Introduction to Optimal Transport and Applications - ι

Graduate course at Taught Course Centre

Practical Information

Duration: 16h
Schedule: 2h on Monday afternoon (4-6pm), from Feb 6th to Mar 20th, 2016.
Where: Imperial College, room 6m42. Classes will be live broadcasted and accessible in Bath, Bristol, Oxford and Warwick.
Examination: according to the number of students who want to take it, schedule and form to be fixed.
Enrollment: students should register by emailing graduate.studies@maths.ox.ac.uk regardless of whether they intend on taking the course for assessment. Also, writing me an email is appreciated, in order to have a list of attending student.
Prerequisites: some functional analysis and some knowledge of basic PDEs (the main points will be recalled).
News: The program has been rescheduled on a total of 8 lessons.

Program

There will be 8 classes of 2h each, tentatively organized as follows.

  • Lecture 1 (6/2) Monge and Kantorovich problems, existence of optimal plans, dual problem, and existence of Kantorovich potentials.
  • Lecture 2 (13/2) The example of the marriage market. Strong duality (inf sup = sup inf). Existence of optimal maps and the Monge-Ampère equation. The quadratic and the distance cases.
  • Lecture 3 (20/2) Optimal flow formulation of the optimal transport problem. Traffic congestion variants.
  • Lecture 4 (27/2) Wasserstein distances, curves in the Wasserstein spaces and connections with the continuity equation.
  • Lecture 5 (6/3) Geodesics in the Wasserstein spaces and the Benamou-Brenier method for numerical computations.
  • Lecture 6 (13/3) Introduction to gradient flows in metric spaces; the JKO minimization scheme for some parabolic equation. Geodesically convex functionals.
  • Lecture 7 (20/3) Heat and Fokker-Planck equations as gradient flows in the Wasserstein space (convergence of the scheme).
  • Lecture 8 (27/3) Porous media, chemotaxis and other population dynamics PDEs. Estimates by using the JKO scheme.

    • References:

    All the material covered in the course is contained in the recent book Optimal Transport for Applied Mathematicians (see here or here for a non-official version), and in particular Chapter 1, Sections 4.2 and 4.4.1, Chapter 5, Section 6.1, Chapter 8.
    Of course, there are more classical references: the first book by Cédric Villani Topics in Optimal Transportation (Am. Math. Soc., GSM, 2003) on the general theory, and Gradient Flows in Metric Spaces and in the Space of Probabiliy Measures, by Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré (Birkhäuser, 2005).
    For the last lectures, on gradient flows, useful material is also contained in a survey I recently wrote, accessible here.
    Precise references class by class:
    6/2: OTAM, sections 1.1 +1.2
    13/2: OTAM, sections 1.3, 1.6.1, 1.6.3, 1.7.3
    20/2: OTAM, sections 4.1, 4.2, 4.3, 4.4.1
    27/2: OTAM, sections 5.1, 5.2, 5.3.1, 5.3.2, 5.3.3
    6/3: OTAM, sections 5.3.4, 5.4, 6.1
    13/3: Survey, sections 2.1, 4.3, 4.5

    Exercises

    Exercises are available at the end of the book OTAM. A list of exercises related to each class will be given later on.