A phase-field approximation of the Steiner problem in dimension two
A phase-field approximation of the Steiner problem in dimension two
复制标题
二维 Steiner 问题的相场近似
DOI：
10.1515/acv-2016-0034
复制
发表时间：
2019
影响因子：
1.7
通讯作者：
B. Merlet
中科院分区：
文献类型：
--
作者：
A. Chambolle;L. Ferrari;B. Merlet
Abstract In this paper we consider the branched transportation problem in two dimensions associated with a cost per unit length of the form 1 + β θ {1+\beta\,\theta} , where θ denotes the amount of transported mass and β > 0 {\beta>0} is a fixed parameter (notice that the limit case β = 0 {\beta=0} corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals ( { ℱ ε } ε > 0 {\{\mathcal{F}_{\varepsilon}\}_{\varepsilon>0}} ) which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the Γ-convergence of { ℱ ε } {\{\mathcal{F}_{\varepsilon}\}} as ε ↓ 0 {\varepsilon\downarrow 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.
Abstract In this paper we consider the branched transportation problem in two dimensions associated with a cost per unit length of the form 1 + β θ {1+\beta\,\theta} , where θ denotes the amount of transported mass and β > 0 {\beta>0} is a fixed parameter (notice that the limit case β = 0 {\beta=0} corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals ( { ℱ ε } ε > 0 {\{\mathcal{F}_{\varepsilon}\}_{\varepsilon>0}} ) which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the Γ-convergence of { ℱ ε } {\{\mathcal{F}_{\varepsilon}\}} as ε ↓ 0 {\varepsilon\downarrow 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.