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

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 full -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 -dimensional Euclidean space or to more general manifold ambients, as shown in the companion paper [11].

1 Introduction

Connected one dimensional structures play a crucial role in very different areas like discrete geometry (graphs, networks, spanning and Steiner trees), structural mechanics (crack formation and propagation), inverse problems (defects identification, contour segmentation), etc. The modeling of these structures is a key problem both from the theoretical and the numerical point of view. Most of the difficulties encountered in studying such one dimensional objects are related to the fact that they are not canonically associated to standard mathematical quantities. In this article we plan to bridge the gap between the well-established methods of multi-phase modeling and the world of one dimensional connected sets or networks. Whereas we strongly believe that our approach may lead to new points of view in quite different contexts, we restrict here our exposition to the study of two standard problems in the Calculus of Variations which are respectively the classical Steiner tree problem and the Gilbert-Steiner problem (also called the irrigation problem).

The Steiner Tree Problem (STP) [22] can be described as follows: given points in a metric space , (e.g. a graph, with assigned vertices), find a connected (sub-)graph containing the points and having minimal length. Such an optimal graph turns out to be a tree and is thus called a Steiner Minimal Tree (SMT). In case , endowed with the Euclidean metric, one refers often to the Euclidean or geometric STP, while for endowed with the (Manhattan) distance or for contained in a fixed grid one refers to the rectilinear STP.

Here we will adopt the general metric space formulation of [31]: given a metric space , and given a compact (possibly infinite) set of terminal points , find

\text{STP}

where indicates the 1-dimensional Hausdorff measure on . 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. , connected).

Problems like (STP) 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 . 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 is theoretically well understood: given points a SMT connecting them always exists, the solution being in general not unique (think for instance to symmetric configurations of the endpoints ). The SMT is a union of segments connecting the endpoints, possibly meeting at in at most further branch points, called Steiner points.

Nonetheless, the quest of computationally tractable approximating schemes for (STP) has recently attracted a lot of attention in the Calculus of Variations community, due to different variational interpretations of (STP) as respectively a size minimization problem for 1-dimensional connected sets [27, 20], an optimal branched transport problem [10, 16], or even a Plateau problem in a suitable class of vector distributions endowed with some algebraic structure [27, 24], to be solved by finding suitable calibrations [25]. Several authors have proposed different approximations of the problem, whose validity is essentially limited to the planar case, mainly using a phase field based approach together with some coercive regularization, see e.g. [12, 19, 29, 13].

Our aim is to propose a variational approximation for (STP) and for the Gilbert-Steiner irrigation problem (in the equivalent formulations of [34, 23]) in the Euclidean case , . In this paper we focus on the planar case and prove a genuine -convergence result (see Theorem 3.9 and Proposition 3.3) by considering integral functionals of Modica-Mortola type [26]. In the companion paper [11] we rigorously prove that certain integral functionals of Ginzburg-Landau type (see [1]) yield a variational approximation for (STP) and of the irrigation problem valid in any dimension . This approach is related to the interpretation of (STP) as a Plateau problem in a cobordism class of integral currents with multiplicities in a suitable normed group as studied by Marchese and Massaccesi in [24] (see also [27] for the planar case). Our method is quite general and may be easily adapted to a variety of situations (e.g. in manifolds or more general metric space ambients, with densities or anisotropic norms, etc.).

The plan of the paper is as follows: in Section 2 we reformulate (STP) and the irrigation problem as a suitable modification of the optimal partition problem in the planar case. In section 3, we state and prove our main -convergence results, respectively Proposition 3.3 and Theorem 3.9. Inspired by [18], we introduce in section 4 a convex relaxation of the corresponding energies. In Section 5 we present our approximating scheme for (STP) and for the Gilbert-Steiner problem and illustrate its flexibility in different situations, showing how our convex formulation is able to recover multiple solutions whereas -relaxation detects any locally minimizing configuration. Finally, in Section 6 we propose some examples and generalizations that are extensively studied in the companion paper [11].

2 Irrigation-type problems for Euclidean graphs and optimal partitions

In this section we describe some optimization problems on Euclidean graphs with fixed endpoints set , like (STP) or irrigation-type problems, following the approach of [24, 23], and we rephrase them as optimal partition-type problems in the planar case .

2.1 Acyclic graphs and rank one tensor valued measures

Let , , be a given set of distinct points, with . Define to be the set of acyclic graphs connecting the endpoints set such that can be described as the union , where are simple rectifiable curves with finite length having as initial point and as final point, oriented by -measurable unit vector fields satisfying for -a.e. (i.e. the orientation of is coherent with that of on their intersection).

For , if we identify the curves with the vector measures , all the information concerning this acyclic graph is encoded in the rank one tensor valued measure , where the -measurable vector field carrying the orientation of the graph satisfies spt, , -a.e. on , and the -measurable vector field has components satisfying , with the total variation measure of the vector measure . Observe that a.e. for any .

