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

偏微分方程（PDEs）描述的许多物理系统受到微观变化的影响，这些变化可能被描述为随机的。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。研究者将研究带有随机系数的偏微分方程。在某些情况下，宏观系统可以用一个更简单的“有效方程”的解很好地近似。然而，在其他情况下，宏观有效方程不足以表征系统的行为，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，理解偏离平均行为是很重要的。项目的第一部分是描述反应扩散方程和一些相关方程的类波解和类脉冲解的随机涨落。由于系数的随机波动，解是随机的，理解解的统计特性如何取决于介质的统计特性是很重要的。第二个目标是研究高维界面运动模型解的随机波动。项目的第三个目标是研究随机系数椭圆方程解的涨落。了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一个重要问题。当只知道模型参数的一些统计特性时，如何量化模型预测中的不确定性？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在湍流燃烧、种群生态学和神经科学等现象模型中出现的反应扩散方程。该项目还将考虑在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，诸如磁导率或电导率之类的材料特性随机变化，并且这些参数的一些统计特性是已知的。

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.