An adaptive finite element approach for lifted branched transport problems

Abstract: 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.