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Nonlinear Partial Differential Equations and Applications

非线性偏微分方程及其应用

基本信息

批准号：
1900599
负责人：
Panagiotis Souganidis
金额：
$ 29.79万
依托单位：
University of Chicago
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1900599&HistoricalAwards=false
关键词：
Nonlinear Partial Differential Equations Applications

项目摘要

Many applications in Physical and Social Sciences and Engineering, like porous media, composite material, turbulence and combustion, traffic models, spread of crime, climate modeling and prediction, agent models and others, involve heterogeneous media described by partial differential equations, which, typically, depend on many parameters and vary randomly on a small scale. In addition, often the available information, as, for example, in weather prediction, is not exact (deterministic) but statistical (random) with large fluctuations. On macroscopic scales that are much larger than the ones of the heterogeneities, the models often show an effective deterministic behavior, which is much simpler than the original one. The process of the averaging is known as homogenization. Mathematically, this means that the original random and inhomogeneous problem is replaced by a deterministic and homogeneous one. When this averaging is not possible, which is typically the case when the fluctuations are too strong (wild), it is necessary to deal with stochastic media (stochastic partial differential equations), which have rather singular behavior in space and time. The mathematical study of both the stochastic averaging and the stochastic partial differential equations requires original ideas and the development of new methodologies, since both topics fall outside the traditional theories of averaging and partial differential equations. Another burgeoning area of research is the theory of mean field games. Applications that have been so far looked at range from complex socio-economical topics, regulatory financial issues, crowd movement, meaningful big data and advertising to engineering contexts involving "decentralized intelligence'" and machine learning. Mean field games are the ideal mathematical structures to study the quintessential problems in the social-economical sciences, which differ from physical settings because of the forward looking behavior on the part of individual agents. Concrete examples of applications in this direction include the modeling of the macroeconomy and conflicts in the modern era. In both cases, a large number of agents interact strategically in a stochastically evolving environment, all responding to partly common and partly idiosyncratic incentives, and all trying to simultaneously forecast the decision of others. Training of graduate students is an integral part of this research project.The project is about developing general methodologies to study random homogenization, nonlinear stochastic partial differential equations and applications to front propagation, phase transitions, and mean field games. Random environments are much more general than periodic ones. The latter are basically fixed translations of a certain equation while the former can be thought as all the possible equations. This leads to considerable issues of lack of compactness. It is therefore necessary to develop novel tools that combine both the differential and random structures of the media. In this setting, the equation is the random variable and the special dependence signifies the location in space where the equation is observed. The PI and his collaborators were the first to consider stochastic homogenization in stationary ergodic environments. A large part of the project is about the further development of the theory. Stochastic partial differential equations have coefficients with very singular (Brownian) behavior. In the linear context, this can be typically handled by known methods like the classical martingale approach. The latter is based on the linear character of the higher order part of the equation and thus cannot be used for nonlinear problems, where it is necessary to find appropriate notions of solutions. In the context of first- and second-order nonlinear equations, these are the stochastic viscosity and pathwise entropy solutions, introduced by the PI and his collaborators. A part of the project is about the study of the qualitative behavior/properties of these solutions. In the context of mean field games, the PI will concentrate on the role of inhomogeneities at the several level of the game up to the master equation and models with common noise. This will require the development of novel techniques to understand the behavior of the problem past singularities and the role of the averaging.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

物理和社会科学及工程中的许多应用，如多孔介质、复合材料、湍流和燃烧、交通模型、犯罪蔓延、气候建模和预测、代理模型等，都涉及由偏微分方程描述的异质介质，这些介质通常取决于许多参数并在小范围内随机变化。此外，可用的信息（例如天气预报）通常不是精确的（确定性的），而是具有较大波动的统计信息（随机的）。在比异质性大得多的宏观尺度上，模型通常表现出有效的确定性行为，这比原始模型简单得多。平均的过程称为均质化。从数学上讲，这意味着原来的随机和非齐次问题被确定性和齐次问题所取代。当无法进行这种平均时（通常是波动太强（剧烈）时的情况），有必要处理随机介质（随机偏微分方程），它在空间和时间上具有相当奇异的行为。随机平均和随机偏微分方程的数学研究需要原创思想和新方法的开发，因为这两个主题都不属于平均和偏微分方程的传统理论。另一个新兴的研究领域是平均场博弈理论。迄今为止所研究的应用范围从复杂的社会经济主题、监管金融问题、人群流动、有意义的大数据和广告到涉及“去中心化智能”和机器学习的工程环境。平均场博弈是研究社会经济科学中典型问题的理想数学结构，由于个体主体的前瞻性行为，它与物理环境不同。该方向应用的具体例子包括现代宏观经济和冲突的建模。在这两种情况下，大量智能体在随机演变的环境中进行战略性交互，所有智能体都对部分常见和部分特殊的激励做出反应，并且都试图同时预测其他人的决策。研究生培训是该研究项目的一个组成部分。该项目旨在开发研究随机均匀化、非线性随机偏微分方程及其在前向传播、相变和平均场博弈中的应用的通用方法。随机环境比周期性环境更普遍。后者基本上是某个方程的固定翻译，而前者可以被认为是所有可能的方程。这导致了相当大的缺乏紧凑性的问题。因此，有必要开发结合媒体的差异结构和随机结构的新颖工具。在此设置中，方程是随机变量，特殊相关性表示观察方程的空间位置。 PI 和他的合作者是第一个考虑静态遍历环境中的随机均质化的人。该项目的很大一部分是关于理论的进一步发展。随机偏微分方程的系数具有非常奇异（布朗）行为。在线性环境中，这通常可以通过已知方法（例如经典鞅方法）来处理。后者基于方程高阶部分的线性特征，因此不能用于需要找到适当解概念的非线性问题。在一阶和二阶非线性方程的背景下，这些是由 PI 及其合作者引入的随机粘度和路径熵解。该项目的一部分是关于这些解决方案的定性行为/属性的研究。在平均场博弈的背景下，PI 将专注于博弈的多个级别上的不均匀性的作用，直至主方程和具有共同噪声的模型。这将需要开发新颖的技术来理解过去奇点问题的行为以及平均的作用。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。