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Cover Art
PRINTED BOOKS
Author Bernot, Marc.

Title Optimal transportation networks : models and theory / Marc Bernot, Vicent Caselles, Jean M. Morel.

Published Berlin ; New York : Springer-Verlag, [2009]
©2009

Copies

Location Call No. Status
 UniM ERC  510 LECT  no.1955    AVAILABLE
Edition 1. ed.
Physical description x, 200 pages : illustrations ; 24 cm.
Series Lecture notes in mathematics, 0075-8434 ; 1955.
Lecture notes in mathematics (Springer-Verlag) ; 1955.
Bibliography Includes bibliographical references (pages 193-197) and index.
Contents 1 Introduction: The Models 1 -- 2 Mathematical Models 11 -- 2.1 Monge-Kantorovich Problem 11 -- 2.2 Gilbert-Steiner Problem 12 -- 2.3 Three Continuous Extensions of the Gilbert-Steiner Problem 13 -- 2.5 Related Problems and Models 19 -- 3 Traffic Plans 25 -- 3.1 Parameterized Traffic Plans 27 -- 3.2 Stability Properties of Traffic Plans 29 -- 3.3 Application to the Monge-Kantorovich Problem 34 -- 3.4 Energy of a Traffic Plan and Existence of a Minimizer 35 -- 4 Structure of Optimal Traffic Plans 39 -- 4.1 Speed Normalization 39 -- 4.2 Loop-Free Traffic Plans 41 -- 4.3 Generalized Gilbert Energy 42 -- 4.4 Appendix: Measurability Lemmas 44 -- 5 Operations on Traffic Plans 47 -- 5.1 Elementary Operations 47 -- 5.2 Concatenation 48 -- 5.3 A Priori Properties on Minimizers 51 -- 6 Traffic Plans and Distances between Measures 55 -- 6.1 All Measures can be Irrigated for [alpha] > 1 - 1/N 56 -- 6.2 Stability with Respect to [mu superscript +] and [mu superscript -] 58 -- 6.3 Comparison of Distances between Measures 59 -- 7 Tree Structure of Optimal Traffic Plans and their Approximation 65 -- 7.1 Single Path Property 65 -- 7.2 Tree Property 70 -- 7.3 Decomposition into Trees and Finite Graphs Approximation 71 -- 7.4 Bi-Lipschitz Regularity 77 -- 8 Interior and Boundary Regularity 79 -- 8.1 Connected Components of a Traffic Plan 79 -- 8.2 Cuts and Branching Points of a Traffic Plan 81 -- 8.3 Interior Regularity 82 -- 8.4 Boundary Regularity 91 -- 9 Equivalence of Various Models 95 -- 9.1 Irrigating Finite Atomic Measures (Gilbert-Steiner) and Traffic Plans 95 -- 9.2 Patterns (Maddalena et al.) and Traffic Plans 96 -- 9.3 Transport Paths (Qinglan Xia) and Traffic Plans 97 -- 9.4 Optimal Transportation Networks as Flat Chains 100 -- 10 Irrigability and Dimension 105 -- 10.1 Several Concepts of Dimension of a Measure and Irrigability Results 105 -- 10.2 Lower Bound on d([mu]) 111 -- 10.3 Upper Bound on d([mu]) 112 -- 10.4 Remarks and Examples 114 -- 11 Landscape of an Optimal Pattern 119 -- 11.2 A General Development Formula 122 -- 11.3 Existence of the Landscape Function and Applications 124 -- 11.4 Properties of the Landscape Function 128 -- 11.5 Holder Continuity under Extra Assumptions 131 -- 12 Gilbert-Steiner Problem 135 -- 12.1 Optimum Irrigation from One Source to Two Sinks 135 -- 12.2 Optimal Shape of a Traffic Plan with given Dyadic Topology 143 -- 12.3 Number of Branches at a Bifurcation 145 -- 13 Dirac to Lebesgue Segment: A Case Study 151 -- 13.1 Analytical Results 152 -- 13.2 A "T Structure" is not Optimal 153 -- 13.3 Boundary Behavior of an Optimal Solution 155 -- 13.4 Can Fibers Move along the Segment in the Optimal Structure? 159 -- 13.5 Numerical Results 159 -- 13.6 Heuristics for Topology Optimization 160 -- 14 Application: Embedded Irrigation Networks 169 -- 14.1 Irrigation Networks made of Tubes 169 -- 14.2 Getting Back to the Gilbert Functional 172 -- 14.3 A Consequence of the Space-filling Condition 175 -- 14.4 Source to Volume Transfer Energy 176 -- 15 Open Problems 179 -- 15.1 Stability 179 -- 15.2 Regularity 179 -- 15.3 Who goes where Problem 180 -- 15.4 Dirac to Lebesgue Segment 180 -- 15.5 Algorithm or Construction of Local Optima 181 -- 15.6 Structure 182 -- 15.7 Scaling Laws 183 -- 15.8 Local Optimality in the Case of Non Irrigability 183 -- A Skorokhod Theorem 185 -- B Flows in Tubes 189 -- B.1 Poiseuille's Law 189 -- B.2 Optimality of the Circular Section 190.
Other author Caselles, Vicent, 1960-
Morel, Jean M.
Subject Transportation -- Mathematical models.
Traffic engineering -- Mathematical models.
Transportation engineering -- Mathematical models.
ISBN 9783540693147
3540693149