Gradient flows in metric spaces and in the space of probability measures:
Looking at the theory of gradient flows in spaces which are not necessarily endowed with a natural linear or differentiable structure, this text covers gradient flows in the space of probability measures on a separable Hilbert space, endowed with the Kantorovich-Rubinstein-Wasserstein distance and g...
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel ; Boston ; Berlin
Birkhäuser
2008
|
| Edition: | Second Edition |
| Series: | Lectures in mathematics ETH Zürich
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | Looking at the theory of gradient flows in spaces which are not necessarily endowed with a natural linear or differentiable structure, this text covers gradient flows in the space of probability measures on a separable Hilbert space, endowed with the Kantorovich-Rubinstein-Wasserstein distance and gradient flows in metric spaces. |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | vii, 334 Seiten |
| ISBN: | 9783764387211 |
Staff View
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| 100 | 1 | |a Ambrosio, Luigi |d 1963- |0 (DE-588)133791408 |4 aut | |
| 245 | 1 | 0 | |a Gradient flows in metric spaces and in the space of probability measures |c Luigi Ambrosio ; Nicola Gigli ; Giuseppe Savaré |
| 250 | |a Second Edition | ||
| 264 | 1 | |a Basel ; Boston ; Berlin |b Birkhäuser |c 2008 | |
| 300 | |a vii, 334 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Lectures in mathematics ETH Zürich | |
| 500 | |a Includes bibliographical references and index | ||
| 520 | 3 | |a Looking at the theory of gradient flows in spaces which are not necessarily endowed with a natural linear or differentiable structure, this text covers gradient flows in the space of probability measures on a separable Hilbert space, endowed with the Kantorovich-Rubinstein-Wasserstein distance and gradient flows in metric spaces. | |
| 650 | 4 | |a Measure theory | |
| 650 | 4 | |a Metric spaces | |
| 650 | 4 | |a Differential equations, Parabolic | |
| 650 | 4 | |a Monotone operators | |
| 650 | 4 | |a Evolution equations, Nonlinear | |
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Record in the Search Index
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| adam_text | Contents
Preface
to the Second Edition
ix
Introduction
1
Notation
18
I Gradient Flow in Metric Spaces
21
1
Curves and Gradients in Metric Spaces
23
1.1
Absolutely continuous curves and metric derivative
......... 23
1.2
Upper gradients
............................. 26
1.3
Curves of maximal slope
........................ 30
1.4
Curves of maximal slope in Hilbcrt and Banach spaces
....... 32
2
Existence of Curves of Maximal Slope
39
2.1
Main topological assumptions
..................... 42
2.2
Solvability of the discrete problem and compactness of discrete tra¬
jectories
................................. 44
2.3
Generalized minimizing movements and curves of maximal slope
. 45
2.4
The (geodcsically) convex case
.................... 49
3
Proofs of the Convergence Theorems
59
3.1
Moreau-Yosida approximation
..................... 59
3.2
A priori estimates for the discrete solutions
............. 66
3.3
A compactness argument
....................... 69
3.4
Conclusion of the proofs of the convergence theorems
........ 71
4
Generation of Contraction Semigroups
75
4.1
Cauchy-type estimates for discrete solutions
............. 82
4.1.1
Discrete variational inequalities
................ 82
4.1.2
Piecewise
affine
interpolation and comparison results
.... 84
vi
Contents
4.2
Convergence of discrete solutions
................... 89
4.2.1
Convergence when the initial datum
щ Є
D{ò)
....... 89
4.2.2
Convergence when the initial datum
«о Є -О(ф)........
92
4.3
Regularizing effect, uniqueness and the semigroup property
.... 93
4.4
Optimal error estimates
........................ 97
4.4.1
The case
λ
= 0......................... 97
4.4.2
The case A ^
0......................... 99
II Gradient Flow in the Space of Probability Measures
103
5
Preliminary Results on Measure Theory
105
5.1
Narrow convergence, tightness, and uniform integrability
...... 106
5.1.1
Unbounded and l.s.c. integrands
............... 109
5.1.2
Hubert spaces and weak topologies
.............. 113
5.2
Transport of measures
......................... 118
5.3
Measure-valued maps and disintegration theorem
.......... 121
5.4
Convergence of plans and convergence of maps
........... 124
5.5
Approximate differentiability and area formula in Euclidean spaces
128
6
The Optimal Transportation Problem
133
6.1
Optimality conditions
......................... 135
6.2
Optimal transport maps and their regularity
............ 139
6.2.1
Approximate differentiability of the optimal transport map
142
6.2.2
The infinite dimensional case
................. 147
6.2.3
The quadratic case
ρ
= 2................... 149
7
The
Wasserstein
Distance and its Behaviour along Geodesies
151
7.1
The
Wasserstein
distance
....................... 151
7.2
Interpolation and geodesies
...................... 158
7.3
The curvature properties of B^2{X)
