Variational Approximation of Functionals Defined on 1-dimensional Connected Sets: The Planar Case

Abstract

In this paper we consider variational problems involving 1-dimensional connected sets in the Euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica-Mortola type energies proving a Γ-convergence result. We also introduce a suitable convex relaxation and develop the corresponding numerical implementations. The proposed methods are quite general and the results we obtain can be extended to n-dimensional Euclidean space or to more general manifold ambients, as shown in the companion paper [11].F turns out to be a tree and is thus called a Steiner Minimal Tree (SMT). In case X = R d , d ≥ 2 endowed with the Euclidean 2 metric, one refers often to the Euclidean or geometric STP, while for X = R d endowed with the 1 (Manhattan) distance or for X contained in a fixed grid G ⊂ R d one refers to the rectilinear STP. Here we will adopt the general metric space formulation of [31]: given a metric space X, and given a compact (possibly infinite) set of terminal points A ⊂ X , findwhere H 1 indicates the 1-dimensional Hausdorff measure on X. Existence of solutions for (STP) relies on Golab's compactness theorem for compact connected sets, and it holds true also in generalized cases (e.g. inf H 1 (S), S ∪ A connected). The Gilbert-Steiner problem, or α-irrigation problem [10,37] consists in finding a network S along which to flow unit masses located at the sources P 1 , . . . , P N −1 to the target point P N . Such a network S can be viewed as S = ∪ N −1 i=1 γ i , with γ i a path connecting P i to P N , corresponding to the trajectory of the unit mass located at P i . To favour branching, one is led to consider a cost to be minimized by S which is a sublinear (concave) function of the mass density θ(Notice that (I 1 ) corresponds to the Monge optimal transport problem, while (I 0 ) corresponds to (STP). As for (STP) a solution to (I α ) is known to exist and the optimal network S turns out to be a tree [10]. Problems like (STP) or (I α ) are relevant for the design of optimal transport channels or networks connecting given endpoints, for example the optimal design of net routing in VLSI circuits in the case d = 2, 3. The Steiner Tree Problem has been widely studied from the theoretical and numerical point of view in order to efficiently devise constructive solutions, mainly through combinatoric optimization techniques. Finding a Steiner Minimal Tree is known to be a NP hard problem (and even NP complete in certain cases), see for instance [6,7] for a comprehensive survey on PTAS algorithms for (STP).The situation in the Euclidean case for (STP) is theoretically well understood: given N points P i ∈ R d a SMT connecting them always exists, the solution being in general not unique (think for instance to symmetric configurations of the endpoints P i ). The SMT is a union of segments connecting the endpoints, possibly meeting at 120 • in at most N − 2 further branch points, called St...

Optimal synchronization problem for a multi-agent system

“…We precise that in this paper we are dealing with the case of non renewable resources and non-interacting particles (see Example 1 for a physic application to a fluid depuration problem). We leave these nontrivial considerations to further improvements of the present work which is also connected with possible applications to the so called irrigation (Gilbert-Steiner) problem [3,5].…”

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

“…Eventually, we remark that another way to approach the problem is to investigate possible convex relaxations of the limiting functional, as already pointed out in [4] and then further extended in [5], so as to include more general irrigation-type problems (with multiple sources/sinks) and even problems for 1-d structures on manifolds.…”

Variational approximation of size-mass energies fork-dimensional currents

“…More recently this tool has been used to approximate energies depending on one dimensional sets, for instance in [20] the authors take advantage of a functional similar to the one from Modica and Mortola defined on vector valued measures to approach the branched transportation problem [6]. With similar techniques approximations of the Steiner minimal tree problem ( [2], [17] and [21]) have been proposed in [8,7].…”