A Variational Principle Based Study of Random Front Speeds

摘要：DMS-0506766，J Xin，德克萨斯大学标题：基于变分原理的随机波阵面速度研究该项目基于变分原理对随机介质中的反应扩散波阵面速度进行分析和计算，直接计算速度系综既昂贵又不准确。由于前沿速度是随机反应扩散方程解的大时间(大尺度)行为的一部分，因此需要较大的区域大小、足够的前沿结构分辨率和许多随机样本的实现。变分原理是利用解的解析和概率性质建立的。实现了显著的降维，因此只需解决相关的线性问题即可找到主特征值或Lyapunov指数。变分原理除了是速度统计分析的有用工具外，还有助于生成快速高效的计算算法。该项目将探索这种方法来研究各种空间/时间随机介质中的前沿速度、速度渐近规律和随机介质的统计相关性。该项目的动机是环境(森林或建筑火灾)和车辆内燃机中的火焰前沿，其中流体(空气或液体)运动可以显著改变燃烧过程的速度。流体运动通常包含不确定性，可以最好地描述为随机介质。对发动机前缘速度的科学认识和有效计算，有助于控制火焰的蔓延，最大限度地减少内燃机废气对环境的影响。该项目正在开发的方法将有助于理解和计算随机前沿，这是一个在很大程度上处于初级阶段的学科。

Abstract: DMS-0506766, J Xin, University of TexasTitle: A Variational Principle Based Study of Random Front Speeds The project pursues analysis and computation of reaction-diffusionfront speeds in random media based on their variational principles.Direct computation of the speed ensemble can be both expensiveand less accurate. Because front speeds are part of the large time(large scale) behavior of solutions to stochastic reaction-diffusion equations, one needs a large domain size, sufficient resolution of front structures, and many realizations of random samples. The variational principles are establishedby exploiting the analytical and probabilistic properties of solutions. Significant dimensional reductions are achieved so that onlyassociated linear problems need to be solved to find principaleigenvalues or Lyapunov exponents. Besides being a useful tool for analysis of speed statistics, the variational principles also help to generate fast and efficient computational algorithms. The project willexplore this approach to study front speeds in various space/time random media, the speed asymptotic laws and the dependence on statistics of random media.The project is motivated by flame fronts in the environment (forest or building fires) and internal combustion engines of vehicles, where fluid (air or liquid) motion could alter the speed of burning processsignificantly. The fluid motion often contains uncertainties andcan be best described as random media. Scientific understandingand efficient computing of front speeds can help to control thespread of flames, and minimize the waste gases from the combustion enginesto benefit the environment. The methods being developed in the project will contribute to both the understanding and computingof random fronts, a subject largely in its infancy.