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

摘要

In this paper we consider the branched transportation problem in two dimensions associated with a cost per unit length of the form 1 + beta theta, where theta denotes the amount of transported mass and beta > 0 is a fixed parameter (notice that the limit case beta = 0 corresponds to the classical Steiner problem). Motivated by the numerical approximation of this problem, we introduce a family of functionals ({f(epsilon)}(epsilon>0) ) which approximate the above branched transport energy. We justify rigorously the approximation by establishing the equicoercivity and the T-convergence of {f(epsilon)} as epsilon down arrow 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.