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：其几何形状和应用程序重点是分析围绕最佳运输主题的变异优化和动态演化问题的分析 - 每当Evolution在本地保存标量时，它进入动态设置。中心问题可以用如下概述：鉴于在整个农村的铁矿分布，以及需要铁矿石的工厂分布，确定哪些地雷应向每个工厂供应矿石，以最大程度地降低总运输成本。 在这里，函数C（X，Y）指定了从X矿山运输到工厂的每吨矿石的成本 - 因此，问题可以作为线性程序提出。 但是，当矿山和工厂在整个欧几里得空间中连续分配或具有障碍物的弯曲景观 - 成本与该景观的距离有关时，该问题与几何结构具有丰富的结构，与几何形状和非线性PDE有着深厚的联系，而这些景观只有开始探索。 该问题的化身嵌入了当前模型中，以实现令人惊讶的多样化现象。 除了有关最佳映射的结构和定性特征的基本问题外，拟议的研究还涉及大气中的前部形成模型，动力学理论中的耗散平衡，流体流动，晶体和颗粒状材料，几何学和动态不等式的弹性和动态性不等式，以及在主要的框架中形成的范围或范围内构成的构图，这些框架的范围或构成范围的构成，这些框架构成了构成的构图，以设计构成的范围，这些框架构成了构成的构图，以设计构成的构图，以设计为构成的构图，以设计范围内的构图，以设计范围内的构图，以设计范围。不对称。在半个世纪的数学忽视之后，过去的十年见证了对最佳运输的兴趣，并观察到它蓬勃发展到一个肥沃的调查领域，以及一种探索充满活力的应用程序，用于探索数学内部和超越数学的多样化应用程序。转型是部分问题，因为长期存在的问题可能会出现问题，因为它可能会解决任何问题，因为地理上是在链接中发现的，因为地理上是链接的，因为这些链接是在链接中发现的，因为这些链接都可以解决，因为这些链接是这些问题，因为这些链接是因为这些链接而发现的，因为这些链接是在链接，因为这些链接是这些问题，因为这些链接都可以解决。视觉，部分微分方程，地球科学和经济学。在国际规模上进行协作努力以探索现有联系和发掘新的时期，同时开发了最佳图像的基本理论，并向学生和同事介绍该领域的挑战和希望 - 因此，为了建立了一个以这些目标的重点研究小组，就可以探索现有的联系。 我们计划的核心是安排小组成员之间和周围成员之间的持续互动，除了科学合作外，他们还将在未来几年内共同努力，以创造研究环境和人力，以使运输研究蓬勃发展。为了实现这一目标，我们计划在我们的几个家庭机构中组织一系列三个学期的长时间重点，并在该主题的不同方面进行两个研讨会。 此外，我们计划通过利用赠款的资金来支持年轻的研究人员，同时允许他们在家庭机构和重点学期之间分配时间，同时分享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获得各种各样的观点和专业知识。此外，我们认为，一个为期三年的培养窗口，让年轻的研究人员学习该主题并参与其中 - 如果现在建立 - 最终将在该领域的进步超过十年。