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Cover: 9783540693147 | Optimal Transportation Networks | Models and Theory | Bernot (u. a.)
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Optimal Transportation Networks
Models and Theory
Taschenbuch von Marc Bernot (u. a.)
Sprache: Englisch

58,84 €*

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Beschreibung
The transportation problem can be formalized as the problem of ?nding + ? the optimal paths to transport a measure ? onto a measure ? 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 ?ow ? in a tube or a road or a wire, the transportation cost per unit length ? is supposed to be proportional to ? with 0<?< 1. For the Monge- Kantorovich energy,? = 1 so that it is equivalent to have two roads with ?ow 1/2oralargeronewith?ow1. Ifinstead0<?< 1,aroadwith?ow? +? is 1 2 ? ? ? preferable to two individual roads ? and ? because (? +? ) 1? N N such structures can irrigate a whole bounded open set ofR . The aim of this set of lectures is to give a mathematical proof of s- eral existence, structure and regularity properties empirically observed in transportation networks. This will be done in a simple mathematical fra- work (measures on the set of paths) unifying several di?erent approaches and results due to Brancolini, Buttazzo, Devillanova, Maddalena, Pratelli, Santambrogio, Solimini, Stepanov, Xia and the authors.
The transportation problem can be formalized as the problem of ?nding + ? the optimal paths to transport a measure ? onto a measure ? 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 ?ow ? in a tube or a road or a wire, the transportation cost per unit length ? is supposed to be proportional to ? with 0<?< 1. For the Monge- Kantorovich energy,? = 1 so that it is equivalent to have two roads with ?ow 1/2oralargeronewith?ow1. Ifinstead0<?< 1,aroadwith?ow? +? is 1 2 ? ? ? preferable to two individual roads ? and ? because (? +? ) 1? N N such structures can irrigate a whole bounded open set ofR . The aim of this set of lectures is to give a mathematical proof of s- eral existence, structure and regularity properties empirically observed in transportation networks. This will be done in a simple mathematical fra- work (measures on the set of paths) unifying several di?erent approaches and results due to Brancolini, Buttazzo, Devillanova, Maddalena, Pratelli, Santambrogio, Solimini, Stepanov, Xia and the authors.
Zusammenfassung

Includes supplementary material: [...]

Inhaltsverzeichnis
Introduction: The Models.- The Mathematical Models.- Traffic Plans.- The Structure of Optimal Traffic Plans.- Operations on Traffic Plans.- Traffic Plans and Distances between Measures.- The Tree Structure of Optimal Traffic Plans and their Approximation.- Interior and Boundary Regularity.- The Equivalence of Various Models.- Irrigability and Dimension.- The Landscape of an Optimal Pattern.- The Gilbert-Steiner Problem.- Dirac to Lebesgue Segment: A Case Study.- Application: Embedded Irrigation Networks.- Open Problems.
Details
Erscheinungsjahr: 2008
Fachbereich: Analysis
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 220
Reihe: Lecture Notes in Mathematics
Inhalt: x
200 S.
53 s/w Illustr.
5 farbige Illustr.
200 p. 58 illus.
5 illus. in color.
ISBN-13: 9783540693147
ISBN-10: 3540693149
Sprache: Englisch
Herstellernummer: 12318872
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Bernot, Marc
Morel, Jean-Michel
Caselles, Vicent
Auflage: 2009
Hersteller: Springer-Verlag GmbH
Springer Berlin Heidelberg
Lecture Notes in Mathematics
Maße: 235 x 155 x 13 mm
Von/Mit: Marc Bernot (u. a.)
Erscheinungsdatum: 23.09.2008
Gewicht: 0,341 kg
preigu-id: 101791830
Zusammenfassung

Includes supplementary material: [...]

Inhaltsverzeichnis
Introduction: The Models.- The Mathematical Models.- Traffic Plans.- The Structure of Optimal Traffic Plans.- Operations on Traffic Plans.- Traffic Plans and Distances between Measures.- The Tree Structure of Optimal Traffic Plans and their Approximation.- Interior and Boundary Regularity.- The Equivalence of Various Models.- Irrigability and Dimension.- The Landscape of an Optimal Pattern.- The Gilbert-Steiner Problem.- Dirac to Lebesgue Segment: A Case Study.- Application: Embedded Irrigation Networks.- Open Problems.
Details
Erscheinungsjahr: 2008
Fachbereich: Analysis
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Seiten: 220
Reihe: Lecture Notes in Mathematics
Inhalt: x
200 S.
53 s/w Illustr.
5 farbige Illustr.
200 p. 58 illus.
5 illus. in color.
ISBN-13: 9783540693147
ISBN-10: 3540693149
Sprache: Englisch
Herstellernummer: 12318872
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Bernot, Marc
Morel, Jean-Michel
Caselles, Vicent
Auflage: 2009
Hersteller: Springer-Verlag GmbH
Springer Berlin Heidelberg
Lecture Notes in Mathematics
Maße: 235 x 155 x 13 mm
Von/Mit: Marc Bernot (u. a.)
Erscheinungsdatum: 23.09.2008
Gewicht: 0,341 kg
preigu-id: 101791830
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