Optimal transportation networks: models and theory
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2009
|
| Series: | Lecture notes in mathematics
1955 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 193 - 197 |
| Physical Description: | X, 200 S. Ill., graph. Darst. |
| ISBN: | 9783540693147 |
Staff View
MARC
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| 100 | 1 | |a Bernot, Marc |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Optimal transportation networks |b models and theory |c Marc Bernot ; Vicent Caselles ; Jean-Michel Morel |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2009 | |
| 300 | |a X, 200 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in mathematics |v 1955 | |
| 500 | |a Literaturverz. S. 193 - 197 | ||
| 650 | 4 | |a Circulation, Technique de la - Modèles mathématiques | |
| 650 | 4 | |a Transport - Technologie - Modèles mathématiques | |
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Traffic engineering |x Mathematical models | |
| 650 | 4 | |a Transportation engineering |x Mathematical models | |
| 650 | 4 | |a Transportation |x Mathematical models | |
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| 700 | 1 | |a Morel, Jean-Michel |d 1953- |e Verfasser |0 (DE-588)136525229 |4 aut | |
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Record in the Search Index
| _version_ | 1804138044359442432 |
|---|---|
| adam_text | Table
of
Contents
1
Introduction: The Models
................................ 1
2
The Mathematical Models
................................ 11
2.1
The Monge-Kantorovich Problem
......................... 11
2.2
The
Gilbert-Steiner
Problem
............................. 12
2.3
Three Continuous Extensions of the
Gilbert-Steiner
Problem
............................................... 13
2.3.1
Xia s Transport Paths
............................ 13
2.3.2
Maddalena-Solimini s Patterns
..................... 14
2.3.3
Traffic Plans
..................................... 14
2.4
Questions and Answers
.................................. 16
2.4.1
Plan
............................................ 17
2.5
Related Problems and Models
............................ 19
2.5.1
Measures on Sets of Paths
......................... 19
2.5.2
Urban Transportation Models with more than One
Transportation Means
............................ 20
3
Traffic Plans
.............................................. 25
3.1
Parameterized Traffic Plans
.............................. 27
3.2
Stability Properties of Traffic Plans
....................... 29
3.2.1
Lower Semicontinuity of Length, Stopping Time,
Averaged Length and Averaged Stopping Time
....... 30
3.2.2
Multiplicity of a Traffic Plan and its Upper
Semicontinuity
................................... 31
3.2.3
Sequential Compactness of Traffic Plans
............. 33
3.3
Application to the Monge-Kantorovich Problem
............ 34
3.4
Energy of a Traffic Plan and Existence of a Minimizer
....... 35
4
The Structure of Optimal Traffic Plans
................... 39
4.1
Speed Normalization
.................................... 39
4.2
Loop-Free Traffic Plans
................................. 41
4.3
The Generalized Gilbert Energy
.......................... 42
4.3.1
Rectifiability of Traffic Plans with Finite Energy
..... 44
4.4
Appendix: Measurability Lemmas
........................ 44
VII
VIII Table of
Contents
5
Operations on Traffic Plans
............................... 47
5.1
Elementary Operations
.................................. 47
5.1.1
Restriction, Domain of a Traffic Plan
............... 47
5.1.2
Sum of Traffic Plans
(or Union of their Parameterizations)
............... 48
5.1.3
Mass Normalization
.............................. 48
5.2
Concatenation
......................................... 48
5.2.1
Concatenation of Two Traffic Plans
................. 48
5.2.2
Hierarchical Concatenation (Construction of Infinite
Irrigating Trees or Patterns)
....................... 49
5.3
A Priori Properties on Minimizers
........................ 51
5.3.1
An Assumption on
μ+, μ~
and
π
Avoiding Fibers
with Zero Length
................................. 51
5.3.2
A Convex Hull Property
.......................... 53
6
Traffic Plans and Distances between Measures
............ 55
6.1
All Measures can be Irrigated for
α
> 1 —
^
.............. 56
6.2
Stability with Respect to
μ+
and
μ~
..................... 58
6.3
Comparison of Distances between Measures
............... 59
7
The Tree Structure of Optimal Traffic Plans
and their Approximation
................................. 65
7.1
The Single Path Property
............................... 65
7.2
The 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
8.3.1
The Main Lemma
................................ 82
8.3.2
Interior Regularity when supp(/x+)
Π
supp(/x~)
= 0 ... 85
8.3.3
Interior Regularity when
μ+
is a Finite Atomic
Measure
......................................... 89
8.4
Boundary Regularity
.................................... 91
8.4.1
Further Regularity Properties
...................... 93
9
The 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
Table
of Contents IX
10
Irrigability and Dimension
................................105
10.1
Several Concepts of Dimension of a Measure
and Irrigability Results
..................................105
10.2
Lower Bound on
Αμ)
...................................
