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.
Highlights
Branched transportation problems constitute a special class of optimal transport problems that have recently attracted lots of interest
Given two probability measures μ+ and μ− on some domain ⊂ Rn, representing an initial and a final mass distribution, respectively, one seeks the most cost-efficient way to transport the mass from the initial to the final distribution
A divergence measure vector field is a measure F ∈ M( ; R2), whose weak divergence is a Radon measure, div F ∈ M( ), where the weak divergence is defined as ψ d div F = − ∇ψ · dF for all ψ ∈ C1(R2) with compact support
Summary
Branched transportation problems constitute a special class of optimal transport problems that have recently attracted lots of interest (see for instance [26, §4.4.2] and the references therein). Given two probability measures μ+ and μ− on some domain ⊂ Rn, representing an initial and a final mass distribution, respectively, one seeks the most cost-efficient way to transport the mass from the initial to the final distribution. Unlike in classical optimal transport, the cost of a transportation scheme does depend on initial and final position of each mass particle, and takes into account how many particles travel together. The transportation cost per transport distance is typically not proportional to the transported mass m, but rather a concave, nondecreasing function τ : [0, ∞) → [0, ∞), m → τ (m).
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