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

Abstract

In this paper we consider the branched transportation problem in 2D associated with a cost per unit length of the form $1+\hat{I}\pm m$ where $m$ denotes the amount of transported mass and $\hat{I}\pm >0$ is a fixed parameter (notice that the limit case $\hat{I}\pm =0$ corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals $(\{F_{\hat{I}}\mu {\}}_{\hat{I}\mu >0})$ which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the $\hat{I}“$-convergence of $\{F_{\hat{I}}\mu \}$ as $\hat{I}\mu \downarrow 0$. 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.

24 pages, 8 figures