April  2011, 30(1): 365-374. doi: 10.3934/dcds.2011.30.365

Limit theorems for optimal mass transportation and applications to networks

1. 

Department of Mathematics, Technion, Haifa 32000, Israel

Received  November 2009 Revised  August 2010 Published  February 2011

It is shown that optimal network plans can be obtained as a limit of point allocations. These problems are obtained by minimizing the mass transportation on the set of atomic measures of a prescribed number of atoms.
Citation: Gershon Wolansky. Limit theorems for optimal mass transportation and applications to networks. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 365-374. doi: 10.3934/dcds.2011.30.365
References:
[1]

G. Buttazzo and E. Stepanov, "Optimal Urban Networks via Mass Transportation,", Lec. Notes in Math., 1961 (1961).   Google Scholar

[2]

E. N. Gilbert, Minimum cost communication networks,, Bell Sys. Tech. J., 46 (1967), 2209.   Google Scholar

[3]

E. N. Gilbert and H. O. Pollak, Steiner minimal trees,, SIAM J. Appl. Math., 16 (1968), 1.  doi: 10.1137/0116001.  Google Scholar

[4]

F. K. Hwang, D. S. Richards and P. Winter, "The Steiner Tree Problem,", Elsevier, (1992).   Google Scholar

[5]

M. Bernot, V. Caselles and J. M. Morel, "Optimal Transportation Networks,", Lec. Notes in Math., 1955 (1955).   Google Scholar

[6]

F. Maddalena, S. Solimini and J. J. Morel, A variational model of irrigation pattern,, Interfaces and Free Boundaries, 5 (2003), 391.  doi: 10.4171/IFB/85.  Google Scholar

[7]

E. Paolini and E. Stepanov, Optimal transportation networks as flat chains,, Interfaces Free Bound., 8 (2006), 393.  doi: 10.4171/IFB/149.  Google Scholar

[8]

E. Paolini and E. Stepanov, Connecting measures by means of branched transportation networks at finite cost,, J. Math. Sci. (N. Y.), 157 (2009), 858.  doi: 10.1007/s10958-009-9362-x.  Google Scholar

[9]

F. Santambrogio, Optimal channel networks, landscape function and branched transport,, Interfaces and Free Boundaries, 9 (2007), 149.   Google Scholar

[10]

Q. Xia, Optimal paths related to transport problems,, Communications in Contemporary Math, 5 (2003), 251.  doi: 10.1142/S021919970300094X.  Google Scholar

[11]

G. Wolansky, Limit theorems for optimal mass transportation, to appear in Cal. Var PDE., \arXiv{0903.0145}., ().   Google Scholar

show all references

References:
[1]

G. Buttazzo and E. Stepanov, "Optimal Urban Networks via Mass Transportation,", Lec. Notes in Math., 1961 (1961).   Google Scholar

[2]

E. N. Gilbert, Minimum cost communication networks,, Bell Sys. Tech. J., 46 (1967), 2209.   Google Scholar

[3]

E. N. Gilbert and H. O. Pollak, Steiner minimal trees,, SIAM J. Appl. Math., 16 (1968), 1.  doi: 10.1137/0116001.  Google Scholar

[4]

F. K. Hwang, D. S. Richards and P. Winter, "The Steiner Tree Problem,", Elsevier, (1992).   Google Scholar

[5]

M. Bernot, V. Caselles and J. M. Morel, "Optimal Transportation Networks,", Lec. Notes in Math., 1955 (1955).   Google Scholar

[6]

F. Maddalena, S. Solimini and J. J. Morel, A variational model of irrigation pattern,, Interfaces and Free Boundaries, 5 (2003), 391.  doi: 10.4171/IFB/85.  Google Scholar

[7]

E. Paolini and E. Stepanov, Optimal transportation networks as flat chains,, Interfaces Free Bound., 8 (2006), 393.  doi: 10.4171/IFB/149.  Google Scholar

[8]

E. Paolini and E. Stepanov, Connecting measures by means of branched transportation networks at finite cost,, J. Math. Sci. (N. Y.), 157 (2009), 858.  doi: 10.1007/s10958-009-9362-x.  Google Scholar

[9]

F. Santambrogio, Optimal channel networks, landscape function and branched transport,, Interfaces and Free Boundaries, 9 (2007), 149.   Google Scholar

[10]

Q. Xia, Optimal paths related to transport problems,, Communications in Contemporary Math, 5 (2003), 251.  doi: 10.1142/S021919970300094X.  Google Scholar

[11]

G. Wolansky, Limit theorems for optimal mass transportation, to appear in Cal. Var PDE., \arXiv{0903.0145}., ().   Google Scholar

[1]

David Kinderlehrer, Adrian Tudorascu. Transport via mass transportation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 311-338. doi: 10.3934/dcdsb.2006.6.311

[2]

