A Variational Principle Based Study of Random Front Speeds

摘要/ Abstract摘要：DMS-0506766， J Xin，美国德克萨斯大学。题目：基于变分原理的随机前速度研究。该项目基于随机介质中反应扩散前速度的变分原理进行分析与计算。直接计算速度集合既昂贵又不准确。由于前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，因此需要大的域尺寸、足够的前结构分辨率和许多随机样本的实现。通过利用解的解析性和概率性，建立了变分原理。实现了显著的降维，因此只需要解决相关的线性问题来找到原理值或李亚普诺夫指数。变分原理除了是分析速度统计数据的有用工具外，还有助于生成快速有效的计算算法。本项目将探索这种方法来研究不同时空随机介质中的前速度、速度渐近规律和随机介质对统计量的依赖。该项目的动力来自于环境中的火焰前沿（森林或建筑火灾）和车辆的内燃机，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。对前方速度的科学理解和有效计算有助于控制火焰的蔓延，并最大限度地减少内燃机产生的废气，从而有利于环境。该项目中正在开发的方法将有助于理解和计算随机前沿，这一学科在很大程度上还处于起步阶段。

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.