Multi-marginal optimal transportation and applications.

多边际最优运输和应用。

• 批准号: 412779-2012
• 负责人: Pass, Brendan
• 金额: $ 1.24万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635303
• 关键词: Multi; marginal; optimal; transportation; applications.

This proposal revolves around the geometry and smoothness of solutions to multi-marginal optimal transportation problems. This is the problem of aligning several prescribed mass distributions, or marginals, as efficiently as possible, relative to a given surplus function. To help illustrate the problem, suppose a large construction company employs, for example, carpenters, electricians and plumbers in order to build a large number of houses. The company, then, needs to make teams; a team consists of one of each type of trades person and will be responsible for building one house. Of course, different combinations of workers will be more or less efficient and it is in the company's best interests to construct its teams such that the overall process is as efficient as possible. My research focuses on the structure of this optimal alignment. Is it optimal, for example, for each carpenter to work with exactly one electrician and one plumber, or is it better for some carpenters to work part-time on several different teams? In the first case, is the assignment of, say, electricians to the carpenters they work with smooth? What happens as the size of each team (ie, the number of types of workers) becomes very large? Surprising, problems of this type interact deeply with mathematical fields such as partial differential equations, geometry and probability and have numerous potential applications in economics, mathematical finance, statistical physics and image processing. However, despite great progress on similar problems with only two types of workers, relatively little is known in the case where teams consist of three or more types of tradespeople. The primary goal of this research is the advancement of scientific knowledge; however, there are many promising, potential applications emerging in the areas mentioned above.

该提议围绕了多核心最佳运输问题的几何形状和平稳性。这是相对于给定的剩余功能，将几个规定的质量分布或边际分配对齐。为了帮助说明这个问题，假设一家大型建筑公司雇用例如木匠，电工和水管工来建造大量房屋。因此，公司需要建立团队；一个团队由每种类型的交易人员组成，将负责建造一所房屋。当然，工人的不同组合将或多或少有效，构建其团队的最大利益符合公司的最大利益，以使整个过程尽可能高效。我的研究重点是这种最佳一致性的结构。例如，每个木匠都可以与一名电工和一名水管工一起工作，还是有些木匠在几个不同的团队上兼职工作是更好的？在第一种情况下，将电工的分配是将他们与他们合作的木匠运转吗？随着每个团队的大小（即工人类型）的规模非常大？令人惊讶的是，这种类型的问题与数学领域（例如部分微分方程，几何图形和概率）有着深入的相互作用，并且在经济学，数学金融，统计物理和图像处理中具有许多潜在的应用。但是，尽管在类似问题的类似问题上取得了长足的进步，但在团队由三种或更多类型的商人组成的情况下，相对较少的知识。这项研究的主要目标是进步科学知识。但是，在上述领域中有许多有希望的潜在应用。