A. Daducci, A. Marigonda, G. Orlandi, R. Posenato. Neuronal Fiber--tracking via optimal mass transportation. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2157-2177. doi: 10.3934/cpaa.2012.11.2157

[3]

Massimiliano Caramia, Giovanni Storchi. Evaluating the effects of parking price and location in multi-modal transportation networks. Networks & Heterogeneous Media, 2006, 1 (3) : 441-465. doi: 10.3934/nhm.2006.1.441

[4]

Eva Barrena, Alicia De-Los-Santos, Gilbert Laporte, Juan A. Mesa. Transferability of collective transportation line networks from a topological and passenger demand perspective. Networks & Heterogeneous Media, 2015, 10 (1) : 1-16. doi: 10.3934/nhm.2015.10.1

[5]

G.S. Liu, J.Z. Zhang. Decision making of transportation plan, a bilevel transportation problem approach. Journal of Industrial & Management Optimization, 2005, 1 (3) : 305-314. doi: 10.3934/jimo.2005.1.305

[6]

Alexander V. Kolesnikov. Hessian metrics, $CD(K,N)$-spaces, and optimal transportation of log-concave measures. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1511-1532. doi: 10.3934/dcds.2014.34.1511

[7]

Lorenzo Brasco, Filippo Santambrogio. An equivalent path functional formulation of branched transportation problems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 845-871. doi: 10.3934/dcds.2011.29.845

[8]

Jun Pei, Panos M. Pardalos, Xinbao Liu, Wenjuan Fan, Shanlin Yang, Ling Wang. Coordination of production and transportation in supply chain scheduling. Journal of Industrial & Management Optimization, 2015, 11 (2) : 399-419. doi: 10.3934/jimo.2015.11.399

[9]

Ş. İlker Birbil, Kerem Bülbül, J. B. G. Frenk, H. M. Mulder. On EOQ cost models with arbitrary purchase and transportation costs. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1211-1245. doi: 10.3934/jimo.2015.11.1211

[10]

Paulina Ávila-Torres, Fernando López-Irarragorri, Rafael Caballero, Yasmín Ríos-Solís. The multimodal and multiperiod urban transportation integrated timetable construction problem with demand uncertainty. Journal of Industrial & Management Optimization, 2018, 14 (2) : 447-472. doi: 10.3934/jimo.2017055

[11]

Sangkyu Baek, Jinsoo Park, Bong Dae Choi. Performance analysis of transmission rate control algorithm from readers to a middleware in intelligent transportation systems. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 357-375. doi: 10.3934/naco.2012.2.357

[12]

Biswajit Sarkar, Bijoy Kumar Shaw, Taebok Kim, Mitali Sarkar, Dongmin Shin. An integrated inventory model with variable transportation cost, two-stage inspection, and defective items. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1975-1990. doi: 10.3934/jimo.2017027

[13]

Xia Zhao, Jianping Dou. Bi-objective integrated supply chain design with transportation choices: A multi-objective particle swarm optimization. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1263-1288. doi: 10.3934/jimo.2018095

[14]

Marco Arieli Herrera-Valdez, Maytee Cruz-Aponte, Carlos Castillo-Chavez. Multiple outbreaks for the same pandemic: Local transportation and social distancing explain the different "waves" of A-H1N1pdm cases observed in México during 2009. Mathematical Biosciences & Engineering, 2011, 8 (1) : 21-48. doi: 10.3934/mbe.2011.8.21

[15]

Boris P. Belinskiy. Optimal design of an optical length of a rod with the given mass. Conference Publications, 2007, 2007 (Special) : 85-91. doi: 10.3934/proc.2007.2007.85

[16]

Amir Adibzadeh, Mohsen Zamani, Amir A. Suratgar, Mohammad B. Menhaj. Constrained optimal consensus in dynamical networks. Numerical Algebra, Control & Optimization, 2019, 9 (3) : 349-360. doi: 10.3934/naco.2019023

[17]

Giovanni Alberti, Giuseppe Buttazzo, Serena Guarino Lo Bianco, Édouard Oudet. Optimal reinforcing networks for elastic membranes. Networks & Heterogeneous Media, 2019, 14 (3) : 589-615. doi: 10.3934/nhm.2019023

[18]

Yangang Chen, Justin W. L. Wan. Numerical method for image registration model based on optimal mass transport. Inverse Problems & Imaging, 2018, 12 (2) : 401-432. doi: 10.3934/ipi.2018018

[19]

Rudy R. Negenborn, Peter-Jules van Overloop, Tamás Keviczky, Bart De Schutter. Distributed model predictive control of irrigation canals. Networks & Heterogeneous Media, 2009, 4 (2) : 359-380. doi: 10.3934/nhm.2009.4.359

[20]

Qinglan Xia. An application of optimal transport paths to urban transport networks. Conference Publications, 2005, 2005 (Special) : 904-910. doi: 10.3934/proc.2005.2005.904

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]