Optimal Micropatterns in 2D Transport Networks and Their Relation to Image Inpainting

Abstract

We consider two different variational models of transport networks: the so-called branched transport problem and the urban planning problem. Based on a novel relation to Mumford–Shah image inpainting and techniques developed in that field, we show for a two-dimensional situation that both highly non-convex network optimization tasks can be transformed into a convex variational problem, which may be very useful from analytical and numerical perspectives. As applications of the convex formulation, we use it to perform numerical simulations (to our knowledge this is the first numerical treatment of urban planning), and we prove a lower bound for the network cost that matches a known upper bound (in terms of how the cost scales in the model parameters) which helps better understand optimal networks and their minimal costs.