A phase-field approximation of the Steiner ...
Document type :
Compte-rendu et recension critique d'ouvrage
DOI :
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]
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
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 >
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
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