Using Witten Laplacians to locate index-1 saddle points

Tony Lelièvre; Panos Parpas

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arXiv · 2212.10135

Using Witten Laplacians to locate index-1 saddle points

Tony Lelièvre·Panos Parpas

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Abstract

We introduce a new stochastic algorithm to locate the index-1 saddle points of a function $V:\mathbb R^d \to \mathbb R$, with $d$ 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 $V$) 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.

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Tony Lelièvre, Panos Parpas. 2022-12-20. Using Witten Laplacians to locate index-1 saddle points. https://arxiv.org/abs/2212.10135

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