Some variational problems related to optimal transportation

本文的目的是研究一些由最优运输引起的几何变分问题。PI提出了与此目标有关的项目，重点是：(a)研究具有分支运输技术的数学经济学中出现的最佳分配问题；(b)采用Monge-Kantorovich最优输运方法模拟泥浆裂缝的动态形成；(c)利用分枝优化运输模型模拟胎盘血管树结构。最优运输问题的目的是寻找一种有效的分配方案，将一些商品从来源运输到目标。特别是，分枝最优运输被用来模拟在群体运输中广泛观察到的规模运输经济，在自然界（如树木，河道网络）和有效设计的分支结构运输系统（如铁路配置和邮资投递网络）中都有。所提出的最优分配问题是数学经济学中出现的一类问题（如最优存储问题）的原型。它的目的是找到一个最优的分配方案，以及相关的最优分配路径，以最小化商品从工厂到家庭的运输总成本。此外，所提出的泥浆开裂动力学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（例如预测地震），从而提高公共安全。此外，在使用分支运输的胎盘模型中发现的数量可以潜在地用作预测胎盘异常新生儿健康问题的诊断工具的一部分。

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.