An $hp$-Adaptive Minimum Action Method Based on a Posteriori Error Estimate (Communications in Computational Physics, 2018)
In this work, we develop an hp-adaptivity strategy for the minimum actionmethod (MAM) using a posteriori error estimate. MAM plays an important role inminimizing the Freidlin-Wentzell action functional, which is the central object of theFreidlin-Wentzell theory of large deviations for noise-induced transitions in stochasticdynamical systems. Because of the demanding computation cost, especially in spatiallyextended systems, numerical efficiency is a critical issue for MAM. Difficultiescome from both temporal and spatial discretizations. One severe hurdle for the applicationof MAM to large scale systems is the global reparametrization in time direction,which is needed in most versions of MAM to achieve accuracy. We recently introduceda new version of MAM in [22], called tMAM, where we used some simple heuristic criteriato demonstrate that tMAM can be effectively coupled with $h$-adaptivity, i.e., theglobal reparametrization can be removed. The target of this paper is to integrate $hp$-adaptivityinto tMAM using a posteriori error estimation techniques, which provides ageneral adaptive MAM more suitable for parallel computing. More specifically, we usethe zero-Hamiltonian constraint to define an indicator to measure the error induced bylinear time scaling, and the derivative recovery technique to construct an error indicatorand a regularity indicator for the transition paths approximated by finite elements.Strategies for $hp$-adaptivity have been developed. Numerical results are presented.
Keywords: Large deviation principle; small random perturbations; minimum action method; rare events; uncertainty quantification.