Print syllabusOptimal transport in evolution equations
General data
| Course ID: | 1000-1M23OTE |
| Erasmus code / ISCED: |
11.1
|
| Course title: | Optimal transport in evolution equations |
| Name in Polish: | Optymalny transport w równaniach ewolucyjnych |
| Organizational unit: | Faculty of Mathematics, Informatics, and Mechanics |
| Course groups: |
Elective courses for 2nd stage studies in Mathematics |
| ECTS credit allocation (and other scores): |
6.00
|
| Language: | English |
| Type of course: | elective monographs |
| Requirements: | Functional Analysis 1000-135AF |
| Prerequisites (description): | The participant should know the basic theorems of functional analysis. Advanced knowledge of measure theory is not required, but not advisable. The basics of measure theory related to such lectures as Mathematical Analysis and Probability should be sufficient. The participant can expect short excursions into differential equations. |
| Short description: |
The lecture aims to familiarize participants with the theory of optimal transport, in particular by deriving Wasserstein metrics. Finally, we will introduce the notion of a gradient flow in with respect to the Wasserstein metric and notice how many physical phenomena can be treated as gradient flows. If time permits, we'll finally talk about the mean-field limit for the Vlasov equation. |
| Full description: |
Most of the lecture follows the script of L. Ambrosio, N. Gigli ''A user's guide to optimal transport'' limiting itself to the case of Euclidean space. Part I: Optimal transportation problem. 1. Formulating the problem of optimal transport according to Monge and Kantorowicz. 2. Conditions equivalent to optimality of the transport plan. 3. The existence of optimal mappings. Part II: Wassestein Metrics. 1. Introduction of Wasserstein metrics. W2 metric. 2. Basic properties of the space of measures with the W2 metric. 3. Measures with the W2 metric as geodetic space. 4. Absolutely continuous curves and the continuity equation. 5. Weakly-Riemannian structure of the measure space with the W2 metric. Part III: Gradient flows on metric spaces 1. The notion of a gradient flow on Hilbert and metric spaces. 2. Three definitions of a gradient flow and the relationships between them. 3. Gradient flows of geodesically convex functionals. 4. Three classic examples of gradient flow. Supplementary material: 1. The boundary of the mean field for the Vlasov equation. The limit in the deterministic variant. |
| Bibliography: |
Ambrosio, Gigli ''A user's guide to optimal transport'', Ambrosio, Gigli, Savare ''Gradient flows: In metric spaces and in the space of probability measures'', François Golse, ''Mean-Field Limits in Statistical Dynamics''. |
| Learning outcomes: |
The students know and understand the concept of gradient flow and its relationship with transport of mass and the continuity equation. The students know where to look for extensions of the material from the lecture and have a sufficient understanding of the subject to continue studying on their own. |
| Assessment methods and assessment criteria: |
Two to choose from: activity in exercises, one fairly extensive homework, oral exam. |
Classes in period "Winter semester 2023/24" (past)
| Time span: | 2023-10-01 - 2024-01-28 |
 
MO TU W TH FR WYK
CW
|
| Type of class: |
Classes, 30 hours
Lecture, 30 hours
|
|
| Coordinators: | Jan Peszek | |
| Group instructors: | Jan Peszek | |
| Students list: | (inaccessible to you) | |
| Examination: | Examination |
Copyright by University of Warsaw.
