TY  - JOUR
A1  - Cvetković, Nada
A1  - Conrad, Tim
A1  - Lie, Han Cheng
T1  - A convergent discretization method for transition path theory for diffusion processes
JF  - Multiscale modeling & simulation : a SIAM interdisciplinary journal
N2  - Transition path theory (TPT) for diffusion processes is a framework for analyzing the transitions of multiscale ergodic diffusion processes between disjoint metastable subsets of state space. Most methods for applying TPT involve the construction of a Markov state model on a discretization of state space that approximates the underlying diffusion process. However, the assumption of Markovianity is difficult to verify in practice, and there are to date no known error bounds or convergence results for these methods. We propose a Monte Carlo method for approximating the forward committor, probability current, and streamlines from TPT for diffusion processes. Our method uses only sample trajectory data and partitions of state space based on Voronoi tessellations. It does not require the construction of a Markovian approximating process. We rigorously prove error bounds for the approximate TPT objects and use these bounds to show convergence to their exact counterparts in the limit of arbitrarily fine discretization. We illustrate some features of our method by application to a process that solves the Smoluchowski equation on a triple-well potential.
KW  - ergodic diffusion processes
KW  - transition paths
KW  - rare events
KW  - Monte Carlo
KW  - methods
Y1  - 2021
U6  - https://doi.org/10.1137/20M1329354
SN  - 1540-3459
SN  - 1540-3467
VL  - 19
IS  - 1
SP  - 242
EP  - 266
PB  - Society for Industrial and Applied Mathematics
CY  - Philadelphia
ER  -