Optimal transportation: geometry and dynamics

最佳交通：几何和动力学

• 批准号: 217006-2008
• 负责人: McCann, Robert
• 金额: $ 3.06万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=546496
• 关键词: Optimal; transportation; geometry; dynamics

This proposal focuses on the analysis of a family of optimization and evolution problems having applications in physics, geometry, medical imaging, partial differential equations, and economics. Linking them together is a transportation problem --- which comes into play whenever the dynamics conserve a local quantity. It can be caricatured 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). However, when the mines and factories are distributed continuously over a curved landscape, then the problem has a rich structure and deep connections to geometry, nonlinear partial differential equations and the topology of the cost function, which have only begun to be explored.

该建议的重点是分析一个家庭的优化和进化问题，在物理学，几何学，医学成像，偏微分方程和经济学的应用。 将它们联系在一起的是一个运输问题-每当动力学保持一个局部量时，这个问题就开始起作用。 它可以被漫画化如下：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，从x点的矿山运输到y点的工厂的每吨矿石的成本由函数c（x，y）指定。 然而，当矿山和工厂连续分布在一个弯曲的景观，那么这个问题有一个丰富的结构和深刻的联系，几何，非线性偏微分方程和拓扑的成本函数，这才刚刚开始探索。