Ill
10.3
Upper Bound on
ά(μ)
...................................112
10.4
Remarks and Examples
.................................114
11
The Landscape of an Optimal Pattern
....................119
11.1
Introduction
...........................................119
11.1.1
Landscape Equilibrium and OCNs in Geophysics
.....119
11.2
A General Development Formula
.........................122
11.3
Existence of the Landscape Function
and Applications
.......................................124
11.3.1
Well-Definedness of the Landscape Function
.........124
11.3.2
Variational Applications
...........................127
11.4
Properties of the Landscape Function
....................128
11.4.1
Semicontinuity
...................................128
11.4.2
Maximal Slope in the Network Direction
............129
11.5
Holder Continuity under Extra Assumptions
..............131
11.5.1
Campanaio
Spaces by Medians
.....................131
11.5.2
Holder Continuity of the Landscape Function
........132
12
The
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
12.2.1
Topology of a Graph
..............................143
12.2.2
A Recursive Construction of an Optimum with Full
Steiner
Topology
.................................144
12.3
Number of Branches at a Bifurcation
.....................145
13
Dirac to Lebesgue Segment: A Case Study
................151
13.1
Analytical Results
......................................152
13.1.1
The Case of a Source Aligned with the Segment
......152
13.2
A T Structure is not Optimal
..........................153
13.3
The 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
13.5.1
Coding of the Topology
...........................159
13.5.2
Exhaustive Search
................................160
13.6
Heuristics for Topology Optimization
.....................160
13.6.1
Multiscale Method
...............................161
13.6.2
Optimality of Subtrees
............................164
13.6.3
Perturbation of the Topology
......................165
X Table of Contents
14
Application: Embedded Irrigation Networks
..............169
14.1
Irrigation Networks made of Tubes
.......................169
14.1.1
Anticipating some Conclusions
.....................171
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
14.5
Final Remarks
.........................................177
15
Open Problems
...........................................179
15.1
Stability
...............................................179
15.2
Regularity
.............................................179
15.3
The 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
В
Flows in Tubes
...........................................189
B.I Poiseuille s Law
........................................189
B.2 Optimality of the Circular Section
........................190
С
Notations
.................................................191
References
....................................................193
Index
.........................................................199
|
| adam_txt |
Table
of
Contents
1
Introduction: The Models
. 1
2
The Mathematical Models
. 11
2.1
The Monge-Kantorovich Problem
. 11
2.2
The
Gilbert-Steiner
Problem
. 12
2.3
Three Continuous Extensions of the
Gilbert-Steiner
Problem
. 13
2.3.1
Xia's Transport Paths
. 13
2.3.2
Maddalena-Solimini's Patterns
. 14
2.3.3
Traffic Plans
. 14
2.4
Questions and Answers
. 16
2.4.1
Plan
. 17
2.5
Related Problems and Models
. 19
2.5.1
Measures on Sets of Paths
. 19
2.5.2
Urban Transportation Models with more than One
Transportation Means
. 20
3
Traffic Plans
. 25
3.1
Parameterized Traffic Plans
. 27
3.2
Stability Properties of Traffic Plans
. 29
3.2.1
Lower Semicontinuity of Length, Stopping Time,
Averaged Length and Averaged Stopping Time
. 30
3.2.2
Multiplicity of a Traffic Plan and its Upper
Semicontinuity
. 31
3.2.3
Sequential Compactness of Traffic Plans
. 33
3.3
Application to the Monge-Kantorovich Problem
. 34
3.4
Energy of a Traffic Plan and Existence of a Minimizer
. 35
4
The Structure of Optimal Traffic Plans
. 39
4.1
Speed Normalization
. 39
4.2
Loop-Free Traffic Plans
. 41
4.3
The Generalized Gilbert Energy
