We focus on analysis and data-driven algorithms 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. The scheme
automatically incorporates the manifold structure and enjoys
lots of fine properties such as stability and convergence.
An optimal controlled random walk on point clouds is then
constructed, which enables efficient Monte Carlo simulation
for conformational transitions.Zoom notes: Email Yuan Gao for the Zoom link.