Article Dans Une Revue Advances in Calculus of Variation Année : 2019

A phase-field approximation of the Steiner problem in dimension two

Résumé

In this paper we consider the branched transportation problem in 2D associated with a cost per unit length of the form $1 + αm$ where $m$ denotes the amount of transported mass and $α > 0$ is a fixed parameter (notice that the limit case $α = 0$ corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of func-tionals $({F ε } ε>0)$ which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the $Γ$-convergence of ${F ε } as ε ↓ 0$. Our functionals are modeled on the Ambrosio-Tortorelli functional and are easy to optimize in practice. We present numerical evidences of the efficiency of the method.

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Dates et versions

hal-01359483 , version 1 (02-09-2016)
hal-01359483 , version 2 (27-06-2017)

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Antonin Chambolle, Luca Alberto Davide Ferrari, Benoît Merlet. A phase-field approximation of the Steiner problem in dimension two. Advances in Calculus of Variation, 2019, 12 (2), pp.157-179. ⟨10.1515/acv-2016-0034⟩. ⟨hal-01359483v2⟩
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