no. 6
Motivations, ideas and applications of ramified optimal transportation
Xia, Qinglan1
1 University of California at Davis, Department of Mathematics, One Shields Ave, Davis, CA, 95616, USA.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 6, pp. 1791-1832.

In this survey article, the author summarizes the motivations, key ideas and main applications of ramified optimal transportation that the author has studied in recent years.

Reçu le :
DOI : 10.1051/m2an/2015028
Classification : 90B10, 49Q10, 49Q20
Mots clés : Optimal transportation, transport path, branching network, directed graph, ramified transportation
Xia, Qinglan 1

1 University of California at Davis, Department of Mathematics, One Shields Ave, Davis, CA, 95616, USA. 
@article{M2AN_2015__49_6_1791_0,
     author = {Xia, Qinglan},
     title = {Motivations, ideas and applications of ramified optimal transportation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1791--1832},
     publisher = {EDP-Sciences},
     volume = {49},
     number = {6},
     year = {2015},
     doi = {10.1051/m2an/2015028},
     mrnumber = {3423276},
     zbl = {1331.49067},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2015028/}
}
TY  - JOUR
AU  - Xia, Qinglan
TI  - Motivations, ideas and applications of ramified optimal transportation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2015
SP  - 1791
EP  - 1832
VL  - 49
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2015028/
DO  - 10.1051/m2an/2015028
LA  - en
ID  - M2AN_2015__49_6_1791_0
ER  - 
%0 Journal Article
%A Xia, Qinglan
%T Motivations, ideas and applications of ramified optimal transportation
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2015
%P 1791-1832
%V 49
%N 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2015028/
%R 10.1051/m2an/2015028
%G en
%F M2AN_2015__49_6_1791_0
Xia, Qinglan. Motivations, ideas and applications of ramified optimal transportation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 6, pp. 1791-1832. doi : 10.1051/m2an/2015028. http://www.numdam.org/articles/10.1051/m2an/2015028/

L. Ambrosio, Lecture notes on Optimal Transport Problems. Mathematical Aspects of Evolving Interfaces (Funchal, 2000). In vol. 1812 of Lect. Notes Math. Springer, Berlin (2003) 1–52. | MR | Zbl

Y. Brenier, Décomposition polaire et ré arrangement monotone des champs de vecteurs [Polar decomposition and increasing rearrangement of vector fields]. C. R. Acad. Sci. Paris Sér. I Math. 305 (1987) 805–808. | MR | Zbl

D. Burago, Y. Burago and S. Ivanov, A Course in Metric Geometry. American Mathematical Society (2001). | MR | Zbl

L.A. Caffarelli, M. Feldman and R.J. Mccann, Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs. J. Amer. Math. Soc. 15 (2002) 1–26. | DOI | MR | Zbl

R. Coifman and G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes. Vol. 242 of Lect. Notes Math. Springer-Verlag (1971). | MR | Zbl

G. Devillanova and S. Solimini, On the dimension of an irrigable measure. Rend. Semin. Mat. Univ. Padova 117 (2007) 1–49. | Numdam | MR | Zbl

L.C. Evans and W. Gangbo, Differential equations methods for the Monge−Kantorovich mass transfer problem. Mem. Amer. Math. Soc. 137 (1999) 653. | Zbl

H. Federer, Geometric measure theory. In vol. 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York Inc. (1969). | Zbl

M. Feldman and R.J. Mccann, Uniqueness and transport density in Monge’s mass transportation problem. Calc. Var. Partial Differ. Equ. 15 (2002) 81–113. | DOI | Zbl

W. Fleming, Flat chains over a finite coefficient group. Trans. Amer. Math. Soc. 121 (1966) 160-186. | DOI | Zbl

W. Gangbo and R.J. Mccann, The geometry of optimal transportation. Acta Math. 177 (1996) 113–161. | DOI | Zbl

T.C. Halsey, Diffusion-Limited Aggregation: A Model for Pattern Formation. Phys. Today 53 (2000) 36–41. | DOI

