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

本研究致力于分析包含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。这种系统的例子在自然界中无处不在：气候和天气发展、海洋学和地震学，仅举几例。同样，从股票市场行为到通过社交媒体传播的信息，具有内置不确定性的多尺度过程在许多大规模社会现象中都很常见。数学主题是具有随机和高振荡系数的偏微分方程（PDE），该项目旨在推导出包含随机性和尺度分离影响的有效模型。该项目预计将产生与多尺度算法、逆问题、成像和不确定性量化的联系，并在气候预测、地球物理和材料科学等学科中得到应用。该项目涉及三项调查，重点是与随机均质化、随机介质中的波和随机偏微分方程有关的问题：(i)研究随机均质化和随机介质中波传播的宏观模型的数量方面，包括收敛率和统计波动；（ii）分析随机性的不同相关特性对定性和定量的影响；（iii）研究从确定性模型到随机模型的转变，并研究均匀化范围以外的随机PDE的收敛性。

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.