Optimal micropatterns in 2D transport networks and their relation to image inpainting

TL;DR

This paper introduces a convex reformulation of two complex 2D transport network models, enabling new analytical and numerical insights, including the first numerical simulations for urban planning networks and energy scaling law bounds.

Contribution

It establishes a novel convex variational framework linking transport networks to image inpainting, facilitating analysis and simulation of optimal networks.

Findings

Convex reformulation of non-convex network problems

First numerical simulations of urban planning networks

Proved lower bounds for energy scaling laws

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 the lower bound of an energy scaling law which helps better understand optimal networks and their minimal energies.