Optimal Transportation Networks: Models and Theory / Edition 1

by Marc Bernot, Vicent Caselles, Jean-Michel Morel

Overview

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.

Product Details

ISBN-13:	9783540693147
Publisher:	Springer Berlin Heidelberg
Publication date:	11/17/2008
Series:	Lecture Notes in Mathematics , #1955
Edition description:	2009
Pages:	200
Product dimensions:	6.10(w) x 9.20(h) x 0.50(d)

Table of Contents

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.

Editorial Reviews

From the reviews: “The book aims to give a unified mathematical theory of branched transportation (or irrigation) networks. … The logical structure of the book makes it easy to learn the theory. … this book, in addition to being a great source of references is also extremely suitable for a study from scratch. The theory is presented while avoiding useless complications and keeping the language simple. I would also suggest this book to graduate students who want to enter this very interesting research field.” (Luigi De Pascale, Mathematical Reviews, Issue 2010 e)

From the Publisher