Variational problems in mathematics and the sciences

数学和科学中的变分问题

• 批准号: RGPIN-2015-04383
• 负责人: Mccann, Robert
• 金额: $ 2.99万
• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=688404
• 关键词: Variational; problems; mathematics; sciences

***This proposal addresses the analysis of a collection of variational ***problems with applications in mathematics and the sciences.***Central among them is an optimal transportation problem***of Monge and Kantorovich, which can be caricatured as follows:***given a collection of iron mines and a collection of factories ***consuming iron ore, decide which mine should supply***which factory in order to minimize the total transportation costs.***When the mines and factories are distributed continuously throughout***a curved landscape, with geometry reflected in the transport cost,***the problem has deep connections to***geometry, topology, dynamical systems, and nonlinear partial differential equations.***Despite much recent progress, fundamental questions remain to be resolved. ***We target these, and striking applications of this problem ***to models in economics and in physics.******The last decade has seen enormous advances in our understanding of such***issues. The answers depend heavily on***the choice of cost. Topological restrictions on the cost guarantee***that there is only one solution and every mine supplies a single factory; ***under further geometric restrictions this correspondence is smooth. ***However for smooth transportation costs on compact manifolds such as the***spherical surface of the earth one or both conditions must fail.***Characterizing discontinuities then becomes an interesting question,***The resulting transport patterns are much more complicated: we plan to investigate***uniqueness of solutions, and what can be said about dimension,***branching, and smoothness of their support. ******At the same time, we tackle variants such as: how does congestion affect the situation? ***Generically, the solution saturates the congestion bound***wherever transport occurs. Thus it is completely determined by a separation of***active from inactive transportation routes. We plan to investigate the analytic and geometric***properties of this separation. ******Finally, we target select applications to physics and to economics. The discontinuities***described above correspond to pressure fronts in the semigeostrophic theory***of atmospheric motion. We seek to analyze their formation and evolution.***Nonlinear diffusions model phenomena ranging from electronic transport***in semiconductors to the dynamics of thin films; we seek to analyze their long time***behaviour using a gradient flow formulation of the dynamics with respect to***transportation costs. Stable marriage, optimal-pricing and team-building***problems also involve the solution of continuous linear programs.***We seek an analysis of the models which improves our understanding***of the underlying natural phenomena, and leads to better decision-making principles***concerning allocation of resources, matching problems, team-building, and***the ability to influence, explain, exploit, and control economic phenomena.**

* 这个建议解决了一系列变分 * 问题的分析，并在数学和科学中应用。其中最重要的是Monge和Kantorovich的最优运输问题，可以漫画化如下：给定一组铁矿和一组消耗铁矿石的工厂，决定哪个矿山应该供应哪个工厂，以使总运输成本最小化。当矿山和工厂连续分布在一个弯曲的景观中时，几何学反映在运输成本中，这个问题与几何学、拓扑学、动力系统和非线性偏微分方程有着深刻的联系。尽管最近取得了很大进展，但一些根本问题仍有待解决。* 我们的目标是这些，以及这个问题在经济学和物理学模型中的惊人应用。在过去的十年里，我们对这些问题的理解有了巨大的进步。 答案很大程度上取决于成本的选择。 对成本的拓扑限制保证 * 只有一个解，每个矿都供应一个工厂; * 在进一步的几何限制下，这种对应是光滑的。 * 然而，为了在紧致流形（如地球的球面）上实现平滑的运输成本，其中一个或两个条件必须失败。描述不连续性然后成为一个有趣的问题，* 由此产生的输运模式要复杂得多：我们计划调查 * 唯一性的解决方案，以及可以说什么尺寸，* 分支，和光滑的支持。 ** 与此同时，我们解决了各种问题，例如：拥堵如何影响情况？ * 一般来说，无论传输发生在何处，该解决方案都会使拥塞边界 * 饱和。 因此，它完全由 * 活动运输路线与非活动运输路线的分离决定。 我们计划研究这种分离的分析和几何性质。 * 最后，我们选择应用于物理学和经济学。 上述间断点 * 对应于大气运动的半地转理论 * 中的压力锋。 我们试图分析它们的形成和演变。非线性扩散模型的现象，从电子传输 * 在半导体薄膜的动态，我们试图分析他们的长期 * 行为使用梯度流制定的动态相对于 * 运输成本。 稳定的婚姻，最优定价和团队建设 * 问题也涉及连续线性规划的解决方案。我们寻求对模型的分析，以提高我们对潜在自然现象的理解，并导致更好的决策原则，涉及资源分配，匹配问题，团队建设，以及影响，解释，利用和控制经济现象的能力。