Optimal transportation networks: Models and theory

Marc Bernot, Vicent Caselles, Jean-Michel Morel (auth.)3540693149, 9783540693147

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 supply networks as minima of an energy functional whose essential feature is to favour wide roads. Such a branched structure is observable in ground transportation networks, in draining and irrigation systems, in electrical power supply systems and in natural counterparts such as blood vessels or the branches of trees.
These lectures provide mathematical proof of several existence, structure and regularity properties empirically observed in transportation networks. The link with previous discrete physical models of irrigation and erosion models in geomorphology and with discrete telecommunication and transportation models is discussed. It will be mathematically proven that the majority fit in the simple model sketched in this volume.

Table of contents :
Front Matter….Pages I-X
Introduction: The Models….Pages 1-9
The Mathematical Models….Pages 11-23
Traffic Plans….Pages 25-37
The Structure of Optimal Traffic Plans….Pages 39-45
Operations on Traffic Plans….Pages 47-54
Traffic Plans and Distances between Measures….Pages 55-63
The Tree Structure of Optimal Traffic Plans and their Approximation….Pages 65-78
Interior and Boundary Regularity….Pages 79-93
The Equivalence of Various Models….Pages 95-104
Irrigability and Dimension….Pages 105-117
The Landscape of an Optimal Pattern….Pages 119-134
The Gilbert-Steiner Problem….Pages 135-149
Dirac to Lebesgue Segment: A Case Study….Pages 151-168
Application: Embedded Irrigation Networks….Pages 169-177
Open Problems….Pages 179-183
Back Matter….Pages 185-206