The geometric minimum action method: A least action principle on the space of curves

Matthias Heymann, Eric Vanden-Eijnden

Research output: Contribution to journalArticle

Abstract

Freidlin-Wentzell theory of large deviations for the description of the effect of small random perturbations on dynamical systems is exploited as a numerical tool. Specifically, a numerical algorithm is proposed to compute the quasi-potential in the theory, which is the key object to quantify the dynamics on long time scales when the effect of the noise becomes ubiquitous: the equilibrium distribution of the system, the pathways of transition between metastable states and their rate, etc., can all be expressed in terms of the quasi-potential. We propose an algorithm to compute these quantities called the geometric minimum action method (gMAM), which is a blend of the original minimum action method (MAM) and the string method. It is based on a reformulation of the large deviations action functional on the space of curves that allows one to easily perform the double minimization of the original action required to compute the quasi-potential. The theoretical background of the gMAM in the context of large deviations theory is discussed in detail, as well as the algorithmic aspects of the method. The gMAM is then illustrated on several examples: a finite-dimensional system displaying bistability and modeled by a nongradient stochastic ordinary differential equation, an infinite-dimensional analogue of this system modeled by a stochastic partial differential equation, and an example of a bistable genetic switch modeled by a Markov jump process.

Original languageEnglish (US)
Pages (from-to)1052-1117
Number of pages66
JournalCommunications on Pure and Applied Mathematics
Volume61
Issue number8
DOIs
StatePublished - Aug 2008

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Curve
Ordinary differential equations
Partial differential equations
Dynamical systems
Switches
Large Deviations
Stochastic Ordinary Differential Equations
Large Deviation Theory
Markov Jump Processes
Random Perturbation
Metastable States
Equilibrium Distribution
Bistability
Stochastic Partial Differential Equations
Reformulation
Small Perturbations
Numerical Algorithms
Pathway
Switch
Time Scales

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

The geometric minimum action method : A least action principle on the space of curves. / Heymann, Matthias; Vanden-Eijnden, Eric.

In: Communications on Pure and Applied Mathematics, Vol. 61, No. 8, 08.2008, p. 1052-1117.

Research output: Contribution to journalArticle

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