TY - JOUR

T1 - The geometric minimum action method

T2 - A least action principle on the space of curves

AU - Heymann, Matthias

AU - Vanden-Eijnden, Eric

PY - 2008/8

Y1 - 2008/8

N2 - 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.

AB - 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.

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U2 - 10.1002/cpa.20238

DO - 10.1002/cpa.20238

M3 - Article

AN - SCOPUS:49449085444

VL - 61

SP - 1052

EP - 1117

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 8

ER -