In questa rassegna si traccia un incontro con la teoria del trasporto ottimo, fornendo alcune notizie sulla sua nascita, sulla sua rinnovata riscoperta e conseguenti inattese nuove applicazioni. Si parlerà di trasporto ottimo per il sistema arterioso, della legge di Kleiber, delle radici degli alberi, e infine, della diffusione della muffa Physarum Polycephalum e delle sue caratteristiche geodetiche. Il filo conduttore delle sezioni, apparentemente scollegate, è in realtà sempre il problema di Monge-Kantorovich, a partire dalla sua lettura originale statica, passando per una lettura dinamica di tipo stazionario, fino ad una lettura dinamica di tipo non stazionario (nel caso delle muffe), tendente asintoticamente nel tempo a Monge-Kantorovich. L'esposizione che segue si accompagna ad un utilizzo di tecniche di tipo basilare. Scogli in teoria della misura e analisi funzionale sono solo accennati e ci si cimenta nel mantenere fruibile il racconto per più possibili lettori interessati o incuriositi.
This review concerns the optimal transport theory, providing some news about its birth, its renewed rediscovery and the resulting unexpected new applications. We are talking about optimal transport for the arterial system, Kleiber's law, tree roots, and finally the diffusion of the Physarum Polycephalum mold and its geodetic features. The central thread of our tale is the problem of Monge-Kantorovich, starting from its original static formulation, passing through a stationary dynamic reading, up to a dynamic non stationary transient phase (in the case of molds), tending asymptotically in time to the stationary Monge-Kantorovich setting. The exposition is accompanied by the use of very basic techniques. Measure theory and functional analysis are merely mentioned and we try to keep the story as readable as possible to interested people.
@article{RUMI_2017_1_2_3_327_0,
author = {Franco Cardin},
title = {Trasporto ottimo, sistemi viventi},
journal = {Matematica, Cultura e Societ\`a. Rivista dell'Unione Matematica Italiana},
volume = {2},
year = {2017},
pages = {327-341},
mrnumber = {3753848},
language = {it},
url = {http://dml.mathdoc.fr/item/RUMI_2017_1_2_3_327_0}
}
Cardin, Franco. Trasporto ottimo, sistemi viventi. Matematica, Cultura e Società. Rivista dell'Unione Matematica Italiana, Tome 2 (2017) pp. 327-341. http://gdmltest.u-ga.fr/item/RUMI_2017_1_2_3_327_0/
[1] (2010). Physarum Machines: Computers from Slime Mold. World Scientific series on nonlinear science. World Scientific. | MR 3558779
[2] , Lecture notes on optimal transport problems. Mathematical aspects of evolving interfaces (Funchal, 2000), 1-52, Lecture Notes in Math., 1812, Springer, Berlin, 2003. | MR 2011032
[3] , , Existence and stability results in the theory of optimal transportation, Optimal transportation and applications (Martina Franca, 2001), 123-160, Lecture Notes in Math., 1813, Springer, Berlin, 2003. | MR 2006307 | Zbl 1065.49026
[4] , Mémoire contenant l'application de la théorie. Journal de l'École Polytechnique, (1820). | Zbl 27.0480.03
[5] , Metodi matematici della meccanica classica. Editori Riuniti, 478 pp., 1979.
[6] , , , Optimal transport in living system, proposto per la pubblicazione, 2017.
[7] , , , Size and form in efficient transportation networks, Nature, 399, 130-132 (1999).
[8] , , A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math.84 (2000), no. 3, 375-393. | MR 1738163 | Zbl 0968.76069
[9] , , Optimal transportation networks. Models and theory. Lecture Notes in Mathematics, 1955. Springer-Verlag, Berlin, 2009. x+200 pp. | MR 2449900 | Zbl 1163.90001
[10] , , and (2012). Physarum can compute shortest paths. Journal of Theoretical Biology, 309:121-133. | MR 2948520
[11] , , Principles of optics: Electromagnetic theory of propagation, interference and di raction of light. Third edition Pergamon Press, 1965 xxviii+808 pp. | MR 198807
[12] , , and , Shape optimization solutions via Monge-Kantorovich equation. C. R. Acad. Sci. Paris Sér. I Math.324 (1997), no. 10, 1185-1191. | MR 1451945 | Zbl 0884.49023
[13] , Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math.44 (1991), no. 4, 375-417. | MR 1100809 | Zbl 0738.46011
[14] , Evolution models for mass transportation problems. Milan J. Math.80 (2012), no. 1, 47-63. | MR 2984109 | Zbl 1255.49076
[15] , Problemi di ottimizzazione in teoria del trasporto ottimo. Boll. Unione Mat. Ital. (9) 1 (2008), no. 2, 401-427. | MR 2424301
[16] , and , Constructing optimal mass for Monge's transport problem as a limit of strictly convex costs, J. Amer. Math. Soc. (2002), 1-26. | MR 1862796 | Zbl 1053.49032
[17] , A proof of Sudakov theorem with strictly convex norms. Math. Z.268 (2011), no. 1-2, 371-407. | MR 2805441 | Zbl 1229.49050
[18] , , , Optimal transport from a point-like source, proposto per la pubblicazione, 2017.
