A convergent discretization method for transition path theory for diffusion processes
- 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 someTransition 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.…
Author details: | Nada CvetkovićORCiDGND, Tim ConradORCiDGND, Han Cheng LieORCiDGND |
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DOI: | https://doi.org/10.1137/20M1329354 |
ISSN: | 1540-3459 |
ISSN: | 1540-3467 |
Title of parent work (English): | Multiscale modeling & simulation : a SIAM interdisciplinary journal |
Publisher: | Society for Industrial and Applied Mathematics |
Place of publishing: | Philadelphia |
Publication type: | Article |
Language: | English |
Date of first publication: | 2021/02/04 |
Publication year: | 2021 |
Release date: | 2024/02/27 |
Tag: | Monte Carlo; ergodic diffusion processes; methods; rare events; transition paths |
Volume: | 19 |
Issue: | 1 |
Number of pages: | 25 |
First page: | 242 |
Last Page: | 266 |
Funding institution: | Einstein Center for Mathematics Berlin (ECMath) [CH14]; DFG Research Center Matheon/Mathematics for key technologies in BerlinGerman Research Foundation (DFG); German Ministry of Research and Education (BMBF)Federal Ministry of Education & Research (BMBF) [3FO18501]; Deutsche Forschungsgemeinschaft (DFG)German Research Foundation (DFG) [SFB1294/1 -318763901] |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer review: | Referiert |