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El. knyga: Lectures on Optimal Transport

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  • Formatas: EPUB+DRM
  • Serija: UNITEXT 130
  • Išleidimo metai: 22-Jul-2021
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030721626
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Lectures on Optimal TransportDidesnis paveikslėlis
  • Formatas: EPUB+DRM
  • Serija: UNITEXT 130
  • Išleidimo metai: 22-Jul-2021
  • Leidėjas: Springer Nature Switzerland AG
  • Kalba: eng
  • ISBN-13: 9783030721626
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This textbook is addressed to PhD or senior undergraduate students in mathematics, with interests in analysis, calculus of variations, probability and optimal transport. It originated from the teaching experience of the first author in the Scuola Normale Superiore, where a course on optimal transport and its applications has been given many times during the last 20 years. The topics and the tools were chosen at a sufficiently general and advanced level so that the student or scholar interested in a more specific theme would gain from the book the necessary background to explore it. After a large and detailed introduction to classical theory, more specific attention is devoted to applications to geometric and functional inequalities and to partial differential equations.

Recenzijos

This book is particularly suited for students who desire to learn from a text which closely follows the organization of a course, as well as for researchers and professors looking for inspiration for their own lecturers on the topic. The exposition is clear and mostly self-contained, with a nice list of examples that show the necessity of the assumptions of some classical results of the theory. (Nicolņ De Ponti, zbMATH 1485.49001, 2022)

This book is very well written and will be accessible to graduate students without background on optimal transport . All in all, this textbook is recommended to graduate students and researchers who want to discover the fundamental theory of optimal transport and its ramifications to several areas of mathematics. It can also easily be used by professors who want to teach a graduate course on thetopic. (Hugo Lavenant, Mathematical Reviews, June, 2022)

