Variational Problems and Dynamics in Spaces of Large Dimensions

大维空间中的变分问题和动力学

基本信息

批准号：
2154578
负责人：
Wilfrid Gangbo
金额：
$ 31.55万
依托单位：
University of California-Los Angeles
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-01 至 2025-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154578&HistoricalAwards=false
关键词：
Variational Problems Dynamics Spaces Large

项目摘要

This project investigates a broad array of challenging problems. One of them is on the theory of optimal transport, which dates back to Gaspard Monge in 1781. The theory studies, for instance, the optimal way to move a pile of sand to an excavation, where optimality is measured against a cost function which may be proportional to the distance traveled. Many works led to connections with partial differential equations, fluid mechanics, geometry, probability theory and functional analysis. Currently, optimal transport enjoys applications in signal and image representation, inverse problems, cancer detection, shape and image registration, and machine learning, to name a few. The project relies on the theory of optimal transport to make advances in games which consist of a large number of players, and keeps a focus on equations such as the so-called master equation pioneered by Lasry and Lions. It also features variational problems and dynamics of systems of finitely many and infinitely many particles and also deals with non-commutative optimal transport theory. The project offers training opportunities for undergraduate students, graduate students, and postdoctoral researchers, as well as learning seminars. The students are involved in a research program that aims to encourage interactions between mathematicians, computer scientists, and engineers. The project studies Hamilton-Jacobi equations with non-local terms, known to be challenging, especially when the non-locality appears in the gradient of the unknown function. In this context, the so-called Lasry-Lions monotonicity condition allowed to obtain the first well-posedness result on the master equation. In a previous research project, an alternative condition was proposed, which made it possible for the first time to handle non-separable Hamiltonians. This prior work relied heavily on the presence of the so-called individual noise. This project explores new ideas for handling non-separable Hamiltonians with only common noise in the master equations. It also initiates studies on inverse optimal transport and inverse Mean Field Games problems, with the goal of achieving desired outcomes by designing appropriate Lagrangians. Working on bounded domains makes it more difficult to predict the Lagrangians needed to produce desirable Nash equilibria in Mean Field Games. The project identifies minimal information on the boundary needed to deal with non-smooth augmented Lagrangians which appear in the resolution of these inverse problems. The project also considers inverse optimal transport problems in unbounded sets. Another part of this project is on partial differential equations on graphs and relies on a metric manifold of discrete densities. The project develops a well-posedness theory as well as a stability property for differential equations on graphs. While the role of optimal transport is well established in classical mechanics, establishing these ideas in the quantum setting, opens doors to several new avenues of research. For instance, one faces the challenge of identifying the non-commutative common noise operator, and studying its properties. To achieve these goals, there is a need to continue unearthing new properties of the Biane-Voiculescu's transport distance, and study non-commutative dynamical systems. Non-commutative optimal transport theory offers the appropriate setting for studying dynamics of random matrices, whose sizes will later tend to infinity.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.

该项目调查了许多具有挑战性的问题。其中之一是最佳运输理论，该理论可以追溯到1781年的加斯帕德·蒙格（Gaspard Monge）。许多作品导致与部分微分方程，流体力学，几何学，概率理论和功能分析的联系。当前，最佳运输享受信号和图像表示，反问题，癌症检测，形状和图像注册以及机器学习的应用。该项目依赖于最佳运输理论来在由大量玩家组成的游戏中取得进步，并将重点放在诸如Lasry和Lions开创的所谓的主方程之类的方程式上。它还具有有限的许多粒子和无限多个颗粒的系统的变异问题和动力学，并且还涉及非共同的最佳运输理论。该项目为本科生，研究生和博士后研究人员以及学习研讨会提供了培训机会。这些学生参与了一项研究计划，旨在鼓励数学家，计算机科学家和工程师之间的互动。该项目研究了具有非本地术语的汉密尔顿 - 雅各比方程，已知是具有挑战性的，尤其是当非本地性出现在未知函数的梯度中时。在这种情况下，所谓的Lasry-Lions单调性条件允许在主方程中获得第一个良好的良好性结果。在先前的研究项目中，提出了另一种条件，这使得第一次有可能处理不可分离的哈密顿人。这项先前的工作在很大程度上取决于所谓的个人噪声的存在。该项目探索了处理不可分割的哈密顿人在主方程中仅具有共同噪音的新想法。它还启动了对逆最佳运输和平均野外游戏问题的研究，目的是通过设计适当的Lagrangians来实现所需的结果。在有限的领域上工作使得更难预测平均野外游戏中产生理想的纳什均衡所需的拉格朗日人。该项目确定了有关处理这些反问题所需的非平滑增强Lagrangians所需的边界的最小信息。该项目还考虑了无限制集合中的最佳最佳运输问题。该项目的另一部分是在图形上的部分微分方程上，并依赖于离散密度的度量。该项目开发了一个适合的理论以及图形上微分方程的稳定性。虽然在古典力学中已经建立了最佳运输的作用，并在量子环境中建立了这些思想，但为几种新的研究途径打开了大门。例如，人们面临识别非交通性的公共噪声操作员并研究其特性的挑战。为了实现这些目标，有必要继续发掘Biane-Voiculescu的运输距离的新特性，并研究非共同的动力学系统。非共同的最佳运输理论为研究随机矩阵的动态提供了适当的设置，其大小后来将倾向于无限。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛影响的评估审查标准来通过评估来支持的。