Mathematical Sciences: The Geometry of Optimal Transportation

数学科学：最优运输的几何

• 批准号: 9622997
• 负责人: Robert McCann
• 金额: $ 7.67万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 1999-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622997&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Optimal; Transportation

Abstract McCann 9622997 This proposal focuses on the analysis of a long-standing problem which arises in economics and operations research as well as probability and statistics. Given two distributions (f and g) of mass over a Riemannian manifold M, the problem is to determine the most efficient way to rearrange the mass of the first distribution to yield the second. Efficiency is measured against a function c(x,y) which specifies the cost per unit mass for transporting material from x to y --- so the problem can be formulated as a linear program. However, when the cost function is related to the metrical distance between pairs of points on M, then the problem has a rich structure and deep connections to geometry and non-linear partial differential equations which have only begun to be explored. For costs which are either convex or concave functions of this distance, the goal would be to find a map from the manifold to itself which minimizes the total transportation costs, among maps pushing the measure f forward to g. Even on the line, this map can be intricate. Recent developments in Euclidean space suggest some promising questions to explore: What conditions guarantee existence and uniqueness of an optimal mapping? What geometrical or analytic properties characterize the optimal maps? Can this geometry be exploited fruitfully in applications? The motivation for this research is illustrated by an example from economics: 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. The mines and factories lie on a curved space M, like the surface of the earth, but general enough to include barriers to transportation such as lakes and mountain ranges; the cost per ton for transporting ore from any mine to factory is determined by the distance measured between them in thi s space. Aside from industrial applications, the solution to this problem should yield new understanding of existing patterns in the economy, and may prove useful for infrastructure planning. Because of its close relationship to several areas of pure mathematics, the research promises to stimulate a fruitful exchange in two directions: powerful mathematics will be brought to bear on problems from the real word, while concrete solutions to those problems should provide new insight into the mathematics.

摘要 McCann 9622997 该提案侧重于分析经济学和运筹学以及概率和统计中出现的长期存在的问题。给定黎曼流形 M 上的两个质量分布（f 和 g），问题是确定重新排列第一个分布的质量以产生第二个分布的最有效方法。效率是根据函数 c(x,y) 来衡量的，该函数指定将材料从 x 运输到 y 的每单位质量的成本——因此该问题可以表述为线性程序。然而，当成本函数与 M 上的点对之间的度量距离相关时，问题就具有丰富的结构，并且与刚刚开始探索的几何和非线性偏微分方程有深刻的联系。对于该距离的凸函数或凹函数的成本，目标是在将度量 f 推向 g 的映射中找到从流形到其自身的映射，该映射使总运输成本最小化。即使在线上，这张地图也可能很复杂。欧几里得空间的最新发展提出了一些有希望探索的问题：什么条件保证最优映射的存在和唯一性？ 最佳地图具有哪些几何或分析属性？ 这种几何形状可以在应用中得到有效利用吗？ 这项研究的动机可以用一个经济学的例子来说明：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矿石，以最小化总运输成本。矿山和工厂位于弯曲的空间M上，就像地球表面一样，但足够普遍，包括湖泊和山脉等交通障碍；将矿石从任何矿山运输到工厂的每吨成本取决于该空间中矿山之间测量的距离。 除了工业应用之外，这个问题的解决方案还应该产生对现有经济模式的新理解，并且可能对基础设施规划有用。 由于其与纯数学的多个领域的密切关系，该研究有望在两个方向上促进富有成效的交流：强大的数学将应用于现实世界中的问题，而这些问题的具体解决方案应该为数学提供新的见解。