AMC-SS: Analysis of Fluctuations for PDEs with Random Coefficients

AMC-SS：具有随机系数的偏微分方程的波动分析

• 批准号: 1007572
• 负责人: James Nolen
• 金额: $ 20.69万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007572&HistoricalAwards=false
• 关键词: AMC; SS; Analysis; Fluctuations; PDEs

Many physical systems described by partial differential equations (PDEs) are influenced by microscopic variations that may be characterized as random. A fundamental scientific and mathematical problem is to understand how these random variations influence the system at a macroscopic level. The investigator will study partial differential equations with random coefficients. In some regimes, the macroscopic system is well-approximated by the solution to a simpler "effective equation." In other regimes, however, the macroscopic effective equation is insufficient to characterize the system's behavior because random microscopic variations produce residual fluctuations in the macroscopic system. In this case, it is important to understand the deviations from the mean behavior. The first part of the project is to characterize the random fluctuations of the wave-like and pulse-like solutions to reaction-diffusion equations and some related equations. Because of the random fluctuation in the coefficients, the solution is random, and it is important to understand how the statistical character of the solution depends on statistical properties of the medium. The second goal is to study random fluctuations of solutions to higher-dimensional models for interface motion. The third goal of the project is to study fluctuations of solutions to elliptic equations with random coefficients.Understanding the effect of random microstructure on a macroscopic mathematical model is an important problem in many scientific and engineering applications. How can one quantify uncertainty in predictions of a model when only some statistical properties of the model parameters are known? The analytical techniques developed through this project will have an impact in the area of uncertainty quantification for PDE-based mathematical models. The specific equations to be studied include reaction-diffusion equations which arise in models of phenomena in turbulent combustion, population ecology, and neuroscience, for example. The project also will consider elliptic equations which arise in applications such as hydrology and materials science. In these applications, material properties like permeability or electrical conductivity vary randomly and only some statistical properties of these parameters are known.

许多由偏微分方程（PDE）描述的物理系统受到微观变化的影响，这些微观变化可以被表征为随机的。 一个基本的科学和数学问题是理解这些随机变化如何在宏观层面上影响系统。研究者将研究具有随机系数的偏微分方程。在某些情况下，宏观系统可以很好地近似为一个更简单的“有效方程”的解。然而，在其他情况下，宏观有效方程不足以描述系统的行为，因为随机的微观变化会在宏观系统中产生残余波动。 在这种情况下，重要的是要了解偏离平均行为。第一部分是研究反应扩散方程及相关方程的波动解和脉冲解的随机涨落。由于系数的随机波动，解是随机的，重要的是要理解解的统计特征如何取决于介质的统计性质。第二个目标是研究界面运动的高维模型解的随机波动。本项目的第三个目标是研究具有随机系数的椭圆方程解的涨落。理解随机微观结构对宏观数学模型的影响是许多科学和工程应用中的重要问题。当仅知道模型参数的一些统计特性时，如何量化模型预测的不确定性？通过该项目开发的分析技术将对基于偏微分方程的数学模型的不确定性量化领域产生影响。 要研究的具体方程包括反应扩散方程，例如，这些方程出现在湍流燃烧、人口生态学和神经科学中的现象模型中。 该项目还将考虑水文学和材料科学等应用中出现的椭圆方程。在这些应用中，材料特性（如磁导率或电导率）随机变化，并且仅知道这些参数的一些统计特性。