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.

在物理和社会科学和工程中的许多应用，例如多孔媒体，复合材料，湍流和燃烧，交通模型，犯罪，气候建模和预测，代理模型等，涉及部分差分方程所描述的媒体，通常依赖许多参数，并取决于许多参数，并且在小规模上随机随机。此外，通常，在天气预测中，通常的可用信息不是准确的（确定性），而是统计（随机），并且波动很大。在宏观尺度上比异质性大得多，模型通常显示出有效的确定性行为，这比原始的行为要简单得多。平均过程称为均匀化。从数学上讲，这意味着原始的随机和不均匀问题被确定性和均匀的问题取代。当这种平均是不可能的（通常是波动太强（野性）时）时，有必要处理随机介质（随机偏微分方程），这在时空和时间上具有相当奇异的行为。对随机平均和随机偏微分方程的数学研究都需要原始思想和新方法的发展，因为这两个主题都符合传统的平均和部分微分方程的理论。研究的另一个新兴领域是平均现场游戏的理论。到目前为止，已经关注的应用程序从复杂的社会经济主题，监管财务问题，人群运动，有意义的大数据和广告到涉及“分散智能”和机器学习的工程环境。平均野外游戏是研究社会经济科学中典型问题的理想数学结构，由于各个代理商的前瞻性行为，它们与物理环境有所不同。在这个方向上应用的具体示例包括宏观经济和现代冲突的建模。在这两种情况下，许多代理商在随机发展的环境中战略性互动，都对部分常见和部分特质的激励措施做出了反应，并且所有人都试图同时预测他人的决定。该研究项目的培训是研究项目的组成部分。该项目旨在开发一般方法来研究随机均质化，非线性随机局部微分方程以及对前繁殖，相变和平均野外游戏的应用。随机环境比周期性环境更一般。后者基本上是特定方程式的固定翻译，而前者可以被认为是所有可能的方程式。这导致缺乏紧凑的问题。因此，有必要开发新的工具，以结合介质的差分结构和随机结构。在这种情况下，方程是随机变量，特殊依赖性表示观察到方程的空间位置。 PI和他的合作者是第一个考虑在固定的厄贡环境中随机均质化的人。该项目的很大一部分是关于该理论的进一步发展。随机部分微分方程具有非常单数（布朗）行为的系数。在线性上下文中，通常可以通过已知的方法来处理，例如经典的Martingale方法。后者基于方程式高阶的线性特征，因此不能用于非线性问题，需要找到适当的解决方案概念。在一阶非线性方程式的背景下，这些是随机粘度和路径熵解决方案，由PI及其合作者介绍。该项目的一部分是关于这些解决方案的定性行为/特性的研究。在平均野外游戏的背景下，PI将集中于在游戏的多个级别上的不均匀性的作用，直到主程组和具有共同噪音的模型。这将需要开发新颖的技术，以了解过去奇点的行为和平均作用。该奖项反映了NSF的法定使命，并使用基金会的知识分子优点和更广泛的影响评估标准，被认为值得通过评估来获得支持。