Asymptotic Problems of Partial Differential Equations with Random Coefficients: Homogenization and Beyond

具有随机系数的偏微分方程的渐近问题：齐次化及其他

• 批准号: 1613301
• 负责人: Yu Gu
• 金额: $ 13.99万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2018-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1613301&HistoricalAwards=false
• 关键词: Asymptotic; Problems; Partial; Differential; Equations

This research is devoted to the analysis of systems with parameters that contain uncertainties and vary on multiple scales, and the goal is to understand their effects on the dynamics of the systems. Examples of such systems are ubiquitous in nature: climate and weather developments, oceanography, and seismology, just to name a few. Similarly, multiscale processes with built-in uncertainty are common in many large-scale societal phenomena, from stock market behavior to propagation of information through social media. The mathematical subject is partial differential equations (PDE) with random and highly oscillatory coefficients, and the project is aimed at deriving effective models that incorporate the impacts from the randomness and the separation of scales. The project is expected to result in connections to multi-scale algorithms, inverse problems, imaging, and uncertainty quantification, as well as to have applications in subjects such as climate prediction, geophysics, and materials science. The project involves three investigations, with a focus on questions related to stochastic homogenization, wave in random media and stochastic PDE: (i) study the quantitative aspects, including convergence rates and statistical fluctuations, of stochastic homogenization and macroscopic models of wave propagation in random media; (ii) analyze the effects of different correlation properties of the randomness on both qualitative and quantitative aspects; and (iii) investigate the transition from deterministic to stochastic models, and study the convergence to stochastic PDE beyond the homogenization regime.

这项研究致力于分析含有不确定性和在多个尺度上变化的参数的系统，目的是了解它们对系统动力学的影响。这类系统的例子在自然界中随处可见：气候和天气发展、海洋学和地震学，仅举几例。同样，具有内在不确定性的多尺度过程在许多大规模社会现象中也很常见，从股市行为到通过社交媒体传播信息。该项目的数学主题是具有随机和高度振荡系数的偏微分方程组(PDE)，该项目的目的是推导包含随机性和尺度分离的影响的有效模型。该项目预计将导致与多尺度算法、反问题、成像和不确定性量化的联系，并将在气候预测、地球物理和材料科学等学科中得到应用。本项目包括三个方面的研究，重点是关于随机齐化、随机介质中的波和随机偏微分方程的问题：(I)研究随机齐化的数量方面，包括收敛速度和统计涨落，以及随机介质中波传播的宏观模型；(Ii)分析不同的随机性相关性在定性和定量方面的影响；以及(Iii)研究从确定性模型到随机模型的过渡，并研究在齐化制度之外向随机偏微分方程的收敛。