Optimal Transport for Applied Mathematicians

1. Front Matter

   Pages i-xxvii

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2. Primal and dual problems

   • Filippo Santambrogio

   Pages 1-57

3. One-dimensional issues

   • Filippo Santambrogio

   Pages 59-85

4. L 1 and L ∞ theory

   • Filippo Santambrogio

   Pages 87-119

5. Minimal flows, divergence constraints, and transport density

   • Filippo Santambrogio

   Pages 121-175

6. Wasserstein distances and curves in the Wasserstein spaces

   • Filippo Santambrogio

   Pages 177-218

7. Benamou-Brenier and other continuous numerical methods

   • Filippo Santambrogio

   Pages 219-248

8. Functionals on the space of probabilities

   • Filippo Santambrogio

   Pages 249-284

9. Gradient flows

   • Filippo Santambrogio

   Pages 285-323

10. Back Matter

    Pages 325-353

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This monograph presents a rigorous mathematical introduction to optimal transport as a variational problem, its use in modeling various phenomena, and its connections with partial differential equations. Its main goal is to provide the reader with the techniques necessary to understand the current research in optimal transport and the tools which are most useful for its applications. Full proofs are used to illustrate mathematical concepts and each chapter includes a section that discusses applications of optimal transport to various areas, such as economics, finance, potential games, image processing and fluid dynamics. Several topics are covered that have never been previously in books on this subject, such as the Knothe transport, the properties of functionals on measures, the Dacorogna-Moser flow, the formulation through minimal flows with prescribed divergence formulation, the case of the supremal cost, and the most classical numerical methods. Graduate students and researchers in both pure and applied mathematics interested in the problems and applications of optimal transport will find this to be an invaluable resource.

• Fluid mechanics
• Fokker-Planck equation
• Gradient Flows
• Measure Theory
• Monge-Ampère equation
• Optimal transport
• Partial differential equations
• ordinary differential equations

“This book offers an excellent, exciting and enjoyable, tour through the theory of optimal transportation, with a very good choice of topics … . It is well written and thorough and provides an excellent introduction to applied mathematicians … . a carefully selected list of exercises, make it ideal either as a textbook for an advanced postgraduate of doctoral level course, or for independent study.” (Athanasios Yannacopoulos, zbMATH 1401.49002, 2019)



“This book is very well written, and the proofs are carefully chosen and adapted. It is suitable for the researcher or the student willing to enter this field as well as for the professor planning a course on this topic. Thanks to the discussions at the end of each chapter and to the rich bibliography it is also a very good reference book.” (Luigi De Pascale, Mathematical Reviews, January, 2017)

• Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, Orsay cedex, France

  Filippo Santambrogio