Collaborative Research: Optimal Transportation: Its Geometry and Applications

合作研究：最优交通：其几何结构和应用

• 批准号: 0074037
• 负责人: Wilfrid Gangbo
• 金额: $ 95万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-08-15 至 2005-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0074037&HistoricalAwards=false
• 关键词: Collaborative; Research; Optimal; Transportation; Its

ABSTRACT:Optimal Transportation: Its Geometry and ApplicationsThis project focuses on the analysis of a collection of variational optimization and dynamical evolution problems centered around the theme of optimal transportation --- which enters the dynamical setting whenever the evolution conserves a scalar locally. The central problem can be sketched as follows: Given a distribution of iron mines throughout the countryside, and a distribution of factories which require iron ore, decide which mines should supply ore to each factory in order to minimize the total transportation costs. Here the cost per ton of ore transported from the mine at x to factory at y is specified by a function c(x,y) --- so the problemcan be formulated as a linear program. However, when the mines and factories are distributed continuously throughout Euclidean space or a curved landscape with obstacles --- and the cost is related to the distance on this landscape, then the problem has a rich structure and deep connections to geometry and non-linear PDE which have only begun to be explored. Incarnations of this problem embed in current models for surprisingly diverse phenomena. Along with basic questions concerning the structure and qualitative features of optimal mappings, the proposed research addresses models for front formation in the atmosphere, dissipative equilibration in kinetic theory, fluid flow, elastic crystals, and granular materials, geometric and dynamical inequalities, and microeconomic decision problems formulated in the principal-agent framework which involve designing price systems, tax structures, or contracts in the face of informational asymmetry.After half a century of mathematical neglect, the past decade witnessed a revival of interest in optimal transportation, and watched as it blossomed into a fertile field of investigation as well as a vibrant tool for exploring diverse applications within and beyond mathematics.The transformation occurred partly because long-standing issues could finally be resolved, but also because unexpected connections were discovered which linked these questions to problems in physics, geometry, computer vision, partial differential equations, earth science and economics. The time is ripe for a collaborative effort on an international scale to explore existing connections and unearth new ones, while simultaneously developing the basic theory of optimalmaps and introducing students and colleagues to the challenges and promise of the field --- thus for the formation of a focused research group with these goals. The core of our plan is to arrange sustained interactions between and around members of the group, who in addition to collaborating scientifically, will work together over the next several years to create the research environment and manpower necessary for transportation research to flourish. To achieve this goal, we plan to organize a series of three semester long periods of emphasis and two workshops on different aspects of the subject in several of our home institutions. Furthermore, we plan to share the responsibilities of training graduate students and postdoctoral fellows, by using funds from the grant to support young researchers while allowing them to divide their time between their home institutions and the semesters of emphasis. This unique arrangement will give participants access to an unusually broad assortment of perspectives and expertise. Moreover, we believe a three-year nurturing window for young researchers to learn the subject and become involved --- if established now - will ultimately advance progress in the field by more than a decade.

摘要：最佳运输：它的几何和应用这个项目的重点是分析一系列的变分优化和动态演化问题的主题为中心的最佳运输-这进入动态设置时，演变保存一个标量局部。中心问题可以概括如下：给定铁矿在农村的分布，以及需要铁矿石的工厂的分布，决定哪些矿山应该向每个工厂供应铁矿石，以使总运输成本最小化。 这里，每吨矿石从x点的矿山运输到y点的工厂的成本由函数c（x，y）确定，所以这个问题可以用线性规划来表示。 然而，当矿山和工厂连续分布在欧几里得空间或有障碍物的弯曲景观中时---成本与景观上的距离有关，那么问题就具有丰富的结构和与几何和非线性PDE的深刻联系，而这些只是开始探索。 这一问题的具体表现体现在目前各种现象的模型中。 沿着关于最优映射的结构和定性特征的基本问题，提出的研究解决了大气中锋面形成的模型，动力学理论中的耗散平衡，流体流动，弹性晶体和颗粒材料，几何和动力学不等式，以及在委托代理框架中制定的微观经济决策问题，其中涉及设计价格体系，税收结构，在信息不对称的情况下，人们对最优运输的兴趣在经过了半个世纪的忽视之后，在过去的十年里又重新燃起，并见证了最优运输发展成为一个肥沃的研究领域，以及一个充满活力的工具，用于探索数学内外的各种应用。这种转变的发生，部分原因是长期存在的问题终于得到了解决，还因为人们发现了意想不到的联系，将这些问题与物理学、几何学、计算机视觉、偏微分方程、地球科学和经济学中的问题联系起来。时机已经成熟，在国际范围内合作努力，探索现有的连接和发掘新的，同时发展optimalmaps的基本理论，并向学生和同事介绍该领域的挑战和前景-从而形成一个有这些目标的重点研究小组。 我们计划的核心是安排小组成员之间和周围的持续互动，他们除了科学合作外，还将在未来几年内共同努力，创造交通研究蓬勃发展所需的研究环境和人力。为了实现这一目标，我们计划在我们的几个家庭机构组织一系列的三个学期长的重点和两个主题的不同方面的研讨会。 此外，我们计划通过使用补助金中的资金来支持年轻研究人员，同时允许他们在家乡机构和重点学期之间分配时间，从而分担培训研究生和博士后研究员的责任。 这种独特的安排将使参与者能够获得异常广泛的观点和专业知识。此外，我们相信，一个为期三年的培养窗口，让年轻研究人员学习这一主题并参与其中-如果现在建立-最终将把该领域的进展推进十多年。