Optimal transportation networks : models and theory /

Saved in:

Bibliographic Details
Author / Creator:	Bernot, Marc.
Edition:	1. ed.
Imprint:	Berlin : Springer, c2009.
Description:	x, 200 p. : ill. (some col.), maps ; 24 cm.
Language:	English
Series:	Lecture notes in mathematics, 0075-8434 ; 1955 Lecture notes in mathematics (Springer-Verlag) ; 1955.
Subject:	Transportation -- Mathematical models. Traffic engineering -- Mathematical models. Transportation engineering -- Mathematical models. Traffic engineering -- Mathematical models. Transportation -- Mathematical models.
Format:	Print Book
URL for this record:	http://pi.lib.uchicago.edu/1001/cat/bib/7363426

Hidden Bibliographic Details
Other authors / contributors:	Caselles, Vicent. Morel, Jean M.
ISBN:	9783540693147 3540693149
Notes:	Includes bibliographical references (p. 193-197) and index.

Table of Contents:

1. Introduction: The Models
2. The Mathematical Models
2.1. The Monge-Kantorovich Problem
2.2. The Gilbert-Steiner Problem
2.3. Three Continuous Extensions of the Gilbert-Steiner Problem
2.3.1. Xia's Transport Paths
2.3.2. Maddalena-Solimini's Patterns
2.3.3. Traffic Plans
2.4. Questions and Answers
2.4.1. Plan
2.5. Related Problems and Models
2.5.1. Measures on Sets of Paths
2.5.2. Urban Transportation Models with more than One Transportation Means
3. Traffic Plans
3.1. Parameterized Traffic Plans
3.2. Stability Properties of Traffic Plans
3.2.1. Lower Semicontinuity of Length, Stopping Time, Averaged Length and Averaged Stopping Time
3.2.2. Multiplicity of a Traffic Plan and its Upper Semicontinuity
3.2.3. Sequential Compactness of Traffic Plans
3.3. Application to the Monge-Kantorovich Problem
3.4. Energy of a Traffic Plan and Existence of a Minimizer
4. The Structure of Optimal Traffic Plans
4.1. Speed Normalization
4.2. Loop-Free Traffic Plans
4.3. The Generalized Gilbert Energy
4.3.1. Rectifiability of Traffic Plans with Finite Energy
4.4. Appendix: Measurability Lemmas
5. Operations on Traffic Plans
5.1. Elementary Operations
5.1.1. Restriction, Domain of a Traffic Plan
5.1.2. Sum of Traffic Plans (or Union of their Parameterizations)
5.1.3. Mass Normalization
5.2. Concatenation
5.2.1. Concatenation of Two Traffic Plans
5.2.2. Hierarchical Concatenation (Construction of Infinite Irrigating Trees or Patterns)
5.3. A Priori Properties on Minimizers
5.3.1. An Assumption on [mu superscript +], [mu superscript -] and [pi] Avoiding Fibers with Zero Length
5.3.2. A Convex Hull Property
6. Traffic Plans and Distances between Measures
6.1. All Measures can be Irrigated for [alpha] > 1 - 1/N
6.2. Stability with Respect to [mu superscript +] and [mu superscript -]
6.3. Comparison of Distances between Measures
7. The Tree Structure of Optimal Traffic Plans and their Approximation
7.1. The Single Path Property
7.2. The Tree Property
7.3. Decomposition into Trees and Finite Graphs Approximation
7.4. Bi-Lipschitz Regularity
8. Interior and Boundary Regularity
8.1. Connected Components of a Traffic Plan
8.2. Cuts and Branching Points of a Traffic Plan
8.3. Interior Regularity
8.3.1. The Main Lemma
8.3.2. Interior Regularity when [characters not reproducible]
8.3.3. Interior Regularity when [mu superscript +] is a Finite Atomic Measure
8.4. Boundary Regularity
8.4.1. Further Regularity Properties
9. The Equivalence of Various Models
9.1. Irrigating Finite Atomic Measures (Gilbert-Steiner) and Traffic Plans
9.2. Patterns (Maddalena et al.) and Traffic Plans
9.3. Transport Paths (Qinglan Xia) and Traffic Plans
9.4. Optimal Transportation Networks as Flat Chains
10. Irrigability and Dimension
10.1. Several Concepts of Dimension of a Measure and Irrigability Results
10.2. Lower Bound on d([mu])
10.3. Upper Bound on d([mu])
10.4. Remarks and Examples
11. The Landscape of an Optimal Pattern
11.1. Introduction
11.1.1. Landscape Equilibrium and OCNs in Geophysics
11.2. A General Development Formula
11.3. Existence of the Landscape Function and Applications
11.3.1. Well-Definedness of the Landscape Function
11.3.2. Variational Applications
11.4. Properties of the Landscape Function
11.4.1. Semicontinuity
11.4.2. Maximal Slope in the Network Direction
11.5. Holder Continuity under Extra Assumptions
11.5.1. Campanato Spaces by Medians
11.5.2. Holder Continuity of the Landscape Function
12. The Gilbert-Steiner Problem
12.1. Optimum Irrigation from One Source to Two Sinks
12.2. Optimal Shape of a Traffic Plan with given Dyadic Topology
12.2.1. Topology of a Graph
12.2.2. A Recursive Construction of an Optimum with Full Steiner Topology
12.3. Number of Branches at a Bifurcation
13. Dirac to Lebesgue Segment: A Case Study
13.1. Analytical Results
13.1.1. The Case of a Source Aligned with the Segment
13.2. A "T Structure" is not Optimal
13.3. The Boundary Behavior of an Optimal Solution
13.4. Can Fibers Move along the Segment in the Optimal Structure?
13.5. Numerical Results
13.5.1. Coding of the Topology
13.5.2. Exhaustive Search
13.6. Heuristics for Topology Optimization
13.6.1. Multiscale Method
13.6.2. Optimality of Subtrees
13.6.3. Perturbation of the Topology
14. Application: Embedded Irrigation Networks
14.1. Irrigation Networks made of Tubes
14.1.1. Anticipating some Conclusions
14.2. Getting Back to the Gilbert Functional
14.3. A Consequence of the Space-filling Condition
14.4. Source to Volume Transfer Energy
14.5. Final Remarks
15. Open Problems
15.1. Stability
15.2. Regularity
15.3. The who goes where Problem
15.4. Dirac to Lebesgue Segment
15.5. Algorithm or Construction of Local Optima
15.6. Structure
15.7. Scaling Laws
15.8. Local Optimality in the Case of Non Irrigability
A. Skorokhod Theorem
B. Flows in Tubes
B.1. Poiseuille's Law
B.2. Optimality of the Circular Section
C. Notations
References
Index

Optimal transportation networks : models and theory /

Similar Items