. 42
4.3.1
Rectifiability of Traffic Plans with Finite Energy
. 44
4.4
Appendix: Measurability Lemmas
. 44
VII
VIII Table of
Contents
5
Operations on Traffic Plans
. 47
5.1
Elementary Operations
. 47
5.1.1
Restriction, Domain of a Traffic Plan
. 47
5.1.2
Sum of Traffic Plans
(or Union of their Parameterizations)
. 48
5.1.3
Mass Normalization
. 48
5.2
Concatenation
. 48
5.2.1
Concatenation of Two Traffic Plans
. 48
5.2.2
Hierarchical Concatenation (Construction of Infinite
Irrigating Trees or Patterns)
. 49
5.3
A Priori Properties on Minimizers
. 51
5.3.1
An Assumption on
μ+, μ~
and
π
Avoiding Fibers
with Zero Length
. 51
5.3.2
A Convex Hull Property
. 53
6
Traffic Plans and Distances between Measures
. 55
6.1
All Measures can be Irrigated for
α
> 1 —
^
. 56
6.2
Stability with Respect to
μ+
and
μ~
. 58
6.3
Comparison of Distances between Measures
. 59
7
The Tree Structure of Optimal Traffic Plans
and their Approximation
. 65
7.1
The Single Path Property
. 65
7.2
The 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
8.3.1
The Main Lemma
. 82
8.3.2
Interior Regularity when supp(/x+)
Π
supp(/x~)
= 0 . 85
8.3.3
Interior Regularity when
μ+
is a Finite Atomic
Measure
. 89
8.4
Boundary Regularity
. 91
8.4.1
Further Regularity Properties
. 93
9
The 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
Table
of Contents IX
10
Irrigability and Dimension
.105
10.1
Several Concepts of Dimension of a Measure
and Irrigability Results
.105
10.2
Lower Bound on
Αμ)
.
Ill
10.3
Upper Bound on
ά(μ)
.112
10.4
Remarks and Examples
.114
11
The Landscape of an Optimal Pattern
.119
11.1
Introduction
.119
11.1.1
Landscape Equilibrium and OCNs in Geophysics
.119
11.2
A General Development Formula
.122
11.3
Existence of the Landscape Function
and Applications
.124
11.3.1
Well-Definedness of the Landscape Function
.124
11.3.2
Variational Applications
.127
11.4
Properties of the Landscape Function
.128
11.4.1
Semicontinuity
.128
11.4.2
Maximal Slope in the Network Direction
.129
11.5
Holder Continuity under Extra Assumptions
.131
11.5.1
Campanaio
Spaces by Medians
.131
11.5.2
Holder Continuity of the Landscape Function
.132
12
The
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
12.2.1
Topology of a Graph
.143
12.2.2
A Recursive Construction of an Optimum with Full
Steiner
Topology
.144
12.3
Number of Branches at a Bifurcation
.145
13
Dirac to Lebesgue Segment: A Case Study
.151
13.1
Analytical Results
.152
13.1.1
The Case of a Source Aligned with the Segment
.152
13.2
A "T Structure" is not Optimal
.153
13.3
The 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
13.5.1
Coding of the Topology
.159
13.5.2
Exhaustive Search
.160
13.6
Heuristics for Topology Optimization
.160
13.6.1
Multiscale Method
.161
13.6.2
Optimality of Subtrees
.164
13.6.3
Perturbation of the Topology
.165
X Table of Contents
14
Application: Embedded Irrigation Networks
.169
14.1
Irrigation Networks made of Tubes
.169
14.1.1
Anticipating some Conclusions
.171
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
14.5
Final Remarks
.177
15
Open Problems
.179
15.1
Stability
.179
15.2
Regularity
.179
15.3
The 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
В
Flows in Tubes
.189
B.I Poiseuille's Law
.189
B.2 Optimality of the Circular Section
.190
С
Notations
.191
References
.193
Index
.199 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Bernot, Marc Caselles, Vicent 1960- Morel, Jean-Michel 1953- |
| author_GND | (DE-588)135629985 (DE-588)136525229 |
| author_facet | Bernot, Marc Caselles, Vicent 1960- Morel, Jean-Michel 1953- |
| author_role | aut aut aut |
| author_sort | Bernot, Marc |
| author_variant | m b mb v c vc j m m jmm |
| building | Verbundindex |
| bvnumber | BV035088492 |
| callnumber-first | Q - Science |
| callnumber-label | QA3 |
| callnumber-raw | QA3 |
| callnumber-search | QA3 |
| callnumber-sort | QA 13 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SI 850 SK 970 |
| classification_tum | MAT 904f MAT 498f |
| ctrlnum | (OCoLC)263687714 (DE-599)DNB989061604 |
| dewey-full | 658.4032 |
| dewey-hundreds | 600 - Technology (Applied sciences) |
