Optimal networks for mass transportation problems

Alessio Brancolini; Giuseppe Buttazzo

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 11, Issue: 1, page 88-101
  • ISSN: 1292-8119

Abstract

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In the framework of transport theory, we are interested in the following optimization problem: given the distributions μ + of working people and μ - of their working places in an urban area, build a transportation network (such as a railway or an underground system) which minimizes a functional depending on the geometry of the network through a particular cost function. The functional is defined as the Wasserstein distance of μ + from μ - with respect to a metric which depends on the transportation network.

How to cite

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Brancolini, Alessio, and Buttazzo, Giuseppe. "Optimal networks for mass transportation problems." ESAIM: Control, Optimisation and Calculus of Variations 11.1 (2005): 88-101. <http://eudml.org/doc/245494>.

@article{Brancolini2005,
abstract = {In the framework of transport theory, we are interested in the following optimization problem: given the distributions $\mu ^+$ of working people and $\mu ^-$ of their working places in an urban area, build a transportation network (such as a railway or an underground system) which minimizes a functional depending on the geometry of the network through a particular cost function. The functional is defined as the Wasserstein distance of $\mu ^+$ from $\mu ^-$ with respect to a metric which depends on the transportation network.},
author = {Brancolini, Alessio, Buttazzo, Giuseppe},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {optimal networks; mass transportation problems; Monge-Kantorovich problem; optimal transportation; optimal transportation networks; mass transfer},
language = {eng},
number = {1},
pages = {88-101},
publisher = {EDP-Sciences},
title = {Optimal networks for mass transportation problems},
url = {http://eudml.org/doc/245494},
volume = {11},
year = {2005},
}

TY - JOUR
AU - Brancolini, Alessio
AU - Buttazzo, Giuseppe
TI - Optimal networks for mass transportation problems
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2005
PB - EDP-Sciences
VL - 11
IS - 1
SP - 88
EP - 101
AB - In the framework of transport theory, we are interested in the following optimization problem: given the distributions $\mu ^+$ of working people and $\mu ^-$ of their working places in an urban area, build a transportation network (such as a railway or an underground system) which minimizes a functional depending on the geometry of the network through a particular cost function. The functional is defined as the Wasserstein distance of $\mu ^+$ from $\mu ^-$ with respect to a metric which depends on the transportation network.
LA - eng
KW - optimal networks; mass transportation problems; Monge-Kantorovich problem; optimal transportation; optimal transportation networks; mass transfer
UR - http://eudml.org/doc/245494
ER -

References

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  1. [1] L. Ambrosio and P. Tilli, Selected Topics on “Analysis on Metric Spaces”. Appunti dei Corsi Tenuti da Docenti della Scuola, Scuola Normale Superiore, Pisa (2000). Zbl1084.28500
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  3. [3] A. Brancolini, Problemi di Ottimizzazione in Teoria del Trasporto e Applicazioni. Master’s thesis, Università di Pisa, Pisa (2002). Available at http://www.sns.it/~brancoli/ 
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  5. [5] G. Buttazzo and L. De Pascale, Optimal Shapes and Masses, and Optimal Transportation Problems, in Optimal Transportation and Applications (Martina Franca, 2001). Lecture Notes in Mathematics, CIME series 1813, Springer-Verlag, Berlin (2003) 11–52. Zbl1087.49033
  6. [6] G. Buttazzo, E. Oudet and E. Stepanov, Optimal Transportation Problems with Free Dirichlet Regions, in Variational Methods for Discontinuous Structures (Cernobbio, 2001). Progress in Nonlinear Differential Equations and their Applications 51, Birkhäuser Verlag, Basel (2002) 41–65. Zbl1055.49029
  7. [7] G. Buttazzo and E. Stepanov, Optimal Transportation Networks as Free Dirichlet Regions for the Monge-Kantorovich Problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 (2003) 631–678. Zbl1127.49031
  8. [8] G. Dal Maso and R. Toader, A Model for the Quasi-Static Growth of Brittle Fractures: Existence and Approximation Results. Arch. Rational Mech. Anal. 162 (2002) 101–135. Zbl1042.74002
  9. [9] K.J. Falconer, The Geometry of Fractal Sets. Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge (1986). Zbl0587.28004MR867284
  10. [10] L.V. Kantorovich, On the Transfer of Masses. Dokl. Akad. Nauk. SSSR (1942). 
  11. [11] L.V. Kantorovich, On a Problem of Monge. Uspekhi Mat. Nauk. (1948). 
  12. [12] G. Monge, Mémoire sur la théorie des Déblais et des Remblais. Histoire de l’Acad. des Sciences de Paris (1781) 666–704. 
  13. [13] S.J.N. Mosconi and P. Tilli, Γ -convergence for the Irrigation Problem. Preprint Scuola Normale Superiore, Pisa (2003). Available at http://cvgmt.sns.it/ Zbl1076.49024MR2135803

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