Lectures on optimal transport / / Luigi Ambrosio, Elia Brué, and Daniele Semola

• Autore: Ambrosio Luigi
• Pubblicazione: Cham, Switzerland : , : Springer, , [2021]; ©2021
• Descrizione fisica: 1 online resource (250 pages)
• Disciplina: 519.6
• Soggetto topico: Mathematical optimization; Optimització matemàtica
• Soggetto genere / forma: Llibres electrònics
• Persona (resp. second.): BruéElia; SemolaDaniele
• Nota di bibliografia: Includes bibliographical references.
• Nota di contenuto: Intro -- Preface -- Contents -- Lecture 1: Preliminary Notions and the Monge Problem -- 1 Notation and Preliminary Results -- 2 Monge's Formulation of the Optimal Transport Problem -- Lecture 2: The Kantorovich Problem -- 1 Kantorovich's Formulation of the Optimal Transport Problem -- 2 Transport Plans Versus Transport Maps -- 3 Advantages of Kantorovich's Formulation -- 4 Existence of Optimal Plans -- Lecture 3: The Kantorovich-Rubinstein Duality -- 1 Convex Analysis Tools -- 2 Proof of Duality via Fenchel-Rockafellar -- 3 The Theory of c-Duality -- 4 Proof of Duality and Dual Attainment for Bounded and Continuous Cost Functions -- Lecture 4: Necessary and Sufficient Optimality Conditions -- 1 Duality and Necessary/Sufficient Optimality Conditions for Lower Semicontinuous Costs -- 2 Remarks About Necessary and Sufficient Optimality Conditions -- 3 Remarks About c-Cyclical Monotonicity, c-Concavity and c-Transforms for Special Costs -- 4 Cost=distance2 -- 5 Cost=Distance -- 6 Convex Costs on the Real Line -- Lecture 5: Existence of Optimal Maps and Applications -- 1 Existence of Optimal Transport Maps -- 2 A Digression About Monge's Problem -- 3 Applications -- 4 Iterated Monotone Rearrangement -- Lecture 6: A Proof of the Isoperimetric Inequality and Stability in Optimal Transport -- 1 Isoperimetric Inequality -- 2 Stability of Optimal Plans and Maps -- Lecture 7: The Monge-Ampére Equation and Optimal Transport on Riemannian Manifolds -- 1 A General Change of Variables Formula -- 2 The Monge-Ampère Equation -- 3 Optimal Transport on Riemannian Manifolds -- Lecture 8: The Metric Side of Optimal Transport -- 1 The Distance W2 in P2(X) -- 2 Completeness of Square Integrable Probabilities -- 3 Characterization of Convergence in the Space of Square Integrable Probabilities. ; Lecture 9: Analysis on Metric Spaces and the Dynamic Formulation of Optimal Transport -- 1 Absolutely Continuous Curves and Their Metric Derivative -- 2 Geodesics and Action -- 3 Dynamic Reformulation of the Optimal Transport Problem -- Lecture 10: Wasserstein Geodesics, Nonbranching and Curvature -- 1 Lower Semicontinuity of the Action A2 -- 2 Compactness Criterion for Curves and Random Curves -- 3 Lifting of Geodesics from X to P2(X) -- Lecture 11: Gradient Flows: An Introduction -- 1 lambda-Convex Functions -- 2 Differentiability of Absolutely Continuous Curves -- 3 Gradient Flows -- Lecture 12: Gradient Flows: The Brézis-Komura Theorem -- 1 Maximal Monotone Operators -- 2 The Implicit Euler Scheme -- 3 Reduction to Initial Conditions with Finite Energy -- 4 Discrete EVI -- Lecture 13: Examples of Gradient Flows in PDEs -- 1 p-Laplace Equation, Heat Equation in Domains, Fokker-Planck Equation -- 2 The Heat Equation in Riemannian Manifolds -- 3 Dual Sobolev Space H-1 and Heat Flow in H-1 -- Lecture 14: Gradient Flows: The EDE and EDI Formulations -- 1 EDE, EDI Solutions and Upper Gradients -- 2 Existence of EDE, EDI Solutions -- 3 Proof of Theorem 14.7 via Variational Interpolation -- Lecture 15: Semicontinuity and Convexity of Energies in the Wasserstein Space -- 1 Semicontinuity of Internal Energies -- 2 Convexity of Internal Energies -- 3 Potential Energy Functional -- 4 Interaction Energy -- 5 Functional Inequalities via Optimal Transport -- Lecture 16: The Continuity Equation and the Hopf-Lax Semigroup -- 1 Continuity Equation and Transport Equation -- 2 Continuity Equation of Geodesics in the Wasserstein Space -- 3 Hopf-Lax Semigroup -- Lecture 17: The Benamou-Brenier Formula -- 1 Benamou-Brenier Formula -- 2 Correspondence Between Absolutely Continuous Curves in the Probabilities and Solutions to the Continuity Equation. ; Lecture 18: An Introduction to Otto's Calculus -- 1 Otto's Calculus -- 2 Formal Interpretation of Some Evolution Equations as Wasserstein Gradient Flows -- 3 Rigorous Interpretation of the Heat Equation as a Wasserstein Gradient Flow -- 4 More Recent Ideas and Developments -- Lecture 19: Heat Flow, Optimal Transport and Ricci Curvature -- 1 Heat Flow on Riemannian Manifolds -- 2 Heat Flow, Optimal Transport and Ricci Curvature -- References.
• Titolo autorizzato: Lectures on Optimal Transport
• ISBN: 3-030-72162-0
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910495195503321
• Lo trovi qui: Univ. Federico II
• Serie: Unitext