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Introduction to Otpimal Transport (WiSe 25)
Inscription via LSF. (if you cannot register on LSF, you can do it on Moodle, key: "wasserstein")The course consists in 2 lectures per week and no exercises (WP 37).
Description:
The theory of optimal transport is the study of the best way
to transport a measure onto another one with respect to some cost function.
The initial problem formalized by Monge in 1781 is the following:
given two probability measures \(\mu\) and \(\nu\) on \( \mathbb{R}^d \),
find a transport map \(T: \mathbb{R}^d \to \mathbb{R}^d\) that pushes \(\mu\) forward onto \(\nu\), that is \( \nu(A) = \mu(T^{-1}(A)) \)
for all \(A \subset \mathbb{R}^d\) measurable, and
that minimizes the total cost
\[
\int_{\mathbb{R}^d} |x-T(x)| d \mu (x).
\]
The problem was (much) later generalized by Kantorovitch in 1942 in a form that ensures the existence of solutions.
It is now an ongoing active research area with many applications in analysis, PDEs, geometry,
image processing, quantum mechanics, etc.
The cost functional above defines a natural distance
on the space of probability measures: the Wasserstein distance. This metric not only serves to compare measures,
it is also allows to interpolate between them through
geodesics in the associated metric space.
In this course, we will define the fundamental concepts of optimal transport and explore several applications. The main References
will be the book "Optimal Transport for Applied Mathematicians" of Filippo Santambrogio.
Gif created with POT.
Audience:
The course will be of interest for Master students of mathematics (WP 37), or motivated Bachelor students. A good background in Functional Analysis is required (FA1 is enough). We will recall quickly all the tools we need, but students are expected to already be familiar with them.Credits:
6 ECTS.Schedule:
- Lectures: Tuesday 12:15-13:45 in Room B 251 and Thursday 14:15-15:45 in Room B 045
References:
- F. Santambrogio, Optimal Transport for Applied Mathematicians, Springer, 2015 (Available for download on the website of the LMU's library and also direclty on Santambrogio's website )
- C. Villani Optimal transport: old and new. Springer, 2008.
- J. Maas, S. Rademacher, T. Titkos & D. Virosztek (Eds.). (2024). Optimal transport on quantum structures. Springer.