On a phase field approximation of the planar Steiner problem: Existence, regularity, and asymptotic of minimizers

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...

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“…Such an optimal graph mainly using a phase field based approach together with some coercive regularization, see e.g. [13,19,29,12].…”

Section: Introductionmentioning

confidence: 99%

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

Bonafini¹,

Orlandi²,

Oudet³

2018

SIAM J. Math. Anal.

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“…Section 2 recalls some results from [5,4] and gives mathematical justifications of the different modified phase field models that we propose. This section is purely theoretical.…”

Section: Outline Of This Papermentioning

confidence: 99%

“…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…”

Section: Introductionmentioning

confidence: 99%

“…Finally, the phase field approach in [12,5,4] relies on the minimization among u ∈ 1 + H 1 0 (Ω) of the following functional:…”

Section: Introductionmentioning

confidence: 99%

“…Besides, the cost of the numerical differentiation of the geodesic distance can become prohibitive if the number of Steiner points N increases. In order to avoid the differentiation of the geodesic distance D w; a 0 , a i ) with respect to the metric w, we adopt the point of view developed in [4], that consists in dissociating the minimization problem (1.8) by introducing an extra variable γ := (γ i ) 1≤i≤N , where each γ i is a Lipschitz curve joining the base point a 0 to the point a i in Ω. Following the notation in [4], we define for any pair of distinct points a, b ∈ Ω the set P(a, b) := γ ∈ Lip ([0, 1]; Ω 0 ) : γ(0) = a and γ(1) = b , and introduce the admissible set P(a 0 , µ) := γ = (γ i ) N i=1 : γ i ∈ P(a 0 , a i ) .…”

Section: Introductionmentioning

confidence: 99%

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

Bonnivard¹,

Bretin²,

Lemenant³

2019

Math. Comp.

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The aim of this work is to present some numerical computations of solutions of the Steiner Problem, based on the recent phase field approximations proposed in [12] and analyzed in [5, 4]. Our strategy consists in improving the regularity of the associated phase field solution by use of higherorder derivatives in the Cahn-Hilliard functional as in [6]. We justify the convergence of this slightly modified version of the functional, together with other technics that we employ to improve the numerical experiments. In particular, we are able to consider a large number of points in dimension 2. We finally present and justify an approximation method that is efficient in dimension 3, which is one of the major novelties of the paper.

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

Bonafini

Orlandi

Oudet

2020

Advances in Calculus of Variations

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In this paper we consider the Euclidean Steiner tree problem and, more generally, (single sink) Gilbert-Steiner problems as prototypical examples of variational problems involving 1-dimensional connected sets in R n . Following the the analysis for the planar case presented in [4], we provide a variational approximation through Ginzburg-Landau type energies proving a Γ-convergence result for n ≥ 3.

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On a phase field approximation of the planar Steiner problem: Existence, regularity, and asymptotic of minimizers

Cited by 8 publications

References 23 publications

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

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

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

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

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On a phase field approximation of the planar Steiner problem: Existence, regularity, and asymptotic of minimizers

Cited by 8 publications

References 23 publications

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

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

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

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

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