Communications in Mathematical Sciences

Volume 15 (2017)

Number 8

Multiscale gentlest ascent dynamics for saddle point in effective dynamics of slow-fast system

Pages: 2279 – 2302



Shuting Gu (Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong)

Xiang Zhou (Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong)


Here we present a multiscale method to calculate the saddle point associated with the effective dynamics arising from a stochastic system which couples slow deterministic drift and fast stochastic dynamics. This problem is motivated by the transition states on free energy surfaces in chemical physics. Our method is based on the gentlest ascent dynamics which couples the position variable and the direction variable and has the local convergence to saddle points. The dynamics of the direction vector is derived in terms of the covariance function with respective to the equilibrium distribution of the fast stochastic process. We apply multiscale numerical methods to efficiently solve the obtained multiscale gentlest ascent dynamics, and discuss the acceleration techniques based on an adaptive idea. The examples of stochastic ordinary and partial differential equations are presented.


saddle point, gentlest ascent dynamics, multiscale method

2010 Mathematics Subject Classification

65K05, 82B05

The research of X.Z. was supported by the grants from the Research Grants Council of the Hong Kong SAR, China (Project No. CityU 11304314, 109113 and 11304715).

Received 15 November 2016

Accepted 23 August 2017

Published 20 December 2017