.................. 160
8 A.C.
Curves in 3^p{X) and the Continuity Equation
167
8.1
The continuity equation in Rrf
..................... 169
8.2
A probabilistic representation of solutions of the continuity equation
178
8.3
Absolutely continuous curves in
0äp{X)............... 182
8.4
The tangent bundle to &P(X)
.................... 189
8.5
Tangent space and optimal maps
................... 194
9
Convex Functional in
£^Р{Х)
201
9.1
A-geodesically convex functionals in 3°P(X)
............. 202
9.2
Convexity along generalized geodesies
................ 205
9.3
Examples of convex functionals in &v{X)
.............. 209
Contents
vu
9.4
Relative entropy and convex fmictkmals of measures
........ 215
9.4.1
Log-concavity and displacement convexity
.......... 220
10
Metric Slope and Subdifferential Calculus in PJ°P{X)
227
10.1
Subdifferential calculus in ^{X): the regular case
......... 229
10.1.1
The case of A-convex functionals along geodesies
...... 231
10.1.2
Regular functionals
....................... 232
10.2
Differentiability properties of the p-
Wasserstein
distance
...... 234
10.3
Subdifferential calculus in
£Pp(X):
the general case
......... 240
10.3.1
The case of
А
-convex
functionals along geodesies
...... 244
10.3.2
Regular functionals
....................... 246
10.4
Example of subdifferentials
...................... 254
10.4.1
Variational integrals: the smooth case
............ 254
10.4.2
The potential energy
...................... 255
10.4.3
The internal energy
...................... 257
10.4.4
The relative internal energy
.................. 265
10.4.5
The interaction energy
..................... 267
10.4.6
The opposite
Wasserstein
distance
.............. 269
10.4.7
The sum of internal, potential and interaction energy
. . . 272
10.4.8
Relative entropy and Fisher information in infinite dimensions276
11
Gradient Flows and Curves of Maximal Slope in P?P{X)
279
11.1
The gradient flow equation and its metric formulations
....... 280
11.1.1
Gradient flows and curves of maximal slope
......... 283
11.1.2
Gradient flows for
А
-convex
functionals
........... 284
11.1.3
The convergence of the Minimizing Movement scheme
. . 286
11.2
Gradient flows for
А
-convex
functionals along generalized geodesies
295
11.2.1
Applications to Evolution PDE s
............... 298
11.3
Gradient flows in
S^p{X)
for regular functionals
........... 304
12
Appendix
307
12.1
Carathéodory
and normal integrands
................. 307
12.2
Weak convergence of plans and disintegrations
........... 308
12.3
PG metric spaces and their geometric tangent cone
......... 310
12.4
The geometric tangent spaces in
Р?ч{Х)
............... 314
Bibliography
331
Index
333
|
| adam_txt |
Contents
Preface
to the Second Edition
ix
Introduction
1
Notation
18
I Gradient Flow in Metric Spaces
21
1
Curves and Gradients in Metric Spaces
23
1.1
Absolutely continuous curves and metric derivative
. 23
1.2
Upper gradients
. 26
1.3
Curves of maximal slope
. 30
1.4
Curves of maximal slope in Hilbcrt and Banach spaces
. 32
2
Existence of Curves of Maximal Slope
39
2.1
Main topological assumptions
. 42
2.2
Solvability of the discrete problem and compactness of discrete tra¬
jectories
. 44
2.3
Generalized minimizing movements and curves of maximal slope
. 45
2.4
The (geodcsically) convex case
. 49
3
Proofs of the Convergence Theorems
59
3.1
Moreau-Yosida approximation
. 59
3.2
A priori estimates for the discrete solutions
. 66
3.3
A compactness argument
. 69
3.4
Conclusion of the proofs of the convergence theorems
. 71
4
Generation of Contraction Semigroups
75
4.1
Cauchy-type estimates for discrete solutions
. 82
4.1.1
Discrete variational inequalities
. 82
4.1.2
Piecewise
affine
interpolation and comparison results
. 84
vi
Contents
4.2
Convergence of discrete solutions
. 89
4.2.1
Convergence when the initial datum
щ Є
D{ò)
. 89
4.2.2
Convergence when the initial datum
«о Є -О(ф).
