Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-28T03:54:12.554Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal networks for mass transportation problems

Published online by Cambridge University Press:  15 December 2004

Alessio Brancolini  and
Giuseppe Buttazzo
Alessio Brancolini
Affiliation:
Alessio Brancolini, Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy; [email protected]
Giuseppe Buttazzo
Affiliation:
Giuseppe Buttazzo, Dipartimento di Matematica, Università di Pisa, Via Buonarroti 2, 56127 Pisa, Italy; [email protected]
Get access

Abstract

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.

Keywords

Optimal networksmass transportation problems.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 1 , January 2005 , pp. 88 - 101
Copyright
© EDP Sciences, SMAI, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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).
Bouchitté, G. and Buttazzo, G., Characterization of Optimal Shapes and Masses through Monge-Kantorovich Equation. J. Eur. Math. Soc. (JEMS) 3 (2001) 139–168.
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/
G. Buttazzo, Semicontinuity, Relaxation and Integral Representation in the Calculus of Variations. Pitman Research Notes in Mathematics Series 207. Longman Scientific & Technical, Harlow (1989).
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.
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.
Buttazzo, G. and Stepanov, E., Optimal Transportation Networks as Free Dirichlet Regions for the Monge-Kantorovich Problem. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2 (2003) 631–678.
Dal Maso, G. and Toader, R., Model, A for the Quasi-Static Growth of Brittle Fractures: Existence and Approximation Results. Arch. Rational Mech. Anal. 162 (2002) 101–135. CrossRef
K.J. Falconer, The Geometry of Fractal Sets. Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge (1986).
L.V. Kantorovich, On the Transfer of Masses. Dokl. Akad. Nauk. SSSR (1942).
L.V. Kantorovich, On a Problem of Monge. Uspekhi Mat. Nauk. (1948).
G. Monge, Mémoire sur la théorie des Déblais et des Remblais. Histoire de l'Acad. des Sciences de Paris (1781) 666–704.
S.J.N. Mosconi and P. Tilli, Γconvergence for the Irrigation Problem. Preprint Scuola Normale Superiore, Pisa (2003). Available at http://cvgmt.sns.it/