A Minimum Action Method with Optimal Linear Time Scaling (Communications in Computational Physics, 2020)
In this work, we develop a minimum action method (MAM) with optimallinear time scaling, called tMAM for short. The main idea is to relax the integrationtime as a functional of the transition path through optimal linear time scaling suchthat a direct optimization of the integration time is not required. The Feidlin-Wentzellaction functional is discretized by finite elements, based on which h-type adaptivity isintroduced to tMAM. The adaptive tMAM does not require reparametrization for thetransition path. It can be applied to dealing with quasi-potential: 1) When the minimalaction path is subject to an infinite integration time due to critical points, tMAM with auniform 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 actionpath is subject to a finite integration time, tMAM with a uniform mesh convergesat the optimal rate since the problem is not singular, and the optimal integration timecan be obtained directly from the minimal action path. Numerical experiments havebeen implemented for both SODE and SPDE examples.