Variational approximation of size-mass energies fork-dimensional currents

Abstract

In this paper we produce a Γ-convergence result for a class of energies F k ε,a modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that F 1 ε,a Γ-converges to a branched transportation energy whose cost per unit length is a function f n−1 a depending on a parameter a > 0 and on the codimension n − 1. The limit cost f a (m) is bounded from below by 1 + m so that the limit functional controls the mass and the length of the limit object. In the limit a ↓ 0 we recover the Steiner energy. We then generalize the approach to any dimension and codimension. The limit objects are now k-currents with prescribed boundary, the limit functional controls both their masses and sizes. In the limit a ↓ 0, we recover the Plateau energy defined on k-currents, k < n. The energies F k ε,a then can be used for the numerical treatment of the k-Plateau problem.1. r 2 → q d (ξ, r 1 , r 2 ) is nonincreasing,

Phase field approximations of branched transportation problems

“…The algorithms were implemented in MATLAB c ○ ; parameters reported in this section refer to the rescaled cost (12). We first present simulation results for a single phase field and no diffuse mass flux, N = 1 and α 0 = ∞.…”

“…Similarly, in [5] Santambrogio et al study a variational approximation to the Steiner minimal tree problem in which the connectedness constraint of the graph is enforced trough the introduction of a geodesic distance depending on the phase field. Our phase field approximations can be viewed as a generalization of recent work by two of the authors [11,12], in which essentially (1) for τ (m) = αm + β with α, β > 0 is approximated by an Ambrosio-Tortorelli-type functional defined as [σ, ϕ] = ∞ otherwise. The function 1 − ϕ may be regarded as a smooth version of the characteristic function of a 1-rectifiable set (the Steiner tree) whose total length is approximated by the Ambrosio-Tortorelli phase field term in parentheses.…”

“…The employed optimization method is similar to the one presented in [12] and updates the variables σ and ϕ alternatingly.…”

Variational approximation of functionals defined on 1-dimensional connected sets in ℝ n

“…As pointed out in the companion paper [4], the Gilbert-Steiner problem represents the basic example of problems defined on 1-dimensional connected sets, and it has recently received a renewed attention in the Calculus of Variations community. In the last years available results focused on variational approximations of the problem mainly in the planar case [8,9,15,7], while higher dimensional approximations have been recently proposed in [10,6].…”

Numerical approximation of the Steiner problem in dimension $2$ and $3$

“…In this paper, we consider the problem from a variational point of view, based on the phase field approximation that has been recently introduced in [12] and analyzed in [5,4] (see also [7,8,3] for different approaches). The model which has been proved to work in dimension 2, consists in coupling a Cahn-Hilliard type functional…”