SIAM Journal on Scientific Computing, Volume 46, Issue 2, Page A770-A797, April 2024.
Abstract. We introduce a new stochastic algorithm to locate the index-1 saddle points of a function [math], with [math] possibly large. This algorithm can be seen as an equivalent of the stochastic gradient descent which is a natural stochastic process to locate local minima. It relies on two ingredients: (i) the concentration properties on index-1 saddle points of the first eigenmodes of the Witten Laplacian (associated with [math]) on 1-forms and (ii) a probabilistic representation of a partial differential equation involving this differential operator. Numerical examples on simple molecular systems illustrate the efficacy of the proposed approach. Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available in https://github.com/pp500/Stochastic-Saddle-Point-Dynamics and in the supplementary materials (Stochastic-Saddle-Point-Dynamics-main.zip [167KB]).

中文翻译：

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使用 Witten Laplacian 来定位 Index-1 鞍点

SIAM 科学计算杂志，第 46 卷，第 2 期，A770-A797 页，2024 年 4 月。
摘要。我们引入一种新的随机算法来定位函数 [math] 的索引 1 鞍点，其中 [math] 可能很大。该算法可以看作是随机梯度下降的等价物，随机梯度下降是定位局部最小值的自然随机过程。它依赖于两个成分：(i) 1-形式上维滕拉普拉斯算子第一本征模态（与[数学]相关）的索引 1 鞍点上的浓度特性，以及 (ii) 涉及的偏微分方程的概率表示这个微分算子。简单分子系统的数值例子说明了所提出方法的有效性。计算结果的再现性。本文被授予“SIAM 再现性徽章：代码和数据可用”，以表彰作者遵循 SISC 和科学计算界所重视的再现性原则。允许读者重现本文结果的代码和数据可在 https://github.com/pp500/Stochastic-Saddle-Point-Dynamics 和补充材料 (Stochastic-Saddle-Point-Dynamics-main.zip [167KB]）。