非梯度系统过渡路径计算的改进自适应最小作用法

Y. Sun, X. Zhou
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引用次数: 9

摘要

最小作用法(MAM)是基于路径空间中作用最小的思想,计算随机扰动随机动力学中最可能的转移路径。不同亚稳态之间的数值路径精度通常受到不动点附近的“聚类问题”的影响。自适应最小作用法(aMAM)利用移动网格策略沿弧长等距重新定位图像点,解决了这一问题。然而,当时间间隔较大时,由于过渡状态下弧长与时间关系的奇异性,路径上的图像仍可能局部被困在过渡状态周围,形成缠结。此外,在大多数非梯度动力学中,在过渡状态下路径的切线方向是不连续的,从而形成几何角,这给aMAM带来了额外的挑战。在本文中,我们改进了aMAM,提出了一个更好的监测函数,不包含导数的数值逼近,并使用欧拉-拉格朗日方程的广义格式来解决最小化问题,从而不存在路径缠绕问题和参数化曲线的非光滑性。为了进一步提高插值精度,我们采用加权本质非振荡(WENO)方法进行插值,以获得更好的插值效果。数值算例说明了该方法的优越性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
An Improved Adaptive Minimum Action Method for the Calculation of Transition Path in Non-gradient Systems
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.
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