92
4.3
Regularizing effect, uniqueness and the semigroup property
. 93
4.4
Optimal error estimates
. 97
4.4.1
The case
λ
= 0. 97
4.4.2
The case A ^
0. 99
II Gradient Flow in the Space of Probability Measures
103
5
Preliminary Results on Measure Theory
105
5.1
Narrow convergence, tightness, and uniform integrability
. 106
5.1.1
Unbounded and l.s.c. integrands
. 109
5.1.2
Hubert spaces and weak topologies
. 113
5.2
Transport of measures
. 118
5.3
Measure-valued maps and disintegration theorem
. 121
5.4
Convergence of plans and convergence of maps
. 124
5.5
Approximate differentiability and area formula in Euclidean spaces
128
6
The Optimal Transportation Problem
133
6.1
Optimality conditions
. 135
6.2
Optimal transport maps and their regularity
. 139
6.2.1
Approximate differentiability of the optimal transport map
142
6.2.2
The infinite dimensional case
. 147
6.2.3
The quadratic case
ρ
= 2. 149
7
The
Wasserstein
Distance and its Behaviour along Geodesies
151
7.1
The
Wasserstein
distance
. 151
7.2
Interpolation and geodesies
. 158
7.3
The curvature properties of B^2{X)
. 160
8 A.C.
Curves in 3^p{X) and the Continuity Equation
167
8.1
The continuity equation in Rrf
. 169
8.2
A probabilistic representation of solutions of the continuity equation
178
8.3
Absolutely continuous curves in
0äp{X). 182
8.4
The tangent bundle to &P(X)
. 189
8.5
Tangent space and optimal maps
. 194
9
Convex Functional in
£^Р{Х)
201
9.1
A-geodesically convex functionals in 3°P(X)
. 202
9.2
Convexity along generalized geodesies
. 205
9.3
Examples of convex functionals in &v{X)
. 209
Contents
vu
9.4
Relative entropy and convex fmictkmals of'measures
. 215
9.4.1
Log-concavity and displacement convexity
. 220
10
Metric Slope and Subdifferential Calculus in PJ°P{X)
227
10.1
Subdifferential calculus in ^{X): the regular case
. 229
10.1.1
The case of A-convex functionals along geodesies
. 231
10.1.2
Regular functionals
. 232
10.2
Differentiability properties of the p-
Wasserstein
distance
. 234
10.3
Subdifferential calculus in
£Pp(X):
the general case
. 240
10.3.1
The case of
А
-convex
functionals along geodesies
. 244
10.3.2
Regular functionals
. 246
10.4
Example of subdifferentials
. 254
10.4.1
Variational integrals: the smooth case
. 254
10.4.2
The potential energy
. 255
10.4.3
The internal energy
. 257
10.4.4
The relative internal energy
. 265
10.4.5
The interaction energy
. 267
10.4.6
The opposite
Wasserstein
distance
. 269
10.4.7
The sum of internal, potential and interaction energy
. . . 272
10.4.8
Relative entropy and Fisher information in infinite dimensions276
11
Gradient Flows and Curves of Maximal Slope in P?P{X)
279
11.1
The gradient flow equation and its metric formulations
. 280
11.1.1
Gradient flows and curves of maximal slope
. 283
11.1.2
Gradient flows for
А
-convex
functionals
. 284
11.1.3
The convergence of the "Minimizing Movement" scheme
. . 286
11.2
Gradient flows for
А
-convex
functionals along generalized geodesies
295
11.2.1
Applications to Evolution PDE's
. 298
11.3
Gradient flows in
S^p{X)
for regular functionals
. 304
12
Appendix
307
12.1
Carathéodory
and normal integrands
. 307
12.2
Weak convergence of plans and disintegrations
. 308
12.3
PG metric spaces and their geometric tangent cone
. 310
12.4
The geometric tangent spaces in
Р?ч{Х)
. 314
Bibliography
331
Index
333 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Ambrosio, Luigi 1963- Gigli, Nicola Savaré, Giuseppe |
| author_GND | (DE-588)133791408 (DE-588)130331236 (DE-588)130331252 |
| author_facet | Ambrosio, Luigi 1963- Gigli, Nicola Savaré, Giuseppe |
| author_role | aut aut aut |
| author_sort | Ambrosio, Luigi 1963- |
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| callnumber-sort | QA 3312 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 110 SK 280 SK 430 |
| classification_tum | MAT 476f MAT 280f MAT 671f |
| ctrlnum | (OCoLC)212432128 (DE-599)BVBBV023280045 |
| dewey-full | 515/.42 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.42 |
| dewey-search | 515/.42 |
| dewey-sort | 3515 242 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | Second Edition |
| format | Book |
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| id | DE-604.BV023280045 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T20:39:25Z |
| indexdate | 2024-07-09T21:14:50Z |
| institution | BVB |