Definition 2.1

Given any graph , we call the above constructed the canonical -valued measure representation of the acyclic graph .

Remark 2.2

Observe that for any the measures verify the property

(2.1)

To any compact connected set with , i.e. to any candidate minimizer for (STP), we associate in a canonical way an acyclic graph connecting such that (see e.g. Lemma 2.1 in [24]). Given such a graph canonically represented by the tensor valued measure , the measure corresponds to the smallest positive measure dominating for , where is the total variation measure of the vector measure . It is thus given by , the supremum of the total variation measures .

Remark 2.3

An equivalent definition of the measure , for positive Radon measures on , can be given by duality: we have, for any positive ,

Remark 2.4 (graphs as -currents)

In [24], the rank one tensor measure identifying a graph in is defined as a current with coefficients in the group . For a smooth compactly supported differential 1-form and a smooth test (vector) function, one sets

Moreover, fixing a norm on , one may define the -total variation of the current as

(2.2)

where is the dual norm to w.r.t. the scalar product on . Remark that (2.2) defines the -total variation for a generic matrix valued measure .

2.2 Irrigation-type functionals

In this section we consider functionals defined on acyclic graphs connecting a fixed set , , by using their canonical representation as rank one tensor valued measures, in order to identify the graph with an irrigation plan from the point sources to the target point . We focus here on suitable energies in order to describe the irrigation problem and the Steiner tree problem in a common framework as in [24, 23]. We observe moreover that the irrigation problem with one point source introduced by Xia [34], in the equivalent formulation of [23], approximates the Steiner tree problem as in the sense of -convergence (see Proposition 2.6).

We first introduce some additional notation: let be given positive measures on , for to form a -valued vector measure . Let , so that with , for , . Accordingly, we denote the supremum measure . For define the measure , with the norm of . We have the coerciveness property

(2.3)

More generally, for a norm on , we define the measure . In particular, we have the characterization

(2.4)

with the dual norm to w.r.t. the Euclidean structure on . The total variation of the measure coincides with the -total variation as defined in (2.2), where for any -measurable unit vector field .

Let be a rank one -valued measure with . For define

(2.5)

and

(2.6)

In other words, , are total variation-type functionals, with respect to the norms and .

When is the canonical representation of an acyclic graph , so that in particular we have and for , we deduce

where , and . We thus recognize that minimizing the functional among graphs connecting to solves the irrigation problem with sources and target (see [23]), while minimizing among graphs with endpoints set solves (STP) in .

Since both and are total variation-type functionals (thanks to the key coerciveness property 2.3), minimizers do exist in the class of rank one tensor valued measures. The fact that the minimization problem within the class of canonical tensor valued measures representing acyclic graphs has a solution in that class is a consequence of compactness properties of Lipschitz maps (in , it follows alternatively by the compactness theorem in the class [5]). Actually, existence of minimizers in the canonically oriented graph class in can be deduced as a byproduct of our -convergence result (see Corollary 3.7 and Corollary 3.8) and in , for , by the parallel -convergence analysis contained in the companion paper [11].

Remark 2.5

A minimizer of (resp. ) among tensor valued measures representing admissible graphs corresponds necessarily to the canonical representation of a minimal graph, i.e. . Indeed if on a connected arc in , with going from to and going from to , this implies that contains a cycle, hence cannot be a minimizer.

We conclude this section by observing in the following proposition that the Steiner tree problem can be seen as the limit of irrigation problems (cf. [29], [23]).

Proposition 2.6

The functional is the -limit, as , of the functionals with respect to the convergence of measures.

Proof: Observe that for any , , and moreover as . Hence, we have that, for the , is a monotonic decreasing sequence as , so that by elementary properties of -convergence, see for instance Remark 1.40 of [15].

2.3 Acyclic graphs and partitions of

This section is dedicated to the two-dimensional case. The following result, which is an instance of the constancy theorem for currents or the Poincaré’s lemma for distributions (see [21]), states that two acyclic graphs having the same endpoints set give rise to a partition of , or equivalently (see [5]), that their oriented difference corresponds to the orthogonal distributional gradient of a piecewise integer valued function having bounded total variation.

Lemma 2.7

Let and let , be simple rectifiable curves from to oriented by -measurable unit vector fields , . Define as above and .

Then there exists a function such that, denoting and respectively the measures representing the gradient and the orthogonal gradient of , we have .

Proof: Consider simple oriented polygonal curves and connecting to such that the Hausdorff distance to respectively and is less than and the length of (resp. ) converges to the length of (resp. ). We can also assume without loss of generality that and intersect only transversally in a finite number of points . Let , be the -measurable unit vector fields orienting , and define the measures and .