F. Maddalena, S. Solimini and J.M. Morel, A variational model of irrigation patterns, Interfaces and Free Boundaries 5 (2003) 391–416. | DOI | Zbl

A. Mas-Colell, M. Whinston and J. Green, Microeconomic Theory. Oxford University Press, New York (1995). | Zbl

P. Meakin, Progress in DLA Research. Physica D 86 (1995) 104–112. | DOI | Zbl

Z.A. Melzak, On the problem of Steiner. Canad. Math. Bull. 4 (1961) 143–148. | DOI | Zbl

F. Santambrogio, Optimal Channel Networks, Landscape Function and Branched Transport. Interfaces Free Bound. 9 (2007) 149–169. | DOI | Zbl

L. Simon, Lectures on geometric measure theory. In vol. 3 of Proc. Centre Math. Anal. Australian National University (1983). | Zbl

D.P. Thierry and R. Hardt, Size minimization and approximating problems. Calc. Var. Partial Differ. Equ. 17 (2003) 405–442. | DOI | Zbl

C. Villani, Topics in Mass Transportation. Vol. 58 of AMS Grad. Stud. Math. 58 (2003). | DOI | Zbl

C. Villani, Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften (2009). | Zbl

B. White, Rectifiability of flat chains. Ann. Math. 150 (1999) 165–184. | DOI | Zbl

T.A. Witten and L.M. Sander, Diffusion-Limited Aggregation, A Kinetic Critical Phenomenon. Phys. Rev. Lett. 47 (1981) 1400–1403. | DOI

Q. Xia, Optimal paths related to transport problems. Commun. Contemp. Math. 5 (2003) 251–279. | DOI | Zbl

Q. Xia, Interior regularity of optimal transport paths. Calc. Var. Partial Differential Equ. 20 (2004) 283–299. | DOI | Zbl

Q. Xia, An application of optimal transport paths to urban transport networks. Discr. Contin. Dyn. Syst., Supp. (2005) 904–910. | Zbl

Q. Xia, The formation of tree leaf. ESAIM: COCV 13 (2007) 359–377. | Numdam | Zbl

Q. Xia, The geodesic problem in quasimetric spaces. J. Geom. Anal. 19 (2009) 452–479. | DOI | Zbl

Q. Xia, Boundary regularity of optimal transport paths. Adv. Calc. Var. 4 (2011) 153–174. | Zbl

Q. Xia, Numerical simulation of optimal transport paths. In vol. 1, Proc. of the Second International Conference on Computer Modeling and Simulation ICCMS 2010 (2010) 521–525. DOI: . | DOI

Q. Xia, Ramified optimal transportation in geodesic metric spaces. Adv. Calc. Var. 4 (2011) 277–307. | Zbl

Q. Xia and A. Vershynina, On the transport dimension of measures. SIAM J. Math. Anal. 41 (2010) 2407–2430. | DOI | Zbl

Q. Xia and D. Unger, Diffusion-limited aggregation driven by optimal transportation. Fractals 18 (2010) 1–7.

Q. Xia and S. Xu, The exchange value embedded in a transport system. Appl. Math. Optim. 62 (2010) 229–252. | DOI | Zbl

Q. Xia and S. Xu, On the ramified optimal allocation problem. Netw. heterog. Media 8 (2013) 591–624. | DOI | Zbl

Q. Xia, On landscape functions associated with transport paths. Discr. Contin. Dyn. Syst. A 34 (2014). | Zbl

Q. Xia and C. Salafia, Transport efficiency of the human placenta. J. Coupled Syst. Multiple Dyn. 2 (2014).

Q. Xia, C. Salafia and M. Simon, Optimal transport and placental function. Vol. 17 of Interdisciplinary Topics Appl. Math., Modeling and Computational Science. Springer Proc. Math. Stat. Springer (2015). DOI: . | DOI

Q. Xia and C. Salafia, Human placentas, Optimal transportation and Autism (submitted).

M. Yampolsky, C.M. Salafia and O. Shlakhter, Probability distributions of placental morphological measurements and origins of variability of placental shapes. Placenta 34 (2013) 493–6. | DOI

Cité par Sources :