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.
DOI : 10.1051/m2an/2015028
Keywords: Optimal transportation, transport path, branching network, directed graph, ramified transportation
Xia, Qinglan 1
@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},
year = {2015},
publisher = {EDP Sciences},
volume = {49},
number = {6},
doi = {10.1051/m2an/2015028},
mrnumber = {3423276},
zbl = {1331.49067},
language = {en},
url = {https://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 - https://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 https://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 , Optimal Transport, Tome 49 (2015) no. 6, pp. 1791-1832. doi: 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
, 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
, and , Constructing optimal maps for Monge’s transport problem as a limit of strictly convex costs. J. Amer. Math. Soc. 15 (2002) 1–26. | MR | Zbl | DOI
R. Coifman and G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes. Vol. 242 of Lect. Notes Math. Springer-Verlag (1971). | MR | Zbl
and , On the dimension of an irrigable measure. Rend. Semin. Mat. Univ. Padova 117 (2007) 1–49. | MR | Zbl | Numdam
and , 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
and , Uniqueness and transport density in Monge’s mass transportation problem. Calc. Var. Partial Differ. Equ. 15 (2002) 81–113. | Zbl | DOI
, Flat chains over a finite coefficient group. Trans. Amer. Math. Soc. 121 (1966) 160-186. | Zbl | DOI
and , The geometry of optimal transportation. Acta Math. 177 (1996) 113–161. | Zbl | DOI
, Diffusion-Limited Aggregation: A Model for Pattern Formation. Phys. Today 53 (2000) 36–41. | DOI
, and , A variational model of irrigation patterns, Interfaces and Free Boundaries 5 (2003) 391–416. | Zbl | DOI
A. Mas-Colell, M. Whinston and J. Green, Microeconomic Theory. Oxford University Press, New York (1995). | Zbl
, Progress in DLA Research. Physica D 86 (1995) 104–112. | Zbl | DOI
, On the problem of Steiner. Canad. Math. Bull. 4 (1961) 143–148. | Zbl | DOI
, Optimal Channel Networks, Landscape Function and Branched Transport. Interfaces Free Bound. 9 (2007) 149–169. | Zbl | DOI
L. Simon, Lectures on geometric measure theory. In vol. 3 of Proc. Centre Math. Anal. Australian National University (1983). | Zbl
and , Size minimization and approximating problems. Calc. Var. Partial Differ. Equ. 17 (2003) 405–442. | Zbl | DOI
, Topics in Mass Transportation. Vol. 58 of AMS Grad. Stud. Math. 58 (2003). | Zbl | DOI
C. Villani, Optimal Transport: Old and New. Grundlehren der mathematischen Wissenschaften (2009). | Zbl
, Rectifiability of flat chains. Ann. Math. 150 (1999) 165–184. | Zbl | DOI
and , Diffusion-Limited Aggregation, A Kinetic Critical Phenomenon. Phys. Rev. Lett. 47 (1981) 1400–1403. | DOI
, Optimal paths related to transport problems. Commun. Contemp. Math. 5 (2003) 251–279. | Zbl | DOI
, Interior regularity of optimal transport paths. Calc. Var. Partial Differential Equ. 20 (2004) 283–299. | Zbl | DOI
Q. Xia, An application of optimal transport paths to urban transport networks. Discr. Contin. Dyn. Syst., Supp. (2005) 904–910. | Zbl
, The formation of tree leaf. ESAIM: COCV 13 (2007) 359–377. | Zbl | Numdam
, The geodesic problem in quasimetric spaces. J. Geom. Anal. 19 (2009) 452–479. | Zbl | DOI
, 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
, Ramified optimal transportation in geodesic metric spaces. Adv. Calc. Var. 4 (2011) 277–307. | Zbl
and , On the transport dimension of measures. SIAM J. Math. Anal. 41 (2010) 2407–2430. | Zbl | DOI
and , Diffusion-limited aggregation driven by optimal transportation. Fractals 18 (2010) 1–7.
and , The exchange value embedded in a transport system. Appl. Math. Optim. 62 (2010) 229–252. | Zbl | DOI
and , On the ramified optimal allocation problem. Netw. heterog. Media 8 (2013) 591–624. | Zbl | DOI
, On landscape functions associated with transport paths. Discr. Contin. Dyn. Syst. A 34 (2014). | Zbl
and , 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).
, and , Probability distributions of placental morphological measurements and origins of variability of placental shapes. Placenta 34 (2013) 493–6. | DOI
Cité par Sources :





