Analysis, simulation, and applications of optimal transport problems

online via zoom

Prof. Dr. Gero Friesecke, TUM

Course content

In its historical forms, optimal transport looked like a very specialized topic, concerned with moving a pile of mortar efficiently to a range of target location on a construction site (Monge, 1781), or transferring the output of an array of steel mines optimally to a network of factories (Kantorovich, 1942). In the past three decades there has been an explosion of interest in the subject as this type of problem was found to arise in amazingly many different fields of mathematics, science, and engineering: fluid dynamics, probability theory, statistics, dissipative PDEs (in the 1980s and 1990s); functional inequalities, curved geometry, traffic flow, urban planning (in the 2000s); many-electron physics, image processing, crowd motion, data analysis, machine learning (in the 2010s).

In this mini-course of six 90-minute lectures, I will explain cornerstone concepts, results and methods of optimal transport such as Wasserstein distances, Kantorovich duality, Brenier's theorem, or the Sinkhorn algorithm, at other than a superficial level. I will build up the mathematical theory rigorously and from scratch, aided by intutitive arguments, informal discussion, and carefully selected applications.

Program:

All sessions take place online via zoom:
Link: https://tum-conf.zoom.us/j/98538258098, Passcode: 749626

Lecture 1: Mon 23 November 15:15-17:00
- Introduction
- Functional analysis and convergence notions for measures on R^d

Exercise class 1: Tue 24 November 12:00-12:45

Lecture 2: Tue 24 November 13:15-15:00
- Existence of optimal plans
- Wasserstein distances

Lecture 3: Thu 26 November, 15:15-17:00
- Fenchel-Rockafellar duality in Banach spaces
- Kantorovich duality

Lecture 4: Mon 30 November 15:15-17:00
- Brenier's theorem
- Generalization to costs satisfying the twist condition

Exercise class 2: Tue 1 December 12:00-12:45

Lecture 5: Tue 1 December 13:15-15:00
- Entropic regularization and the Sinkhorn algorithm
- Matlab implementation and numerical experiments

Lecture 6: Thu 3 December 15:15-17:00
- Selected applications of optimal transport
(students can select from a choice of topics)

Participants can receive a confirmation certificate for 12 credit hours. In order to register for the confirmation certificate, please send a short email of registration to Diane Clayton-Winter (clayton@ma.tum.de) and use your full name when logging in to the zoom conference.



-- DianeClaytonWinter - 17 Nov 2020
 

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