| isbn | 9783764387211 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016464866 |
| oclc_num | 212432128 |
| open_access_boolean | |
| owner | DE-384 DE-355 DE-BY-UBR DE-29T DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-83 DE-11 DE-188 DE-20 |
| owner_facet | DE-384 DE-355 DE-BY-UBR DE-29T DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-83 DE-11 DE-188 DE-20 |
| physical | vii, 334 Seiten |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Lectures in mathematics ETH Zürich |
| spelling | Ambrosio, Luigi 1963- (DE-588)133791408 aut Gradient flows in metric spaces and in the space of probability measures Luigi Ambrosio ; Nicola Gigli ; Giuseppe Savaré Second Edition Basel ; Boston ; Berlin Birkhäuser 2008 vii, 334 Seiten txt rdacontent n rdamedia nc rdacarrier Lectures in mathematics ETH Zürich Includes bibliographical references and index Looking at the theory of gradient flows in spaces which are not necessarily endowed with a natural linear or differentiable structure, this text covers gradient flows in the space of probability measures on a separable Hilbert space, endowed with the Kantorovich-Rubinstein-Wasserstein distance and gradient flows in metric spaces. Measure theory Metric spaces Differential equations, Parabolic Monotone operators Evolution equations, Nonlinear Gradientenfluss (DE-588)4841287-9 gnd rswk-swf Potenzialfeld (DE-588)4126347-9 gnd rswk-swf Metrischer Raum (DE-588)4169745-5 gnd rswk-swf Maßraum (DE-588)4169057-6 gnd rswk-swf Fluss Mathematik (DE-588)4489499-5 gnd rswk-swf Wahrscheinlichkeitsmaß (DE-588)4137556-7 gnd rswk-swf Gradientenfluss (DE-588)4841287-9 s Metrischer Raum (DE-588)4169745-5 s Maßraum (DE-588)4169057-6 s Wahrscheinlichkeitsmaß (DE-588)4137556-7 s DE-604 Potenzialfeld (DE-588)4126347-9 s Fluss Mathematik (DE-588)4489499-5 s b DE-604 Gigli, Nicola (DE-588)130331236 aut Savaré, Giuseppe (DE-588)130331252 aut Erscheint auch als Online-Ausgabe 978-3-7643-8722-8 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016464866&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ambrosio, Luigi 1963- Gigli, Nicola Savaré, Giuseppe Gradient flows in metric spaces and in the space of probability measures Measure theory Metric spaces Differential equations, Parabolic Monotone operators Evolution equations, Nonlinear Gradientenfluss (DE-588)4841287-9 gnd Potenzialfeld (DE-588)4126347-9 gnd Metrischer Raum (DE-588)4169745-5 gnd Maßraum (DE-588)4169057-6 gnd Fluss Mathematik (DE-588)4489499-5 gnd Wahrscheinlichkeitsmaß (DE-588)4137556-7 gnd |
| subject_GND | (DE-588)4841287-9 (DE-588)4126347-9 (DE-588)4169745-5 (DE-588)4169057-6 (DE-588)4489499-5 (DE-588)4137556-7 |
| title | Gradient flows in metric spaces and in the space of probability measures |
| title_auth | Gradient flows in metric spaces and in the space of probability measures |
| title_exact_search | Gradient flows in metric spaces and in the space of probability measures |
| title_exact_search_txtP | Gradient flows in metric spaces and in the space of probability measures |
| title_full | Gradient flows in metric spaces and in the space of probability measures Luigi Ambrosio ; Nicola Gigli ; Giuseppe Savaré |
| title_fullStr | Gradient flows in metric spaces and in the space of probability measures Luigi Ambrosio ; Nicola Gigli ; Giuseppe Savaré |
| title_full_unstemmed | Gradient flows in metric spaces and in the space of probability measures Luigi Ambrosio ; Nicola Gigli ; Giuseppe Savaré |
| title_short | Gradient flows in metric spaces and in the space of probability measures |
| title_sort | gradient flows in metric spaces and in the space of probability measures |
| topic | Measure theory Metric spaces Differential equations, Parabolic Monotone operators Evolution equations, Nonlinear Gradientenfluss (DE-588)4841287-9 gnd Potenzialfeld (DE-588)4126347-9 gnd Metrischer Raum (DE-588)4169745-5 gnd Maßraum (DE-588)4169057-6 gnd Fluss Mathematik (DE-588)4489499-5 gnd Wahrscheinlichkeitsmaß (DE-588)4137556-7 gnd |
| topic_facet | Measure theory Metric spaces Differential equations, Parabolic Monotone operators Evolution equations, Nonlinear Gradientenfluss Potenzialfeld Metrischer Raum Maßraum Fluss Mathematik Wahrscheinlichkeitsmaß |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016464866&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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