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

Luca Alberto Davide Ferrari (1) , Antonin Chambolle (1) , Benoît Merlet (2, 3)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 RAPSODI - Reliable numerical approximations of dissipative systems
3 LPP - Laboratoire Paul Painlevé - UMR 8524

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.

Mots clés: Gamma Convergence; Phase-field app roximations; Steiner Problem

Domaines: Equations aux dérivées partielles [math.AP]; Mathématiques [math]

Luca Alberto Davide Ferrari, Antonin Chambolle, Benoît Merlet. A simple phase-field approximation of the Steiner problem in dimension two. 2016. ⟨hal-01359483v1⟩