Phase field approximations of branched transportation problems

TL;DR

This paper introduces a phase field approximation method for branched transportation problems with piecewise affine costs, modeling hierarchical street networks and proving convergence with numerical simulations.

Contribution

It develops a novel phase field approach for approximating branched transportation networks with multiple street types, including theoretical convergence proof.

Findings

Proved $ ext{Gamma}$-convergence of the approximation

Numerical simulations demonstrating the method's effectiveness

Modeling of hierarchical street networks with multiple street types

Abstract

In branched transportation problems mass has to be transported from a given initial distribution to a given final distribution, where the cost of the transport is proportional to the transport distance, but subadditive in the transported mass. As a consequence, mass transport is cheaper the more mass is transported together, which leads to the emergence of hierarchically branching transport networks. We here consider transport costs that are piecewise affine in the transported mass with N affine segments, in which case the resulting network can be interpreted as a street network composed of N different types of streets. In two spatial dimensions we propose a phase field approximation of this street network using N phase fields and a function approximating the mass flux through the network. We prove the corresponding $\Gamma$-convergence and show some numerical simulation results.