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

Chambolle, Antonin; Ferrari, Luca Alberto Davide; Merlet, Benoît

Document type :

Compte-rendu et recension critique d'ouvrage

DOI :

10.1515/acv-2016-0034

Title :

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

Author(s) :

Chambolle, Antonin [Auteur]
Centre de Mathématiques Appliquées de l'Ecole polytechnique [CMAP]
Ferrari, Luca Alberto Davide [Auteur]
Centre de Mathématiques Appliquées de l'Ecole polytechnique [CMAP]
Merlet, Benoît [Auteur]

Laboratoire Paul Painlevé - UMR 8524 [LPP]
Reliable numerical approximations of dissipative systems [RAPSODI]

Journal title :

Advances in Calculus of Variation

Pages :

157–179

Publisher :

Walter de Gruyter GmbH

Publication date :

2019

ISSN :

1864-8266

English keyword(s) :

Phase-field app roximations
Steiner Problem
Gamma Convergence

HAL domain(s) :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

English abstract : [en]

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 ...
Show more >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.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Théorie géométrique de la mesure et applications
Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :