Equivalent formulations for the branched transport and urban planning problems

Brancolini, Alessio; Wirth, Benedikt

doi:10.1016/j.matpur.2016.03.008

Cited by 19 publications

(27 citation statements)

References 17 publications

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“…Existence of minimising patterns is shown in [BW15] as well as the equivalence of the optimisation problems min χ E ε,a,µ0,µ1…”

Section: An Urban Planning Modelmentioning

confidence: 97%

“…Finally, we will need the fact from [BW15] that patterns can be reparameterised fibrewise without changing the cost. Proposition 2.3.2 (Constant speed reparameterisation of patterns).…”

Section: Branched Transportmentioning

confidence: 99%

“…We may assume χ * to have the single path property (see [BW15,Proposition 3.4.1]). Furthermore, due to the problem symmetry we may assume χ * p (I) to be symmetric across {x n = 1 2 } for all p ∈ Γ.…”

Section: Lower Bound In Nd For N >mentioning

confidence: 99%

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Optimal Energy Scaling for Micropatterns in Transport Networks

Brancolini¹,

Wirth²

2017

SIAM J. Math. Anal.

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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.

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“…Existence of minimising patterns is shown in [BW15] as well as the equivalence of the optimisation problems min χ E ε,a,µ0,µ1…”

Section: An Urban Planning Modelmentioning

confidence: 97%

“…Finally, we will need the fact from [BW15] that patterns can be reparameterised fibrewise without changing the cost. Proposition 2.3.2 (Constant speed reparameterisation of patterns).…”

Section: Branched Transportmentioning

confidence: 99%

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Optimal Energy Scaling for Micropatterns in Transport Networks

Brancolini¹,

Wirth²

2017

SIAM J. Math. Anal.

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“…Remark 3.6. The previous corollary allows us to include in the list of cost functionals for which Theorem 2.4 applies the cost considered in [9], which describes a model for the urban planning (or a discrete version of it, in our case). More precisely the cost is C(z) = min{az; z + b} with a > 1, b > 0, which is clearly concave.…”

Section: Proof Of (Claim 1): Second Reductionmentioning

confidence: 99%

A Multimaterial Transport Problem and its Convex Relaxation via Rectifiable $G$-currents

Marchese¹,

Massaccesi²,

Tione³

2019

SIAM J. Math. Anal.

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In this paper we study a variant of the branched transportation problem, that we call multi-material transport problem. This is a transportation problem, where distinct commodities are transported simultaneously along a network. The cost of the transportation depends on the network used to move the masses, as it is common in models studied in branched transportation. The main novelty is that in our model the cost per unit length of the network does not depend only on the total flow, but on the actual quantity of each commodity. This allows to take into account different interactions between the transported goods. We propose an Eulerian formulation of the discrete problem, describing the flow of each commodity through every point of the network. We prove existence of solutions under minimal assumptions on the cost. Moreover, we prove that, under mild additional assumptions, the problem can be rephrased as a mass minimization problem in a class of rectifiable currents with coefficients in a group, allowing to introduce a notion of calibration. The latter result is new even in the well studied framework of the "single-material" branched transportation.Keywords: Branched transportation, rectifiable currents, calibrations, multi-material transport problem.MSC (2010): 49Q10, 49Q15, 49Q20, 53C38, 90B06, 90B10.1 Nonetheless, to our knowledge only problems involving the transport of one (homogeneous) material have been studied and modeled as variational problems. These models do not apply in planning a network for the transportation of different goods, whose mutual interactions require a formulation which involves several variables. The easiest examples of natural multi-material transport problem concern mixed-use roads (where vehicles of different size and pedestrians are allowed to circulate) and the transport through vehicles for goods and passengers.Another notable example is given by the power line communications technology (PLC, see [26,23]), which uses the electric power distribution network for data transmission. PLC has been introduced in the United States of America more than a century ago and used for the communications on moving trains or, more generally, for maintenance operations of the electric power network. Recently, a special type of PLC, the broadband over power lines (BPL) is being studied and improved for high-speed data transmission, being particularly convenient for isolated areas. Electric power and data signals are impossible to be treated as a homogeneous "material" for several reasons the main one being the fact that the electricity and the Internet supply are subject to different costs, depending on the users' concentration and demands.Similar problems, usually grouped under the name of multi-commodity flow problems, were studied (see e.g. [28,24]) as minimization problems on graphs, often considering also constraints on the capacity of the network. Up to now, the aim of the research in this area was mainly devoted to study the complexity of the problem and to improve the efficiency of algorith...

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“…Example 4 (Urban planning I). Another branched transportation problem from the literature, so-called urban planning [3,1], is given by…”

Section: Examples and Piecewise Linear Approximationmentioning

confidence: 99%

Phase field models for two-dimensional branched transportation problems

Wirth

2019

Calc. Var.

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We analyse the following inverse problem. Given a nonconvex functional (from a specific, but quite general class) of normal, codimension-1 currents (which in two spatial dimensions can be interpreted as transportation networks), find the potential of a phase field energy which approximates the given functional. We prove existence of a solution as well as its characterization via a linear deconvolution problem. We also provide an explicit formula that allows to approximate the solution arbitrarily well in the supremum norm. arXiv:1805.05141v1 [math.OC]

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Equivalent formulations for the branched transport and urban planning problems

Cited by 19 publications

References 17 publications

Optimal Energy Scaling for Micropatterns in Transport Networks

Optimal Energy Scaling for Micropatterns in Transport Networks

A Multimaterial Transport Problem and its Convex Relaxation via Rectifiable $G$-currents

Phase field models for two-dimensional branched transportation problems

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