Galerkin Approximation of Dynamical Quantities using Trajectory Data

Understanding chemical mechanisms requires estimating dynamical statistics such as expected hitting times, reaction rates, and committors. Here, we present a general framework for calculating these dynamical quantities by approximating boundary value problems using dynamical operators with a Galerki...

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Bibliographic Details
Published in:arXiv.org 2019-02
Main Authors: Thiede, Erik H, Giannakis, Dimitrios, Dinner, Aaron R, Weare, Jonathan
Format: Article
Language:English
Subjects:
Boundary conditions
Boundary value problems
Embedded systems
Galerkin method
Lag time
Markov chains
Model accuracy
Organic chemistry
Response time
Online Access:Get full text
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Summary:Understanding chemical mechanisms requires estimating dynamical statistics such as expected hitting times, reaction rates, and committors. Here, we present a general framework for calculating these dynamical quantities by approximating boundary value problems using dynamical operators with a Galerkin expansion. A specific choice of basis set in the expansion corresponds to estimation of dynamical quantities using a Markov state model. More generally, the boundary conditions impose restrictions on the choice of basis sets. We demonstrate how an alternative basis can be constructed using ideas from diffusion maps. In our numerical experiments, this basis gives results of comparable or better accuracy to Markov state models. Additionally, we show that delay embedding can reduce the information lost when projecting the system's dynamics for model construction; this improves estimates of dynamical statistics considerably over the standard practice of increasing the lag time.
ISSN:2331-8422
DOI:10.48550/arxiv.1810.01841