Introduction to Optimal Transport and Applications - ι
Schedule: 2h on Monday afternoon (4-6pm), from Feb 6th to Mar
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
firstname.lastname@example.org 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.
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
Lecture 8 (27/3) Porous media, chemotaxis and other
population dynamics PDEs. Estimates by using the JKO scheme.
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,
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 are available at the end of the book OTAM. A list of
exercises related to each class will be given later on.