| dewey-ones | 658 - General management |
| dewey-raw | 658.4032 |
| dewey-search | 658.4032 |
| dewey-sort | 3658.4032 |
| dewey-tens | 650 - Management and auxiliary services |
| discipline | Mathematik Wirtschaftswissenschaften |
| discipline_str_mv | Mathematik Wirtschaftswissenschaften |
| format | Book |
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| id | DE-604.BV035088492 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:09:45Z |
| indexdate | 2024-07-09T21:21:56Z |
| institution | BVB |
| isbn | 9783540693147 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016756643 |
| oclc_num | 263687714 |
| open_access_boolean | |
| owner | DE-706 DE-824 DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-83 DE-11 DE-92 DE-188 |
| owner_facet | DE-706 DE-824 DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-83 DE-11 DE-92 DE-188 |
| physical | X, 200 S. Ill., graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in mathematics |
| series2 | Lecture notes in mathematics |
| spelling | Bernot, Marc Verfasser aut Optimal transportation networks models and theory Marc Bernot ; Vicent Caselles ; Jean-Michel Morel Berlin [u.a.] Springer 2009 X, 200 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in mathematics 1955 Literaturverz. S. 193 - 197 Circulation, Technique de la - Modèles mathématiques Transport - Technologie - Modèles mathématiques Mathematisches Modell Traffic engineering Mathematical models Transportation engineering Mathematical models Transportation Mathematical models Transportproblem (DE-588)4060694-6 gnd rswk-swf Transportproblem (DE-588)4060694-6 s DE-604 Caselles, Vicent 1960- Verfasser (DE-588)135629985 aut Morel, Jean-Michel 1953- Verfasser (DE-588)136525229 aut Lecture notes in mathematics 1955 (DE-604)BV000676446 1955 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016756643&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bernot, Marc Caselles, Vicent 1960- Morel, Jean-Michel 1953- Optimal transportation networks models and theory Lecture notes in mathematics Circulation, Technique de la - Modèles mathématiques Transport - Technologie - Modèles mathématiques Mathematisches Modell Traffic engineering Mathematical models Transportation engineering Mathematical models Transportation Mathematical models Transportproblem (DE-588)4060694-6 gnd |
| subject_GND | (DE-588)4060694-6 |
| title | Optimal transportation networks models and theory |
| title_auth | Optimal transportation networks models and theory |
| title_exact_search | Optimal transportation networks models and theory |
| title_exact_search_txtP | Optimal transportation networks models and theory |
| title_full | Optimal transportation networks models and theory Marc Bernot ; Vicent Caselles ; Jean-Michel Morel |
| title_fullStr | Optimal transportation networks models and theory Marc Bernot ; Vicent Caselles ; Jean-Michel Morel |
| title_full_unstemmed | Optimal transportation networks models and theory Marc Bernot ; Vicent Caselles ; Jean-Michel Morel |
| title_short | Optimal transportation networks |
| title_sort | optimal transportation networks models and theory |
| title_sub | models and theory |
| topic | Circulation, Technique de la - Modèles mathématiques Transport - Technologie - Modèles mathématiques Mathematisches Modell Traffic engineering Mathematical models Transportation engineering Mathematical models Transportation Mathematical models Transportproblem (DE-588)4060694-6 gnd |
| topic_facet | Circulation, Technique de la - Modèles mathématiques Transport - Technologie - Modèles mathématiques Mathematisches Modell Traffic engineering Mathematical models Transportation engineering Mathematical models Transportation Mathematical models Transportproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016756643&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000676446 |
| work_keys_str_mv | AT bernotmarc optimaltransportationnetworksmodelsandtheory AT casellesvicent optimaltransportationnetworksmodelsandtheory AT moreljeanmichel optimaltransportationnetworksmodelsandtheory |