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

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 functionals $(\{F_ε\}_{ε>0})$ which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the $Γ$-convergence of $\{F_ε\}$ as $ε\downarrow 0$. 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.

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