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

Abstract

AbstractIn this paper we consider the branched transportation problem in two dimensions associated with a cost per unit length of the form1+β⁢θ{1+\beta\,\theta}, where θ denotes the amount of transported mass andβ>0{\beta>0}is a fixed parameter (notice that the limit caseβ=0{\beta=0}corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals ({ℱε}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{ℱε}{\{\mathcal{F}_{\varepsilon}\}}asε↓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.