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Optimal Mass Transport on Euclidean Spaces [Kõva köide]

(University of Texas, Austin)
  • Formaat: Hardback, 345 pages, kõrgus x laius x paksus: 235x159x25 mm, kaal: 630 g, Worked examples or Exercises
  • Sari: Cambridge Studies in Advanced Mathematics
  • Ilmumisaeg: 16-Nov-2023
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1009179705
  • ISBN-13: 9781009179706
Teised raamatud teemal:
Optimal Mass Transport on Euclidean SpacesSuurem pilt
  • Formaat: Hardback, 345 pages, kõrgus x laius x paksus: 235x159x25 mm, kaal: 630 g, Worked examples or Exercises
  • Sari: Cambridge Studies in Advanced Mathematics
  • Ilmumisaeg: 16-Nov-2023
  • Kirjastus: Cambridge University Press
  • ISBN-10: 1009179705
  • ISBN-13: 9781009179706
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9781009179706.html
Märksõnad:
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.

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.

Arvustused

'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 '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

Muu info

A pedagogical introduction to the key ideas and theoretical foundation of optimal mass transport for a graduate course or self-study.
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.
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.