Abstract

We focus on simulations for rare events such as essential conformational transitions in biochemical reactions which are modeled by Langevin dynamics on manifolds. We first reinterpret the observed transition paths from the stochastic optimal control viewpoint, which realizes the transitions almost surely. Then based on collected high dimensional point clouds and nonlinear dimension reduction, we construct an approximated Voronoi tessellation for the reduced manifold and design an upwind scheme for the associated Fokker-Planck equation that automatically incorporates the manifold structure and enjoys lots of fine properties such as stability and convergence. This upwind scheme leads to an optimal controlled random walk on point clouds, which enables the Monte Carlo simulation for conformational transitions. The way finding the observed mean transition paths also provides a finite temperature string method on point clouds.