direction of the path is not continuous at the transition state sothat a geometric corner forms, which brings extra challenges for the aMAM. In thisnote, we improve the aMAM by proposing a better monitor function that does notcontain the numerical approximation of derivatives, and taking use of a generalizedscheme of the Euler-Lagrange equation to solve the minimization problem, so thatboth the path-tangling problem and the non-smoothness in parametrizing the curvedo not exist. To further improve the accuracy, we apply the Weighted Essentially non-oscillatory(WENO) method for the interpolation to achieve better performance. Numericalexamples are presented to demonstrate the advantages of our new method.
Keywords: Rare event; transition path; minimum action method; moving mesh; WENO.