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: 2012
• 资助国家: 加拿大
• 起止时间: 2012-01-01 至 2013-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=508679
• 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.

这一建议围绕着多边际最优运输问题解决方案的几何和平滑性展开。这是相对于一个给定的剩余函数，尽可能有效地调整几个规定的质量分布或边际的问题。为了帮助说明这个问题，假设一家大型建筑公司为了建造大量的房屋，雇佣了木匠、电工和水管工。因此，公司需要组建团队；一个团队由每一种类型的贸易人员组成，并将负责建造一座房子。当然，不同的员工组合将或多或少地提高效率，构建团队以使整个过程尽可能高效符合公司的最佳利益。我的研究重点是这种最佳组合的结构。例如，每个木匠只与一名电工和一名水管工一起工作，还是让一些木匠在几个不同的团队兼职更好？在第一种情况下，把电工分配给与他们一起工作的木匠是否顺利？当每个团队的规模（即工人类型的数量）变得非常大时会发生什么？令人惊讶的是，这种类型的问题与偏微分方程、几何和概率论等数学领域有着深刻的相互作用，并且在经济学、数学金融、统计物理和图像处理中有许多潜在的应用。然而，尽管只有两种类型的工人在类似问题上取得了很大进展，但在由三种或三种以上类型的工人组成的团队中，人们对类似问题的了解相对较少。这项研究的主要目标是科学知识的进步；然而，在上述领域中出现了许多有前途的潜在应用。