Module 24-M-AN-OTAM Optimal Transport and Applications in PDEs and Modelling

Faculty

Person responsible for module

Regular cycle (beginning)

Dieses Modul ist Teil einer langfristigen Gesamtlehrplanung für das Masterprogramm, die sicherstellt, dass in allen fünf Gebieten jedes Jahr jeweils mindestens Module im Umfang von 20 LP angeboten werden. Im Rahmen dieser Gesamtlehrplanung wird das Modul in unregelmäßigen Abständen angeboten.

Credit points and duration

10 Credit points

For information on the duration of the modul, refer to the courses of study in which the module is used.

Competencies

Non-official translation of the module descriptions. Only the German version is legally binding.

Students master advanced content and methods in the field of optimal transport and application in partial differential equations and modelling, i.e. they are able to carry out independently very complex proofs in this field requiring a high level of mathematical expertise. Students are able to present and apply the theory of optimal transport. They are able to present and solve primary, dual and dynamic optimisation problems in a technically appropriate manner. They are able to apply properties of different metrics to probability measures in different contexts: They can model large classes of evolution equations as gradient flows by appropriate choice of metrics and functionals on measures and use this structure to prove statements about existence and long-term behaviour of solutions.

Students are introduced to current research questions in the area of optimal transport and applications in partial differential equations and modelling. They are able to recognise and assess further development opportunities and research goals.
Furthermore, students are able to recognise further-reaching connections to mathematical issues that have already been worked out. They can transfer and apply the knowledge and methods they have learnt so far to deeper mathematical problem areas. Students also expand their mathematical intuition as a result of more intensive study.
In combination with other in-depth modules, they will be able to write their own research papers, e.g. a master's thesis in the field ofOptimal Transport and Gradient Flows.
In the tutorials, students develop their ability to discuss mathematical topics and thus further prepare themselves for the requirements of the Master's module, in particular for the scientific discussion within the Master's seminar presentation and the defence of their Master's thesis.

Content of teaching

The following advanced content of teaching from the field of Optimal Transport and Gradient Flows is compulsory:

  • Primal and dual formulation of optimal transport problems
  • continuity equations and dynamical transport problems
  • distances on the space of probability measures
  • functionals over probabilities
  • gradient flows in metric spaces
  • variational modelling of evolutionary PDEs
  • functional inequalities and longterm behaviour of solutions

In addition, the following content of teaching can be covered, for example:

  • Numerical methods for transport problems
  • further applications of optimal transport

This module prepares the content of a master's thesis.

Recommended previous knowledge

Real analysis, measure and integration, functional analysis, basic PDE theory

Necessary requirements

Explanation regarding the elements of the module

Module structure: 1 SL, 1 bPr 1

Courses

Lecture Optimal Transport and Gradient Flows
Type lecture
Regular cycle Dieses Modul ist Teil einer langfristigen Gesamtlehrplanung für das Masterprogramm, die sicherstellt, dass in allen fünf Gebieten jedes Jahr jeweils mindestens Module im Umfang von 20 LP angeboten werden. Im Rahmen dieser Gesamtlehrplanung wird das Modul in unregelmäßigen Abständen angeboten.
Workload5 60 h (60 + 0)
LP 2 [Pr]
Tutorials Optimal Transport and Gradient Flows
Type exercise
Regular cycle Dieses Modul ist Teil einer langfristigen Gesamtlehrplanung für das Masterprogramm, die sicherstellt, dass in allen fünf Gebieten jedes Jahr jeweils mindestens Module im Umfang von 20 LP angeboten werden. Im Rahmen dieser Gesamtlehrplanung wird das Modul in unregelmäßigen Abständen angeboten.
Workload5 90 h (30 + 60)
LP 3 [SL]

Study requirements

Allocated examiner Workload LP2
Teaching staff of the course Tutorials Optimal Transport and Gradient Flows (exercise)

Regular completion of the exercises, each with a recognisable solution approach, as well as participation in the exercise groups for the module's lecture. As a rule, participation in the exercise group includes presenting solutions to exercises twice after being asked to do so as well as regular contributions to the scientific discussion in the exercise group, for example in the form of comments and questions on the proposed solutions presented. The organiser may replace some of the exercises with face-to-face exercises.

see above see above

Examinations

e-written examination o. written examination o. e-oral examination o. oral examination
Allocated examiner Teaching staff of the course Lecture Optimal Transport and Gradient Flows (lecture)
Weighting 1
Workload 150h
LP2 5

(electronic) written examination in presence of usually 120 minutes, oral examination in presence or remote of usually 40 minutes, A remote electronic written examination is not permitted.

The module is used in these degree programmes:

Degree programme Profile Recom­mended start 3 Duration Manda­tory option 4
Mathematical Economics / Master of Science [FsB vom 28.02.2025] Mathematics 2. o. 3. one semester Compul­sory optional subject
Mathematical Economics / Master of Science [FsB vom 28.02.2025] Economics 2. o. 3. one semester Compul­sory optional subject
Mathematics / Master of Science [FsB vom 28.02.2025] 2. o. 3. one semester Compul­sory optional subject

Automatic check for completeness

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Legend

1
The module structure displays the required number of study requirements and examinations.
2
LP is the short form for credit points.
3
The figures in this column are the specialist semesters in which it is recommended to start the module. Depending on the individual study schedule, entirely different courses of study are possible and advisable.
4
Explanations on mandatory option: "Obligation" means: This module is mandatory for the course of the studies; "Optional obligation" means: This module belongs to a number of modules available for selection under certain circumstances. This is more precisely regulated by the "Subject-related regulations" (see navigation).
5
Workload (contact time + self-study)
SoSe
Summer semester
WiSe
Winter semester
SL
Study requirement
Pr
Examination
bPr
Number of examinations with grades
uPr
Number of examinations without grades
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