Optimal Transportation Networks Models and Theory
The transportation problem can be formalized as the problem of finding the optimal way to transport a given measure into another with the same mass. In contrast to the Monge-Kantorovitch problem, recent approaches model the branched structure of such supp
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1955
Marc Bernot · Vicent Caselles Jean-Michel Morel
Optimal Transportation Networks Models and Theory
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Authors Marc Bernot
Jean-Michel Morel
Unité de mathématiques pures et appliquées ENS Lyon 46, allée d’Italie 69363 Lyon Cedex 7, France [email protected]
CMLA, Ecole Normale Supérieure de Cachan 61 av. du Président Wilson 94235 Cachan Cedex, France [email protected]
Vicent Caselles Dept. de Tecnologies de la Informació i les Comunicacions Pompeu Fabra University Passeig de Circumval.lació 8 08003 Barcelona, Spain [email protected]
ISBN: 978-3-540-69314-7 e-ISBN: 978-3-540-69315-4 DOI: 10.1007/978-3-540-69315-4 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008931162 Mathematics Subject Classification (2000): 49Q10, 90B10, 90B06, 90B20 c 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: SPi Publishing Services Printed on acid-free paper 987654321 springer.com
Preface
The transportation problem can be formalized as the problem of finding the optimal paths to transport a measure μ+ onto a measure μ− with the same mass. In contrast with the Monge-Kantorovich formalization, recent approaches model the branched structure of such supply networks by an energy functional whose essential feature is to favor wide roads. Given a flow ϕ in a tube or a road or a wire, the transportation cost per unit length is supposed to be proportional to ϕα with 0 < α < 1. For the MongeKantorovich energy, α = 1 so that it is equivalent to have two roads with flow 1/2 or a larger one with flow 1. If instead 0 < α < 1, a road with flow ϕ1 +ϕ2 is α preferable to two individual roads ϕ1 and ϕ2 because (ϕ1 + ϕ2 )α < ϕα 1 + ϕ2 . Thus, this very simple model intuitively leads to branched transportation structures. Such a branched structure is observable in ground transportation networks, in draining and irrigation systems, in electric power supply systems and in natural objects like the blood vessels or the trees. When α > 1 − N1 such structures can irrigate a whole bounded open set of RN . The aim of this set of lectures is to give a mathematical proof of several existence, structure and regulari
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