Optimal Transport and Applications
Doctoral course in Trento and Verona
Schedule: 10h in Trento, March 8-11 + 10h in Verona,
For the week in Trento : Tue and Wed 2.30pm-5pm, Thur and Fri
9am-11.30am (aula A102, polo Ferrari).
For the week in Verona : Mon and Wed 4.30pm-6.30pm (aula G), Tue and
Thur 1.30pm-3.30pm and Fri 3.30pm-5.30pm (aula M).
Examination: different forms of examinations (and cominations
of them) are possible:
written exam with exercises; written dissertation or oral presentation
on a related topic; numerical implementation of an alorithm oresented
in class (in groups).
Language: the classes have finally been given in Italian in
Trento and in english in Verona.
Prerequisites: some functional analysis, in particular
existence criteria for minimization problems (semicontinuity,
compactness), measure theory and weak compactness for measures, a
little bit of convex analysis (Legendre transform) and some basic
PDEs. F. Serra Cassano and G. Orlandi gave
some preparatory classes on these topics.
A tentative program for the course is the following.
Lecture TN-1 Monge and Kantorovich problems, existence of optimal plans,
duality, Kantorovich potentials.
Lecture TN-2 Existence of optimal maps and the Monge-Ampère
equation. Cyclical monotonicity, analysis of the 1D case.
Discrete and semi-discrete numerical methods.
Optimal flow formulation of the optimal transport problem. Traffic congestion variants.
Wasserstein distances and Wasserstein spaces.
Curves in the Wasserstein spaces and connections with the continuity equation.
Geodesics in the Wasserstein spaces and the Benamou-Brenier method for numerical computations.
Introduction to gradient flows in metric spaces; the minimization
scheme for the Heat and Fokker-Planck equation.
Heat and Fokker-Planck equations as gradient flows in the Wasserstein
space (convergence of the scheme), porous media, chemotaxis and other population dynamics PDEs.
Due to the bi-location of this course, streaming videos are available
for people interested.
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).
TN-1 : Sections 1.1, 1.2 and 1.6
TN-1 : Sections 1.3, 2.1 and 2.2
TN-3 : Sections 6.4.1 and 6.4.2
TN-4 : Sections 4.2 and 4.4.1
VR-1 : Sections 5.1 and 5.2
VR-2 : Section 5.3
VR-3 : Sections 5.4 and 6.1
VR-4 : Sections 8.1, 8.2 and 8.3
VR-5 : Sections 8.3 and 8.4.2
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