Optimal Transportation Networks

by Bernot, Marc; Caselles, Vicent; Morel, Jean-Michel

• ISBN13: 9783540693147
• ISBN10: 3540693149
• Format: Paperback
• Copyright: 2008-12-04
• Publisher: Springer Verlag

Summary

"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."--BOOK JACKET.

Table of Contents

Introduction: The Models	p. 1
The Mathematical Models	p. 11
The Monge-Kantorovich Problem	p. 11
The Gilbert-Steiner Problem	p. 12
Three Continuous Extensions of the Gilbert-Steiner Problem	p. 13
Xia's Transport Paths	p. 13
Maddalena-Solimini's Patterns	p. 14
Traffic Plans	p. 14
Questions and Answers	p. 16
Plan	p. 17
Related Problems and Models	p. 19
Measures on Sets of Paths	p. 19
Urban Transportation Models with more than One Transportation Means	p. 20
Traffic Plans	p. 25
Parameterized Traffic Plans	p. 27
Stability Properties of Traffic Plans	p. 29
Lower Semicontinuity of Length, Stopping Time, Averaged Length and Averaged Stopping Time	p. 30
Multiplicity of a Traffic Plan and its Upper Semicontinuity	p. 31
Sequential Compactness of Traffic Plans	p. 33
Application to the Monge-Kantorovich Problem	p. 34
Energy of a Traffic Plan and Existence of a Minimizer	p. 35
The Structure of Optimal Traffic Plans	p. 39
Speed Normalization	p. 39
Loop-Free Traffic Plans	p. 41
The Generalized Gilbert Energy	p. 42
Rectifiability of Traffic Plans with Finite Energy	p. 44
Appendix: Measurability Lemmas	p. 44
Operations on Traffic Plans	p. 47
Elementary Operations	p. 47
Restriction, Domain of a Traffic Plan	p. 47
Sum of Traffic Plans (or Union of their Parameterizations)	p. 48
Mass Normalization	p. 48
Concatenation	p. 48
Concatenation of Two Traffic Plans	p. 48
Hierarchical Concatenation (Construction of Infinite Irrigating Trees or Patterns)	p. 49
A Priori Properties on Minimizers	p. 51
An Assumption on [mu superscript +], [mu superscript -] and [pi] Avoiding Fibers with Zero Length	p. 51
A Convex Hull Property	p. 53
Traffic Plans and Distances between Measures	p. 55
All Measures can be Irrigated for [alpha] > 1 - 1/N	p. 56
Stability with Respect to [mu superscript +] and [mu superscript -]	p. 58
Comparison of Distances between Measures	p. 59
The Tree Structure of Optimal Traffic Plans and their Approximation	p. 65
The Single Path Property	p. 65
The Tree Property	p. 70
Decomposition into Trees and Finite Graphs Approximation	p. 71
Bi-Lipschitz Regularity	p. 77
Interior and Boundary Regularity	p. 79
Connected Components of a Traffic Plan	p. 79
Cuts and Branching Points of a Traffic Plan	p. 81
Interior Regularity	p. 82
The Main Lemma	p. 82
Interior Regularity when [characters not reproducible]	p. 85
Interior Regularity when [mu superscript +] is a Finite Atomic Measure	p. 89
Boundary Regularity	p. 91
Further Regularity Properties	p. 93
The Equivalence of Various Models	p. 95
Irrigating Finite Atomic Measures (Gilbert-Steiner) and Traffic Plans	p. 95
Patterns (Maddalena et al.) and Traffic Plans	p. 96
Transport Paths (Qinglan Xia) and Traffic Plans	p. 97
Optimal Transportation Networks as Flat Chains	p. 100
Irrigability and Dimension	p. 105
Several Concepts of Dimension of a Measure and Irrigability Results	p. 105
Lower Bound on d([mu])	p. 111
Upper Bound on d([mu])	p. 112
Remarks and Examples	p. 114
The Landscape of an Optimal Pattern	p. 119
Introduction	p. 119
Landscape Equilibrium and OCNs in Geophysics	p. 119
A General Development Formula	p. 122
Existence of the Landscape Function and Applications	p. 124
Well-Definedness of the Landscape Function	p. 124
Variational Applications	p. 127
Properties of the Landscape Function	p. 128
Semicontinuity	p. 128
Maximal Slope in the Network Direction	p. 129
Holder Continuity under Extra Assumptions	p. 131
Campanato Spaces by Medians	p. 131
Holder Continuity of the Landscape Function	p. 132
The Gilbert-Steiner Problem	p. 135
Optimum Irrigation from One Source to Two Sinks	p. 135
Optimal Shape of a Traffic Plan with given Dyadic Topology	p. 143
Topology of a Graph	p. 143
A Recursive Construction of an Optimum with Full Steiner Topology	p. 144
Number of Branches at a Bifurcation	p. 145
Dirac to Lebesgue Segment: A Case Study	p. 151
Analytical Results	p. 152
The Case of a Source Aligned with the Segment	p. 152
A "T Structure" is not Optimal	p. 153
The Boundary Behavior of an Optimal Solution	p. 155
Can Fibers Move along the Segment in the Optimal Structure?	p. 159
Numerical Results	p. 159
Coding of the Topology	p. 159
Exhaustive Search	p. 160
Heuristics for Topology Optimization	p. 160
Multiscale Method	p. 161
Optimality of Subtrees	p. 164
Perturbation of the Topology	p. 165
Application: Embedded Irrigation Networks	p. 169
Irrigation Networks made of Tubes	p. 169
Anticipating some Conclusions	p. 171
Getting Back to the Gilbert Functional	p. 172
A Consequence of the Space-filling Condition	p. 175
Source to Volume Transfer Energy	p. 176
Final Remarks	p. 177
Open Problems	p. 179
Stability	p. 179
Regularity	p. 179
The who goes where Problem	p. 180
Dirac to Lebesgue Segment	p. 180
Algorithm or Construction of Local Optima	p. 181
Structure	p. 182
Scaling Laws	p. 183
Local Optimality in the Case of Non Irrigability	p. 183
Skorokhod Theorem	p. 185
Flows in Tubes	p. 189
Poiseuille's Law	p. 189
Optimality of the Circular Section	p. 190
Notations	p. 191
References	p. 193
Index	p. 199
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