# An Hp-Adaptive Minimum Action Method Based on a Posteriori Error Estimate

## Abstract

In this work, we develop an hp-adaptive minimum action method (MAM). MAM plays an important role to minimize the Freidlin-Wentzell action functional, which is the central object of the Freidlin-Wentzell theory of large deviations for transitions induced by small random perturbations. Because of the demanding computation cost, especially in spatially extended systems, numerical efficiency is a critical issue for MAM. Difficulties come from discretizations of both time and space. One hurdle for the application of MAM to large scale systems is the global reparametrization in time direction, which is required by most versions of MAM. We recently introduced a new version of MAM in [25], called tMAM, where we used some simple heuristic criteria to demonstrate that tMAM can be effectively coupled with h-adaptivity. The target of this paper is to integrate hp-adaptivity into tMAM using a posterior error estimation techniques. More specifically, we use the arc length constraint given by geometric minimum action method (gMAM) to define an indicator of the effect of linear time scaling in tMAM, and the derivative recovery technique to construct an error indicator and a regularity indicator for hp-refinement. Numerical results are presented.

- Authors:

- Publication Date:

- Research Org.:
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1492405

- Report Number(s):
- PNNL-SA-125890

Journal ID: ISSN 1815-2406

- DOE Contract Number:
- AC05-76RL01830

- Resource Type:
- Journal Article

- Journal Name:
- Communications in Computational Physics

- Additional Journal Information:
- Journal Volume: 23; Journal Issue: 2; Journal ID: ISSN 1815-2406

- Publisher:
- Global Science Press

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Wan, Xiaoliang, Zheng, Bin, and Lin, Guang.
```*An Hp-Adaptive Minimum Action Method Based on a Posteriori Error Estimate*. United States: N. p., 2018.
Web. doi:10.4208/cicp.OA-2017-0025.

```
Wan, Xiaoliang, Zheng, Bin, & Lin, Guang.
```*An Hp-Adaptive Minimum Action Method Based on a Posteriori Error Estimate*. United States. doi:10.4208/cicp.OA-2017-0025.

```
Wan, Xiaoliang, Zheng, Bin, and Lin, Guang. Mon .
"An Hp-Adaptive Minimum Action Method Based on a Posteriori Error Estimate". United States. doi:10.4208/cicp.OA-2017-0025.
```

```
@article{osti_1492405,
```

title = {An Hp-Adaptive Minimum Action Method Based on a Posteriori Error Estimate},

author = {Wan, Xiaoliang and Zheng, Bin and Lin, Guang},

abstractNote = {In this work, we develop an hp-adaptive minimum action method (MAM). MAM plays an important role to minimize the Freidlin-Wentzell action functional, which is the central object of the Freidlin-Wentzell theory of large deviations for transitions induced by small random perturbations. Because of the demanding computation cost, especially in spatially extended systems, numerical efficiency is a critical issue for MAM. Difficulties come from discretizations of both time and space. One hurdle for the application of MAM to large scale systems is the global reparametrization in time direction, which is required by most versions of MAM. We recently introduced a new version of MAM in [25], called tMAM, where we used some simple heuristic criteria to demonstrate that tMAM can be effectively coupled with h-adaptivity. The target of this paper is to integrate hp-adaptivity into tMAM using a posterior error estimation techniques. More specifically, we use the arc length constraint given by geometric minimum action method (gMAM) to define an indicator of the effect of linear time scaling in tMAM, and the derivative recovery technique to construct an error indicator and a regularity indicator for hp-refinement. Numerical results are presented.},

doi = {10.4208/cicp.OA-2017-0025},

journal = {Communications in Computational Physics},

issn = {1815-2406},

number = 2,

volume = 23,

place = {United States},

year = {2018},

month = {1}

}