Optimal Energy Scaling for Micropatterns in Transport Networks

Abstract

We consider two variational models for transport networks, an urban planning and a branched transport model, in which the degree of network complexity and ramification is governed by a small parameter ε > 0. Smaller ε leads to finer ramification patterns, and we analyse how optimal network patterns in a particular geometry behave as ε → 0 by proving an energy scaling law. This entails providing constructions of near-optimal networks as well as proving that no other construction can do better.The motivation of this analysis is twofold. On the one hand, it provides a better understanding of the transport network models; for instance, it reveals qualitative differences in the ramification patterns of urban planning and branched transport. On the other hand, several examples of variational pattern analysis in the literature use an elegant technique based on relaxation and convex duality. Transport networks provide a relatively simple setting to explore variations and refinements of this technique, thereby increasing the scope of its applicability.

Optimal Micropatterns in 2D Transport Networks and Their Relation to Image Inpainting

“…To better understand the model behaviour, an energy scaling law for the network costs has been derived in [BW16b] (and will be reproved in this work, see Theorem 2.2.1) for the following simple problem geometry (Figure 1 left). In two-dimensional Euclidean space one considers as source and sink distribution the following measures concentrated on two lines, Figure 1: Left: Sketch of the considered setting, two measures µ 0 and µ 1 supported on lines at distance 1, as well as an exemplary transport network in between composed of elementary cells (dashed line).…”

Asymptotic Self-Similarity of Minimizers and Local Bounds in a Model of Shape-Memory Alloys

“…Since x > 0, there is n ∈ N such that l x,n+1 < x ≤ l x,n . The assertion with d 1 := c 1 c 2 φ 1/3 follows from (73) using x ≤ l x,n = l x,n+1 /φ and recalling that r n ≤ 8 3 ηL y ≤ 3ηL y , and correspondingly L y − r n ≤ 8 3 ηL y . Now we turn to (10).…”

“…These vectorial generalizations have confirmed that the Kohn-Müller scalar model indeed captures the correct scaling of the energy. A number of other problems have then been addressed with similar tools, including magnetization patterns in uniaxial ferromagnets [14,15,37], diblock copolymers [1,13], flux tubes in type-I superconductors [16,20,22,28], wrinkling in thin elastic films [4,6,7,34], dislocation microstructures [18,19], transport network structures [8,9], and compliance minimization [42].…”

A 2-dimensional shape optimization problem for tree branches

“…Quite often, optimal solutions to problems involving a ramified transportation cost exhibit a fractal structure [2,3,4,12,15,16,17]. In the present note we analyze in more detail the optimization problem for tree branches proposed in [7], in the 2-dimensional case.…”

“…In the formula (4), η(n) accounts for the intensity of light coming from the direction n. Remark 1. According to the above definition, the amount of sunlight S n (µ) captured by the measure µ only depends on its projection µ n on the subspace perpendicular to n. In particular, if a second measure µ is obtained from µ by shifting some of the mass in a direction parallel to n, then S n ( µ) = S n (µ).…”