Phase field approximations of branched transportation problems

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 Γ-convergence and show some numerical simulation results.

An adaptive finite element approach for lifted branched transport problems

“…For the general transportation network problem, a widely used approach was inspired by elliptic approximations of free-discontinuity problems in the sense of Modica-Mortola and Ambrosio-Tortorelli via phase fields. In [29,12,27,17,16,37], corresponding phase field approximations have been presented for the classical branched transport problem, the Steiner tree problem, a variant of the urban planning problem (which is piecewise linear in the amount of transported mass) or more general cost functions, however, all restricted to two space dimensions.…”

Phase field models for two-dimensional branched transportation problems

“…Phase field approximation. One approach to numerically find optimal transportation networks (that is, minimizers of E τ subject to (5)) consists in approximating E τ by a smooth phase field functional E τ ε that is easier to minimize (phase field models for special cases of branched transportation are proposed in [7,6]). In such an approach the flux σ ∈ M(Ω; R d ) is replaced by a smoothed versionσ : Ω → R d , called a phase field, where the degree of smoothing is determined by a small parameter ε > 0 (see fig.…”