Optimal transportation networks as flat chains

Paolini, Emanuele; Stepanov, Eugene

doi:10.4171/ifb/149

Cited by 30 publications

(35 citation statements)

References 20 publications

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“…Observe that this is true for almost any fiber ω ∈ . Since the paths are all contained in a bounded set and T χ (ω)dω < ∞, (20) follows by the dominated convergence theorem.…”

Section: Lemma 64 Let χ Be a Loop-free Traffic Plan With Finite Enermentioning

confidence: 94%

“…Using measures on sets of paths for transport models is a spreading idea. Paolini and Stepanov [20] use normal one-dimensional currents but point out that they can be represented through measures over the metric set of Lipschitz paths in R N . Brenier used them [5] in incompressible fluid mechanics and Buttazzo et al [6,7] in urban Three traffic plans and their associated embedding: a Dirac measure on γ , a tree with one bifurcation, a spread tree irrigating Lebesgue's measure on the segment [0, 1] × {0} of the plane.…”

Section: Traffic Plansmentioning

confidence: 99%

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The structure of branched transportation networks

2007

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The transportation problem can be formalized as the problem of finding the optimal paths to transport a measure µ + onto a measure µ − with the same mass. In contrast with the Monge-Kantorovich formalization, recent approaches model the branched structure of such supply networks by an energy functional whose essential feature is to favor wide roads. Given a flow s in a road or a tube or a wire, the transportation cost per unit length is supposed to be proportional to s α with 0 < α < 1. For the Monge-Kantorovich energy α = 1 so that it is equivalent to have two roads with flow 1/2 or a larger one with flow 1. If instead 0 < α < 1, a road with flow s 1 + s 2 is preferable to two individual roads s 1 and s 2 because (s 1 + s 2 ) α < s α 1 + s α 2 . Thus, this very simple model intuitively leads to branched transportation structures. Such a branched structure is observable in ground transportation networks, in draining and irrigation systems, in electric power supply systems and in natural objects like the blood vessels or the trees. When α > 1 − 1 N such structures can irrigate a whole bounded open set of R N . The aim of this paper is to give a mathematical proof of several structure and regularity properties empirically observed in transportation networks. It is first proven that optimal transportation networks have a tree structure and can be monotonically approximated by finite graphs. An interior regularity result is then proven according to which an optimal network is a finite graph away from the irrigated measure. It is also proven that the branching number of optimal networks has everywhere a universal explicit bound. These results answer questions raised in two recent papers by Xia. M. Bernot UMPA, ENS Lyon, 46, Allée d'Italie, 123 280 M. Bernot et al. Mathematics Subject Classification (2000)Primary: 49Q10; Secondary: 90B10 · 90B06 · 90B20 List of Symbolsor a fixed subset with finite measure of R, set of fiber indices µ + , µ − positive Borel measures in X with equal mass π a probability measure on X × X or "transference plan" t → χ(ω, t), constant for sufficiently large t, Lipschitz path, fiber indexed by ω (ω, t) → χ(ω, t), traffic plan or pattern, set of fibers |χ| := | | the total mass transported by χ P χ the law of ω → χ(ω) ∈ Lip 1 (X ) T (ω)) final point of a fiber π = (τ, σ ) λ transference plan of χ µ + (χ)(A) := |{ω : χ(ω, 0) ∈ A}|, irrigating measure of χ µ + χ = τ λ, irrigating measure of χ µ − (χ)(A) := |{ω : χ(ω, T χ (ω)) ∈ A}|, measure irrigated by χ µ − χ = σ λ, measure irrigated by χ TP(µ + , µ − ) set of traffic plans χ such that µ − χ = µ − and µ + χ = µ + . TP C set of traffic plans such that (G) f (e) α H 1 (e) Gilbert energy M α (µ + , µ − ) minimal transport cost between µ + and µ − for the Xia functional [ω] t ⊂ branch of the fiber ω at time t in a pattern |[ω] t | measure of the branch containing ω at time t E α (χ) =of a traffic plan χ D restriction of χ to a sub-domain of D(χ ) χ x := {ω : x ∈ χ(ω, R)} path class of x in χ x := χ x abbreviation |x| χ = | χx | multiplicity of χ ...

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“…Observe that this is true for almost any fiber ω ∈ . Since the paths are all contained in a bounded set and T χ (ω)dω < ∞, (20) follows by the dominated convergence theorem.…”

Section: Lemma 64 Let χ Be a Loop-free Traffic Plan With Finite Enermentioning

confidence: 94%

Section: Traffic Plansmentioning

confidence: 99%

The structure of branched transportation networks

2007

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“…7 We can obtain the rearrangement χ * of an irrigation pattern χ by applying, in turn, the following operations 1. the good parameterization χ (see Definition 5.2); 2.…”

Section: Corollary 87mentioning

confidence: 99%

Elementary properties of optimal irrigation patterns

Devillanova

Solimini

2006

Calc. Var.

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In this paper we follow the approach in Maddalena et al. (Interfaces and 1 Free Boundaries 5, 391-415, 2003) to the study of the ramified structures and we identify some geometrical properties enjoyed by optimal irrigation patterns. These properties are "elementary" in the sense that they are not concerned with the regularity at the ending points of such structures, where the presumable selfsimilarity properties should take place. This preliminary study already finds an application in G. Devillanova and S. Solimini (Math. J. Univ. Padua, to appear), where it is used in order to discuss the irrigability of a given measure.

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“…The proof is based on the SBV compactness theorem [1]; alternatively it can be proved via rectifiability results in the theory of currents [36], as used in the study of transportation networks, e.g., in [26,31]. …”

Section: Definition 32mentioning

confidence: 99%

Limiting Aspects of Nonconvex ${TV}^{\phi}$ Models

Hintermüller

Valkonen

2015

SIAM J. Imaging Sci.

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Recently, non-convex regularisation models have been introduced in order to provide a better prior for gradient distributions in real images. They are based on using concave energies ϕ in the total variation type functional TV ϕ (u) := ϕ(|∇u(x)|) dx. In this paper, it is demonstrated that for typical choices of ϕ, functionals of this type pose several difficulties when extended to the entire space of functions of bounded variation, BV(Ω). In particular, if ϕ(t) = t q for q ∈ (0, 1) and TV ϕ is defined directly for piecewise constant functions and extended via weak* lower semicontinuous envelopes to BV(Ω), then still TV ϕ (u) = ∞ for u not piecewise constant. If, on the other hand, TV ϕ is defined analogously via continuously differentiable functions, then TV ϕ ≡ 0, (!). We study a way to remedy the models through additional multiscale regularisation and area strict convergence, provided that the energy ϕ(t) = t q is linearised for high values. The fact, that this kind of energies actually better matches reality and improves reconstructions, is demonstrated by statistics and numerical experiments.Mathematics subject classification: 26B30, 49Q20, 65J20.

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Optimal transportation networks as flat chains

Cited by 30 publications

References 20 publications

The structure of branched transportation networks

The structure of branched transportation networks

Elementary properties of optimal irrigation patterns

Limiting Aspects of Nonconvex ${TV}^{\phi}$ Models

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