For a given consider the closed curve oriented by (i.e. we reverse the orientation of ). Fix a direction and a vector so that the line , , intersects transversally at , for . Fix and set , and for in the connected component of containing . For fix and set sign. Extend to be piecewise constant to the connected component of containing . Fix now a new direction and a new vector and repeat the procedure until remains defined on the whole of .

The map is well defined. Indeed suppose and belong to the same connected component of and consider the arc connecting them in the complement of . Let , , be the line passing through and suppose w.l.o.g. that it intersects transversally at , for . By construction we have

On the other hand the arc together with the segment form an oriented boundary , so that by the Green formula we have

where is a regularization of the measure and is the exterior normal to . It follows, passing to the limit as , after direct computations,

and since we have .

We deduce that , hence and by Poincaré’s inequality in . Hence is an equibounded sequence in norm, and by Rellich compactness theorem there exists a subsequence still denoted converging in to a . Taking into account that we have , we deduce in particular that as desired.

Remark 2.8

Let as above. For let be the segment joining to , denote its orientation, and identify with the vector measure . Then is an acyclic graph connecting the endpoints set and , where .

Taking into account Lemma 2.7 we have

Corollary 2.9

Let be a set of terminal points and (for instance the acyclic graph considered in Remark 2.8). For any acyclic graph , denoting (resp. ) the canonical tensor valued representation of (resp. ), we have

for suitable , .

Thus, fixing the family of measures as in Remark 2.8, we are led to consider the minimization for of the functional

We have already seen that to each acyclic graph we can associate a such that . Moreover, for minimizers, we have the following

Remark 2.10

To each minimizer of we can find an acyclic graph connecting the terminal points and such that .

To prove this fact, let be a minimizer of in , and denote . Then and necessarily contains a simple rectifiable curve connecting to since we have (use for instance the decomposition theorem for rectifiable 1-currents, as in [23]). Consider the canonical rank one tensor measure associated to the acyclic subgraph connecting to . Then by Lemma 2.7, there exists such that and in particular .

We thus have a relationship between (STP) and the minimization of over functions in , and a similar connection can be made between the -irrigation problem and minimization over the same space of

The aim of the next section is then to provide an approximation of the functionals in the sense of -convergence through Modica-Mortola type energies.

Remark 2.11

In the case with a convex set, we may choose with connecting to and spt. We deduce by Corollary 2.9 that for any acyclic graph

for suitable such that (in the trace sense) on and elsewhere in , . We recover here an alternative formulation of the optimal partition problem in a convex planar set as studied for instance in [3] and [4].

3 -convergence

In this section we state and prove our main -convergence result, namely Proposition 3.3 and Theorem 3.9.

Lemma 3.1

Let be a norm on , and . Consider positive Radon measures and satisfying

(3.1)

Then, setting and , we have

(3.2)

Proof: Fix , . By (2.4) we deduce that

so that

(3.3)

for any , , whence, again by (2.4), conclusion (3.2) follows.

Lemma 3.2

Let be a norm on , and . Consider positive Radon measures and satisfying

  • and for all ,

  • there exist measurable functions such that

    (3.4)

    and

    (3.5)

Then, setting and , we have

(3.6)

Proof: Fix , . By direct calculations

We consider now Modica–Mortola functionals for functions having a prescribed jump part. Due to summability issues for the absolutely continuous part of the gradient, we work in local spaces.

Proposition 3.3

For any fixed consider the measure defined in Remark 2.8, and consider the Modica-Mortola type functionals

(3.7)

defined for and open, where is a smooth 1-periodic potential vanishing on (take for example, ). Let ( for ). We have:

1. (Compactness and lower bound inequality) For any such that for any open, there exists such that (up to a subsequence) in . Moreover,

(3.8)

for any , .

2. (Upper bound (in)-equality) For any , with a simple rectifiable curve joining to , let , such that . Then there exists a sequence s.t. in and

(3.9)

for any , .

Remark 3.4

Observe that energy-bounded sequences , that are involved in statements 1) and 2) of Proposition 3.3, verify for any , .

Remark 3.5

Proposition 3.3 holds true also in case the measures are associated to oriented simple polyhedral (or even rectifiable) finite length curves joining to .

Remark 3.6

To avoid statements in local energy spaces in Proposition 3.3 one could consider variants of the functionals by relying on suitable smoothings of the measures , with a symmetric mollifier.

Proof of Proposition 3.3 Observe first that by the localization property of -convergence (see [15]) it suffices to prove (3.8) in the case the characteristic function of an arbitrary open set (say a ball) . Then the conclusion follows by approximating the test function by simple functions , with a disjoint family of open sets , thanks to the regularity of the Radon measures involved in the statement.

We thus fix a ball , and we distinguish two cases, according to whether