Using Witten Laplacians to locate index-1 saddle points

We introduce a new stochastic algorithm to locate the index-1 saddle points of a function V:mathbb Rd → 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.