A multi-material transport problem with arbitrary marginals

Marchese, Andrea; Massaccesi, Annalisa; Stuvard, Salvatore; Tione, Riccardo

doi:10.1007/s00526-021-01967-x

Cited by 10 publications

(11 citation statements)

References 26 publications

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“…The lower semicontinuity of the functional E is stated in [41, §6] (see also [18]). In [33], we prove the lower semicontinuity in a more general framework in order to obtain the existence of solutions to a "continuous" version of the MMTP. The only issue is that a minimizing sequence {T l } l∈N for the MMTP does not necessarily have equi-bounded masses, hence it is not possible to apply Theorem 1.10 directly to obtain a minimizer.…”

Section: Theorem 23 (Existence Of Solutions)mentioning

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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Section: Theorem 23 (Existence Of Solutions)mentioning

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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“…The models described above can be used and generalized to describe a variety of problems related to branched transportation: for instance, one can study the mailing problem [2] (for which the first stability result was proved in [18]), the urban planning model [8], including two different regimes of transportation, or the recent multi-material transport problem [32,33], allowing simultaneous transportation of different goods or commodities. Recently, shape optimization problems related to the functional (1.1) were analysed in [41,11] and similar branching structures are observed in superconductivity models and for minimizers of Ginzburg-Landau type functionals, see for instance [27,14,15,16,20].…”

Section: Remark (H-masses)mentioning

confidence: 99%

“…By [44,Section 6] this quantity is lower semi-continuous with respect to the standard notion of convergence in flat norm for currents with coefficients in groups, which by [33,Section 4.6] is equivalent to the joint convergence in flat norm of all components.…”

Section: Definition (Slicing Of 1-rectifiable Currents)mentioning

confidence: 99%

On the well-posedness of branched transportation

Colombo¹,

Rosa²,

Marchese³

2019

Preprint

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“…Several variants and generalizations of the branched transportation problem were proposed and studied by many authors in recent years, see for instance [3,4,6,7,8,9,10,11,13,20,21,25,32]. For the sake of simplicity, we prove the generic uniqueness of minimizers only for the Eulerian formulation introduced in [29].…”

Section: Introductionmentioning

confidence: 98%