Nonsmooth and nonconvex optimal transport problems

非光滑和非凸最优传输问题

• 批准号: 423447095
• 负责人: Professor Dr. Bernhard Schmitzer
• 金额: --
• 依托单位: Institut für Analysis und Numerik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2021-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/423447095?language=en
• 关键词: Nonsmooth; nonconvex; optimal; transport; problems

In recent years a strong interest has developed within mathematics in so-called "branched Transport" models, which allow to describe transportation networks as they occur in road systems, river basins, communication networks, vasculature, and many other natural and artificial contexts. As in classical optimal transport, an amount of material needs to be transported efficiently from a given initial to a final mass distribution. In branched transport, however, the transportation cost is not proportional, but subadditive in the transported mass, modelling an increased transport efficiency if mass is transported in bulk. This automatically favours transportation schemes in which the mass flux concentrates on a complicated, ramified network of one-dimensional lines. The branched transport problem is an intricate nonconvex, nonsmooth variational problem on Radon measures (in fact on normal currents) that describe the mass flux. Various different formulations were developed and analysed (including work by the PIs), however, they all all take the viewpoint of geometric measure theory, working with flat chains, probability measures on the space of Lipschitz curves, or the like. What is completely lacking is an optimization and optimal control perspective (even though some ideas of optimization shimmer through in the existing variational arguments such as regularity analysis via necessary optimality conditions or the concept of calibrations which are related to dual optimization variables). This situation is also reflected in the fact that the field of numerics for branched transport is rather underdeveloped and consists of ad hoc graph optimization methods for special cases and two-dimensional phase field approximations. We will reformulate branched transport in the framework of optimization and optimal control for Radon measures, work out this optimization viewpoint in the variational analysis of branched transport networks, and exploit the results in novel numerical approaches. The new perspective will at the same time help variational analysts, advance the understanding of nonsmooth, nonconvex optimization problems on measures, and provide numerical methods to obtain efficient transport networks.

近年来，数学界对所谓的“分支运输”模型产生了浓厚的兴趣，这些模型可以描述道路系统、河流流域、通信网络、脉管系统和许多其他自然和人工环境中的运输网络。在经典的最佳传输中，需要将一定量的材料从给定的初始质量分布有效地传输到最终质量分布。然而，在分支运输中，运输成本不是成比例的，而是在运输质量中的次加性，如果质量是散装运输，则可以模拟运输效率的提高。这就自动地支持了这样的运输方案，即质量通量集中在复杂的、分叉的一维线路网络上。分支运输问题是一个复杂的非凸，非光滑的Radon措施（实际上是正常的电流），描述的质量通量变分问题。各种不同的配方开发和分析（包括工作的PI），但是，他们都采取的观点，几何测量理论，工作与平面链，概率措施的空间Lipschitz曲线，等等。完全缺乏的是优化和最优控制的观点（即使一些优化的想法在现有的变分论点中闪烁，例如通过必要的最优性条件进行正则性分析或与对偶优化变量相关的校准概念）。这种情况也反映在这样一个事实，即分支运输的数值领域是相当不发达的，包括特殊情况下的特殊图形优化方法和二维相场近似。我们将重新制定分支运输的氡措施的优化和最优控制的框架内，制定出这种优化的观点在分支运输网络的变分分析，并利用新的数值方法的结果。新的视角将在同一时间帮助变分分析，推进非光滑，非凸优化问题的措施的理解，并提供数值方法，以获得有效的运输网络。