This is a graduate-level introduction to the key ideas and theoretical foundation of the vibrant field of optimal mass transport in the Euclidean setting. Taking a pedagogical approach, it introduces concepts gradually and in an accessible way, while also remaining technically and conceptually complete.
This is a graduate-level introduction to the key ideas and theoretical foundation of the vibrant field of optimal mass transport in the Euclidean setting. Taking a pedagogical approach, it introduces concepts gradually and in an accessible way, while also remaining technically and conceptually complete.Hinweis: Dieser Artikel kann nur an eine deutsche Lieferadresse ausgeliefert werden.
Francesco Maggi is Professor of Mathematics at the University of Texas at Austin. His research interests include the calculus of variations, partial differential equations, and optimal mass transport. He is the author of Sets of Finite Perimeter and Geometric Variational Problems published by Cambridge University Press.
Inhaltsangabe
Preface Notation Part I. The Kantorovich Problem: 1. An introduction to the Monge problem 2. Discrete transport problems 3. The Kantorovich problem Part II. Solution of the Monge Problem with Quadratic Cost: the Brenier-McCann Theorem: 4. The Brenier theorem 5. First order differentiability of convex functions 6. The Brenier-McCann theorem 7. Second order differentiability of convex functions 8. The Monge-Ampère equation for Brenier maps Part III. Applications to PDE and the Calculus of Variations and the Wasserstein Space: 9. Isoperimetric and Sobolev inequalities in sharp form 10. Displacement convexity and equilibrium of gases 11. The Wasserstein distance W2 on P2(Rn) 12. Gradient flows and the minimizing movements scheme 13. The Fokker-Planck equation in the Wasserstein space 14. The Euler equations and isochoric projections 15. Action minimization, Eulerian velocities and Otto's calculus Part IV. Solution of the Monge Problem with Linear Cost: the Sudakov Theorem: 16. Optimal transport maps on the real line 17. Disintegration 18. Solution to the Monge problem with linear cost 19. An introduction to the needle decomposition method Appendix A: Radon measures on Rn and related topics Appendix B: Bibliographical Notes Bibliography Index.
Preface Notation Part I. The Kantorovich Problem: 1. An introduction to the Monge problem 2. Discrete transport problems 3. The Kantorovich problem Part II. Solution of the Monge Problem with Quadratic Cost: the Brenier-McCann Theorem: 4. The Brenier theorem 5. First order differentiability of convex functions 6. The Brenier-McCann theorem 7. Second order differentiability of convex functions 8. The Monge-Ampère equation for Brenier maps Part III. Applications to PDE and the Calculus of Variations and the Wasserstein Space: 9. Isoperimetric and Sobolev inequalities in sharp form 10. Displacement convexity and equilibrium of gases 11. The Wasserstein distance W2 on P2(Rn) 12. Gradient flows and the minimizing movements scheme 13. The Fokker-Planck equation in the Wasserstein space 14. The Euler equations and isochoric projections 15. Action minimization, Eulerian velocities and Otto's calculus Part IV. Solution of the Monge Problem with Linear Cost: the Sudakov Theorem: 16. Optimal transport maps on the real line 17. Disintegration 18. Solution to the Monge problem with linear cost 19. An introduction to the needle decomposition method Appendix A: Radon measures on Rn and related topics Appendix B: Bibliographical Notes Bibliography Index.
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