Some variational problems related to optimal transportation

本文的目的是研究由最优运输问题导出的一些几何变分问题。PI提出了与这一目标相关的项目，重点是：（a）研究数学经济学中的最优分配问题与分支运输技术：（B）模拟动态形成的泥浆开裂使用Monge-Kantorovich最优运输;以及（c）使用分枝最优运输模型建立胎盘血管树结构。最优运输问题旨在找到一个有效的分配方案，将某些商品从源头运输到目标的计划。特别是，分枝最优运输被用来模拟运输规模经济的群体运输广泛观察到的性质（如树木，河道网络）和有效设计的分支结构的运输系统（如铁路配置和邮政递送网络）。所提出的最优分配问题是数学经济学中出现的一类问题（例如最优存储问题）的原型。它的目的是寻找一个最优的分配方案以及相关的最优分配路径，以最小化从工厂到家庭的运输商品的总成本。此外，所提出的泥浆开裂的动态模型不仅增加了我们对泥浆开裂的理解，而且可能有助于预测类似材料未来开裂的位置（例如，地震预测），从而提高公共安全。此外，在所提出的使用分支运输的胎盘模型中发现的量可以潜在地用作诊断工具的一部分，用于预测具有异常胎盘的新生儿的健康问题。

The goal of this proposal is to study some geometric variational problems derived from optimal transportation. The PI proposes projects related to this goal with an emphasis on the following: (a) studying the optimal allocation problem arisen in mathematical economics with ramified transport technology; (b) modeling the dynamic formation of mud cracking using Monge-Kantorovich optimal transportation; and (c) modeling vascular tree structures of the placentas using ramified optimal transportation.The optimal transportation problem aims at finding an efficient allocation plan for transporting some commodity from sources to targets. In particular, ramified optimal transportation is used to model the transport economy of scale in group transportation observed widely in both nature (e.g. trees, river channel networks) and efficiently designed transport systems of branching structures (e.g. railway configurations and postage delivery networks). The proposed optimal allocation problem is the prototype for a class of problems (e.g. optimal storage problem) arising in mathematical economics. It aims at finding an optimal allocation plan as well as an associated optimal allocation path to minimize overall cost of transporting commodity from factories to households. Also, the proposed dynamic model for mud cracking not only increases our understanding about mud cracking but also may be useful in predicting the position of future cracking of a similar material (e.g. a prediction of an earth quake), and hence provide increased public safety. Furthermore, the quantities found in the proposed model of placentas using ramified transportation can potentially be used as part of diagnostic tools in predicting healthy problem for newborns with abnormal placentas.