[19] and (1999). Differential equations methods for the Monge-Kantorovich mass transfer problem. Mem. Amer. Math. Soc, 137:653. | MR 1464149 | Zbl 0920.49004
[20] , , Allometric scaling in animals and plants, J. Math. Biol.43, 144-156 (2001). | MR 1860460 | Zbl 0988.92001
[21] , Partial Differential Equations and Monge-Kantorovich Mass Transfer, Current Developments in Mathematics, Int. Press, Boston, MA, 1999. | MR 1698853 | Zbl 0954.35011
[22] and , Differential equations methods for the Monge-Kantorovich mass transfer problem, Mem. Amer. Math. Soc., 653, 1999. | MR 1464149 | Zbl 0920.49004
[23] , , , Towards a stationary Monge-Kantorovich dynamics: the Physarum Polycephalum experience, in stampa sul SIAM Journal on Applied Mathematics (SIAP), https://arxiv.org/pdf/1610.06325.pdf, 2017. | MR 3769710 | Zbl 1385.49012
[24] , , , , Numerical Solution of Monge-Kantorovich equations via a dynamic formulation, proposto per la pubblicazione, 2017.
[25] , , , Branched transportation via dynamical PDE model, proposto per la pubblicazione, 2017.
[26] , La muffa intelligente che può disegnare le "reti" delle città, Venerdì di Repubblica, 24 febbraio 2012.
[27] , , , , A reconstruction of the initial conditions of the Universe by optimal mass transportation, Nature417, 260-262 (16 May 2002).
[28] , , Calculus of variations. I. The Lagrangian formalism. Grundlehren der Mathematischen Wissenschaften, 310. Springer-Verlag, Berlin, 1996. xxx+474 pp. | MR 1368401 | Zbl 0853.49001
[29] , Mathematical methods in the organization and planning of production. Reprint of the book, published in 1939, with introductory paper of L. V. Kantorovich. St. Petersburg, Publishing House of St. Petersburg Univ., (2012) 96 pp. | MR 2181518
[30] , On mass transportation, Doklady Acad. Sci. USSR.37 (7-8) (1942), 227-229 (in Russo).
[31] , On a problem of Monge, Uspekhi Mat. Nauk.3 (1948), 225-226 (in Russo). Tradotto in inglese nel Journal of Mathematical Sciences, Vol. 133, No. 4, 2006, pag. 1383. | MR 98987
[32] , Body Size and Metabolism, Hilgardia, Vol. 6, January, 1932, N. 11, pp.315-353.
[33] , , The classical theory of Fields. Second edition. Course of Theoretical Physics, Vol. 2. Pergamon Press, Oxford-London-Paris-Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass.1962 ix+404 pp. | MR 143451 | Zbl 0178.28704
[34] , , Mathematical foundations of elasticity. Corrected reprint of the 1983 original. Dover Publications, Inc., New York, 1994. xviii+556 pp. | MR 1262126
[35] , Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal.11 (2001), no. 3, 589-608. | MR 1844080 | Zbl 1011.58009
[36] , Mathematical biology. II. Spatial models and biomedical applications. Third edition. Interdisciplinary Applied Mathematics, 18. Springer-Verlag, New York, 2003. xxvi+811 pp. | MR 1952568 | Zbl 1006.92002
[37] , , and (2000). Maze solving by an amoeboid organism. Nature, 407(6803):470.
[38] , Riemannian geometry. Third edition. Graduate Texts in Mathematics, 171. Springer, Cham, 2016. xviii+499 pp. | MR 3469435 | Zbl 06520113
[39] et al., On Feasible Optimality, Istituto Veneto Sci. Lett. Arti, Atti Cl. Sci. Fis. Mat. Natur. 155 (1996-1997), pp. 57-69.
[40] , , Fractal River Networks: Chance and Self-Organization, Cambridge University Press, New York, 1997.
[41] , Optimal channel networks, landscape function and branched transport. Interfaces Free Bound.9 (2007), no. 1, pp.149-169. | MR 2317303 | Zbl 1138.90339
[42] , Optimal transport for applied mathematicians. Calculus of variations, PDEs, and modeling. Progress in Nonlinear Differential Equations and their Applications, 87. Birkhäuser/Springer, Cham, 2015. xxvii+353 pp. | MR 3409718 | Zbl 06457100
[43] , Red Plenty, inside the fties' soviet dream, Faber & Faber (2010). Edizione italiana: L'ultima favola russa, Bollati-Boringhieri (2013), 484 pp.
[44] , Geometric problems of the theory of in nitedimensional probability distributions. (Russian) Trudy Mat. Inst. Steklov. 141 (1976), 191 pp. | MR 431359
[45] , , (2007). A mathematical model for adaptive transport network in path finding by true slime mold. Journal of Theoretical Biology, 244(4):553. | MR 2306350
[46] , , , , , , , and (2010). Rules for biologically inspired adaptive network design. Science, 327(5964):439-442. | MR 2599697 | Zbl 1226.90021
[47] and , On the Monge mass transfer problem, Calc. Var. Partial Differential Equations 13 (2001), 19-31. | MR 1854255 | Zbl 1010.49030
[48] , The Kantorovich metric: the initial history and little-known applications. J. Math. Sci. (N. Y.)133 (2006), no. 4, 1410-1417. | MR 2117883 | Zbl 1090.28009
[49] , Long history of the Monge-Kantorovich transportation problem. Math. Intelligencer35 (2013), no. 4, 1-9. | MR 3133757 | Zbl 1284.01041
[50] , Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften, 338. Springer-Verlag, Berlin, 2009. xxii+973 pp. | MR 2459454 | Zbl 1156.53003
[51] , Optimal paths related to transport problems. Commun. Contemp. Math.5 (2003), no. 2, pp. 251-279. | MR 1966259 | Zbl 1032.90003