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A Minimum Action Method with Optimal Linear Time Scaling

Part of: Numerical methods in calculus of variations and optimal control Stochastic analysis

Published online by Cambridge University Press:  23 November 2015

Xiaoliang Wan
Xiaoliang Wan*
Affiliation:
Department of Mathematics, Center for Computation and Technology, Louisiana State University, Baton Rouge 70803, USA.
*
*Corresponding author. Email address: xlwan@math.lsu.edu(X. Wan)
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Abstract

In this work, we develop a minimum action method (MAM) with optimal linear time scaling, called tMAM for short. The main idea is to relax the integration time as a functional of the transition path through optimal linear time scaling such that a direct optimization of the integration time is not required. The Feidlin-Wentzell action functional is discretized by finite elements, based on which h-type adaptivity is introduced to tMAM. The adaptive tMAM does not require reparametrization for the transition path. It can be applied to deal with quasi-potential: 1) When the minimal action path is subject to an infinite integration time due to critical points, tMAM with a uniform mesh converges algebraically at a lower rate than the optimal one. However, the adaptive tMAM can recover the optimal convergence rate. 2) When the minimal action path is subject to a finite integration time, tMAM with a uniform mesh converges at the optimal rate since the problem is not singular, and the optimal integration time can be obtained directly from the minimal action path. Numerical experiments have been implemented for both SODE and SPDE examples.

Keywords

Large deviation principlesmall random perturbationsminimum action methodrare eventsuncertainty quantification.

MSC classification

Secondary: 49M25: Discrete approximations 60H30: Applications of stochastic analysis (to PDE, etc.) 65K10: Optimization and variational techniques
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 5 , November 2015 , pp. 1352 - 1379
Copyright
Copyright © Global-Science Press 2015 

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