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.

中文翻译：

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提升分支运输问题的自适应有限元方法

我们在两个空间维度上考虑所谓的分支运输及其变体。在这些模型中，人们为给定的大众运输任务寻求最佳运输网络。在二维空间中，它们与 Mumford--Shah 型图像处理问题密切相关，而后者又可以通过所谓的函数提升与某些更高维的凸优化问题相关联。我们检查这些不同模型之间的关系，并利用它通过凸优化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类开发了一种有效的数值处理方法。尽管凸优化问题的高维及其复杂的非局部约束集，该方法允许计算精细解决的最优交通网络。特别是，通过离散化的设计，无限的约束集减少到有限数量的不等式。