Lecture 1 Preliminary Notions and the Monge Problem
1(12)
1 Notation and Preliminary Results
1(5)
2 Monge's Formulation of the Optimal Transport Problem
6(7)
Lecture 2 The Kantorovich Problem
13(10)
1 Kantorovich's Formulation of the Optimal Transport Problem
13(1)
2 Transport Plans Versus Transport Maps
14(2)
3 Advantages of Kantorovich's Formulation
16(2)
4 Existence of Optimal Plans
18(5)
Lecture 3 The Kantorovich-Rubinstein Duality
23(12)
1 Convex Analysis Tools
24(3)
2 Proof of Duality via Fenchel-Rockafellar
27(2)
3 The Theory of c-Duality
29(2)
4 Proof of Duality and Dual Attainment for Bounded and Continuous Cost Functions
31(4)
Lecture 4 Necessary and Sufficient Optimality Conditions
35(1)
1 Duality and Necessary/Sufficient Optimality Conditions for Lower Semicontinuous Costs
35(8)
2 Remarks About Necessary and Sufficient Optimality Conditions
38(1)
3 Remarks About c-Cyclical Monotonicity, c-Concavity and c-Transforms for Special Costs
39(1)
4 Cost = distance2
39(1)
5 Cost = Distance
40(1)
6 Convex Costs on the Real Line
40(3)
Lecture 5 Existence of Optimal Maps and Applications
43(10)
1 Existence of Optimal Transport Maps
43(4)
2 A Digression About Monge's Problem
47(2)
3 Applications
49(2)
4 Iterated Monotone Rearrangement
51(2)
Lecture 6 A Proof of the Isoperimetric Inequality and Stability in Optimal Transport
53(12)
1 Isoperimetric Inequality
53(6)
2 Stability of Optimal Plans and Maps
59(6)
Lecture 7 The Monge-Ampere Equation and Optimal Transport on Riemannian Manifolds
65(12)
1 A General Change of Variables Formula
65(3)
2 The Monge-Ampere Equation
68(4)
3 Optimal Transport on Riemannian Manifolds
72(5)
Lecture 8 The Metric Side of Optimal Transport
77(10)
1 The Distance W2 in P2(X)
77(3)
2 Completeness of (P2(X), W2)
80(2)
3 Characterization of Convergence in (P2(X), W2) and Applications
82(5)
Lecture 9 Analysis on Metric Spaces and the Dynamic Formulation of Optimal Transport
87(8)
1 Absolutely Continuous Curves and Their Metric Derivative
87(4)
2 Geodesies and Action
91(2)
3 Dynamic Reformulation of the Optimal Transport Problem
93(2)
Lecture 10 Wasserstein Geodesies, Nonbranching and Curvature
95(14)
1 Lower Semicontinuity of the Action A2
95(2)
2 Compactness Criterion for Curves and Random Curves
97(2)
3 Lifting of Geodesies from X to p2(X)
99(10)
Lecture 11 Gradient Flows: An Introduction
109(16)
1 A-Convex Functions
110(1)
2 Differentiability of Absolutely Continuous Curves
111(2)
3 Gradient Flows
113(12)
Lecture 12 Gradient Flows: The Brezis-Komura Theorem
125(12)
1 Maximal Monotone Operators
125(1)
2 The Implicit Euler Scheme
126(2)
3 Reduction to Initial Conditions with Finite Energy
128(2)
4 Discrete EVI
130(7)
Lecture 13 Examples of Gradient Flows in PDEs
137(10)
1 p-Laplace Equation, Heat Equation in Domains, Fokker-Planck Equation
138(2)
2 The Heat Equation in Riemannian Manifolds
140(1)
3 Dual Sobolev Space H-1 and Heat Flow in H-1
141(6)
Lecture 14 Gradient Flows: The EDE and EDI Formulations
147(14)
1 EDE, EDI Solutions and Upper Gradients
147(3)
2 Existence of EDE, EDI Solutions
150(3)
3 Proof of Theorem 14.7 via Variational Interpolation
153(8)
Lecture 15 Semicontinuity and Convexity of Energies in the Wasserstein Space
161(22)
1 Semicontinuity of Internal Energies
161(6)
2 Convexity of Internal Energies
167(4)
3 Potential Energy Functional
171(2)
4 Interaction Energy
173(1)
5 Functional Inequalities via Optimal Transport
174(9)
Lecture 16 The Continuity Equation and the Hopf-Lax Semigroup
183(16)
1 Continuity Equation and Transport Equation
183(6)
2 Continuity Equation of Geodesies in the Wasserstein Space
189(1)
3 Hopf-Lax Semigroup
190(9)
Lecture 17 The Benamou-Brenier Formula
199(12)
1 Benamou-Brenier Formula
199(8)
2 Correspondence Between Absolutely Continuous Curves in P2(Kn) and Solutions to the Continuity Equation
207(4)
Lecture 18 An Introduction to Otto's Calculus
211(18)
1 Otto's Calculus
211(1)
2 Formal Interpretation of Some Evolution Equations as Wasserstein Gradient Flows
212(5)
3 Rigorous Interpretation of the Heat Equation as a Wasserstein Gradient Flow
217(6)
4 More Recent Ideas and Developments
223(6)
Lecture 19 Heat Flow, Optimal Transport and Ricci Curvature
229(16)
1 Heat Flow on Riemannian Manifolds
230(5)
2 Heat Flow, Optimal Transport and Ricci Curvature
235(10)
References 245
Prof. Luigi Ambrosio is a Professor of Mathematical Analysis, a former student of the Scuola Normale Superiore and presently its Director. His research interests include calculus of variations, geometric measure theory, optimal transport and analysis in metric spaces. For his scientific achievements, he has been awarded several prizes, in particular the Fermat prize in 2003 and the Balzan Prize in 2019. Dr. Elia Brué is a postdoctoral member at the Institute for Advanced Studies in Princeton. He earned his PhD degree at the Scuola Normale Superiore in 2020. His research interests include geometric measure theory, optimal transport, non-smooth geometry and PDE.





Dr. Daniele Semola is a postdoctoral research assistant at the Mathematical Institute of the University of Oxford. He was a student in Mathematics at the Scuola Normale Superiore, where he earned his PhD degree in 2020.  His research interests lie at the interface between geometric analysis and analysis on metric spaces, mainly with a focus on lower curvature bounds.
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