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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=573443
• 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.

该提案围绕多边际最优运输问题解决方案的几何形状和平滑度展开。 这是相对于给定的剩余函数尽可能有效地对齐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雇用木匠、电工和水管工等人员来建造大量房屋。那么，公司需要组建团队；一个团队由每种类型的行业人员组成，负责建造一栋房子。当然，不同的员工组合会或多或少地提高效率，并且构建团队以使整个流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对齐的结构。 例如，每个木匠只与一名电工和一名水管工一起工作是最佳选择，还是一些木匠在几个不同的团队兼职工作更好？ 第一种情况，比如说，将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，此类问题与偏微分方程、几何和概率等数学领域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。 然而，尽管在只有两种类型的工人的类似问题上取得了很大进展，但对于由三种或更多类型的商人组成的团队的情况却知之甚少。 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景的潜在应用。