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 and Continuous Dynamical Systems, 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, Springer-Verlag, 2009.

[2]

E. N. Gilbert, Minimum cost communication networks, Bell Sys. Tech. J., 46 (1967), 2209-2227.

[3]

E. N. Gilbert and H. O. Pollak, Steiner minimal trees, SIAM J. Appl. Math., 16 (1968), 1-29. doi: 10.1137/0116001.

[4]

F. K. Hwang, D. S. Richards and P. Winter, "The Steiner Tree Problem," Elsevier, North-Holland, 1992.

[5]

M. Bernot, V. Caselles and J. M. Morel, "Optimal Transportation Networks," Lec. Notes in Math., 1955, Springer-Verlag, 2009.

[6]

F. Maddalena, S. Solimini and J. J. Morel, A variational model of irrigation pattern, Interfaces and Free Boundaries, 5 (2003), 391-416. doi: 10.4171/IFB/85.

[7]

E. Paolini and E. Stepanov, Optimal transportation networks as flat chains, Interfaces Free Bound., 8 (2006), 393-436. doi: 10.4171/IFB/149.

[8]

E. Paolini and E. Stepanov, Connecting measures by means of branched transportation networks at finite cost, J. Math. Sci. (N. Y.), 157 (2009), 858-873. doi: 10.1007/s10958-009-9362-x.

[9]

F. Santambrogio, Optimal channel networks, landscape function and branched transport, Interfaces and Free Boundaries, 9 (2007), 149-169.

[10]

Q. Xia, Optimal paths related to transport problems, Communications in Contemporary Math, 5 (2003), 251-279, doi: 10.1142/S021919970300094X.

[11]

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

show all references

References:
[1]

G. Buttazzo and E. Stepanov, "Optimal Urban Networks via Mass Transportation," Lec. Notes in Math., 1961, Springer-Verlag, 2009.

[2]

E. N. Gilbert, Minimum cost communication networks, Bell Sys. Tech. J., 46 (1967), 2209-2227.

[3]

E. N. Gilbert and H. O. Pollak, Steiner minimal trees, SIAM J. Appl. Math., 16 (1968), 1-29. doi: 10.1137/0116001.

[4]

F. K. Hwang, D. S. Richards and P. Winter, "The Steiner Tree Problem," Elsevier, North-Holland, 1992.

[5]

M. Bernot, V. Caselles and J. M. Morel, "Optimal Transportation Networks," Lec. Notes in Math., 1955, Springer-Verlag, 2009.

[6]

F. Maddalena, S. Solimini and J. J. Morel, A variational model of irrigation pattern, Interfaces and Free Boundaries, 5 (2003), 391-416. doi: 10.4171/IFB/85.

[7]

E. Paolini and E. Stepanov, Optimal transportation networks as flat chains, Interfaces Free Bound., 8 (2006), 393-436. doi: 10.4171/IFB/149.

[8]

E. Paolini and E. Stepanov, Connecting measures by means of branched transportation networks at finite cost, J. Math. Sci. (N. Y.), 157 (2009), 858-873. doi: 10.1007/s10958-009-9362-x.

[9]

F. Santambrogio, Optimal channel networks, landscape function and branched transport, Interfaces and Free Boundaries, 9 (2007), 149-169.

[10]

Q. Xia, Optimal paths related to transport problems, Communications in Contemporary Math, 5 (2003), 251-279, doi: 10.1142/S021919970300094X.

[11]

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

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