Optimal Mass Transport on Euclidean Spaces
£49.99
Part of Cambridge Studies in Advanced Mathematics
- Author: Francesco Maggi, University of Texas, Austin
- Date Published: November 2023
- availability: Available
- format: Hardback
- isbn: 9781009179706
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Optimal mass transport has emerged in the past three decades as an active field with wide-ranging connections to the calculus of variations, PDEs, and geometric analysis. This graduate-level introduction covers the field's theoretical foundation and key ideas in applications. By focusing on optimal mass transport problems in a Euclidean setting, the book is able to introduce concepts in a gradual, accessible way with minimal prerequisites, while remaining technically and conceptually complete. Working in a familiar context will help readers build geometric intuition quickly and give them a strong foundation in the subject. This book explores the relation between the Monge and Kantorovich transport problems, solving the former for both the linear transport cost (which is important in geometric applications) and for the quadratic transport cost (which is central in PDE applications), starting from the solution of the latter for arbitrary transport costs.
Read more- Adapts easily to a graduate course on optimal mass transport, with each of the first 14 chapters corresponding to one or two lectures
- Provides detailed and fully rigorous proofs
- Explains the historical and scientific context of the various applications so readers will be able to find connections and unity
- Assumes only a background in undergraduate analysis and familiarity with the basic theory of Radon measures in Rn
Reviews & endorsements
'Francesco Maggi's book is a detailed and extremely well written explanation of the fascinating theory of Monge-Kantorovich optimal mass transfer. I especially recommend Part IV's discussion of the 'linear' cost problem and its subtle mathematical resolution.' Lawrence C. Evans, University of California, Berkeley
See more reviews'Over the last three decades, optimal transport has revolutionized the mathematical analysis of inequalities, differential equations, dynamical systems, and their applications to physics, economics, and computer science. By exposing the interplay between the discrete and Euclidean settings, Maggi's book makes this development uniquely accessible to advanced undergraduates and mathematical researchers with a minimum of prerequisites. It includes the first textbook accounts of the localization technique known as needle decomposition and its solution to Monge's centuries old cutting and filling problem (1781). This book will be an indispensable tool for advanced undergraduates and mathematical researchers alike.' Robert McCann, University of Toronto
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×Product details
- Date Published: November 2023
- format: Hardback
- isbn: 9781009179706
- length: 345 pages
- dimensions: 235 x 159 x 25 mm
- weight: 0.63kg
- availability: Available
Table of Contents
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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