Optimal transport

Teacher: Xavier Fernandez-Real

Assistant: Roberto Colombo

 

Exercise Sessions

The exercise session will have a ~30’ discussion, mainly of an exercise of the previous series or of the current one. This discussion may happen at the beginning or in the middle of the exercise session, depending on the topic to be discussed.
The rest of the time is devoted to Q&A. 
Schedule
Lectures: Monday, 8.15 – 10.00, MAA331
Exercises: Monday, 10.15 – 12.00, MAA331

Hand-in exercises
There will be some series of exercises where the students are invited to hand in the solutions. One exercise (randomly selected) will be corrected and the work will receive an evaluation of 0, 1, or 2 points. These points will contribute to the final evaluation.

Course content  

The theory of optimal transport began in the eighteenth century with the Monge problem (1781), which is to minimize the cost of transporting an amount of material from the given set of origins to the given set of destinations. In the forties, Kantorovitch gave an important reformulation of the problem and, since then, the Monge-Kantorovitch problem has been a classical subject in probability theory, economics, and optimization. More recently, the interplay between optimal transport and various fields such as PDEs (Ricci flow, Euler equations'), fluid mechanics, geometric analysis (isoperimetric and Sobolev inequalities, curvature-dimension conditions), functional analysis, urban planning, and economics has been deeply investigated.

The first part of the course will be devoted to Monge and Kantorovitch's problems, discussing the existence and the properties of the optimal plan under different conditions on the cost. We will exploit the relation with Kantorovitch's duality theorem, with Brenier's polar decomposition theorem, and with the Monge-Ampère equation, a PDE that arises naturally in this context. The second part of the course will be centered on the applications of optimal transport to different problems: after introducing the Wasserstein distance, we will see the connection with some PDEs and with functional/geometric inequalities.

Material and reference books

Students are expected to learn the material discussed in class. The lectures, in turn, will be based mainly on the lecture notes of

A. Figalli and F. Glaudo, An invitation to Optimal Transport, Wasserstein Distances and Gradient Flows

attached below.

Course book

Complementary material: Santambrogio’s and Villani's books are also a precious source of interesting exercises and for deepening your understanding.

• F. Santambrogio, Optimal transport for applied mathematicians
• C. Villani, Optimal transport, old and new

 

 The graded series will receive a grade in [1,6].  Each hand-in series will have one evaluated exercise, which can give a bonus of 0.25. The presentation of one or more exercises during the exercise class gives up to an extra 0.25.

Exercises available for a presentation: Exercise 1.4 point (i); Exercise 2.2.

Presentations:

- Ex. 1.4 point (i): Alejandro Morera Alvarez;

- Ex. 2.2: Shivang Sachar.

Exercises available for a presentation: Exercise 3.3 i) (a) and ii); Exercise 3.5.

Presentations:

-Exercise 3.3 i) (a) and ii): Francesco Maria Guadagnuolo.

-Exercise 3.5: Vadym Koval.

Exercises available for a presentation: Exercise 4.3; Exercise 4.4.

Presentations:

- Ex 4.3: Lennart Kutzschebauch.

- Ex 4.4: Roy Makhlouf.

Exercises available for a presentation: Exercise 5.2; Exercise 5.4.

Presentations:

- Exercise 5.2: Berk Ceylan.

- Exercise 5.4: Erasmo Santini.

-Exercise 5.3: Polina Stankevich.

Exercises available for a presentation: Ex 7.1, 7.2, 7.3.  As they are connected, Exercises 7.2 and 7.3 can also be presented in couple by two students.

- Ex. 7.1: Luca Raffo

- Ex. 7.2: Matteo Picco

- Ex. 7.3: Haocong Li

Exercises available for a presentation: ex. 8.1 and 8.2 (by the same person); ex. 8.3; Alexandrov theorem. 

- ex. 8.1 and 8.2: Konstantin Myasnikov

Exercises available for a presentation: Ex. 9.1, 9.2 and 9.3.

- Ex 9.1: Yuekai Yan;

- Ex 9.2: Blas Ángel Loyens;

- Ex 9.3: Gaëlle Ambeau.

Exercises available for a presentation: Ex. 10.1 and 10.3.

- Ex 10.1: Léo Karlen;

- Ex 10.3: Alain Follonier.

Exercises available for a presentation: all of them.

-Marko Ivanovic: Ex 12.1

-Florian Manzini: Ex 12.2

-Jakob Josef Maisch: Ex. 12.3

-Ryosei Okamoto: Ex. 12.4

All exercises are available for a presentation.  If someone who wants to present remains without exercise, please contact roberto.colombo@epfl.ch, and we will propose some additional exercises.

- Ex 13.1: Jean-Sebastien Delineau;

-Ex 13.2: Mikhail Gorbunov;

-Ex 13.3: Vasco Reigadas;

- Ex 13.4: Vasco Frazão.