论文标题
提升分支运输问题的自适应有限元方法
An adaptive finite element approach for lifted branched transport problems
论文作者
论文摘要
我们在两个空间维度中考虑所谓的分支传输和其变体。在这些模型中，人们寻求一个最佳的运输网络，以执行给定的大规模运输任务。在两个空间维度中，它们与Mumford-Shah型图像处理问题紧密连接，这反过来又可以通过所谓的功能提升与某些高维凸优化问题有关。我们检查了这些不同模型之间的关系，并利用它来通过凸优化来数值求解分支传输模型。为此，我们基于专门设计的自适应有限元素开发有效的数值处理。尽管凸优化问题及其复杂的非局部约束集具有很高的维度，但该方法允许计算精确解决的最佳运输网络。特别是，通过设计离散化，无限的约束集减少到有限数量的不平等现象。
We consider so-called branched transport and variants thereof in two space dimensions. In these models one seeks an optimal transportation network for a given mass transportation task. In two space dimensions, they are closely connected to Mumford--Shah-type image processing problems, which in turn can be related to certain higher-dimensional convex optimization problems via so-called functional lifting. We examine the relation between these different models and exploit it to solve the branched transport model numerically via convex optimization. To this end we develop an efficient numerical treatment based on a specifically designed class of adaptive finite elements. This method allows the computation of finely resolved optimal transportation networks despite the high dimensionality of the convex optimization problem and its complicated set of nonlocal constraints. In particular, by design of the discretization the infinite set of constraints reduces to a finite number of inequalities.
