1. Vacatures
2. Technische Universiteit Delft (TUD)
3. PHD position differential equations on graphs and discrete-to-continuum limits

# PHD position differential equations on graphs and discrete-to-continuum limits

Graph gradient flows: from discrete to continuous
In this project we study continuum limits of gradient flows that are defined on graphs (networks)

ongeveer 2 maanden geleden

## Arbeidsvoorwaarden

Standplaats:
Mekelweg, Delft, Zuid-Holland
Dienstverband:
Tijdelijk contract / Tijdelijke opdracht
Uren per week:
38 - 40 uur
Salarisindicatie:
€ 2434 - € 3111 per maand
Opleidingsniveau:
WO

## Functieomschrijving

Graph gradient flows are commonly used in applications such as image segmentation and data clustering. In such applications the vertices of the graph represent the pixels from the image or the datapoints. It is important to know what happens when more pixels or data are available, i.e. when the number of vertices increases. If the graph gradient flow converges to a continuum gradient flow in this limit, the corresponding segmentation or clustering method is called "consistent". It shows that in the presence of more data, the method does not give very different outcomes, but rather better approximates the solution to a well-defined problem. This is an important requirement to interpret and trust the outcomes of the method.

The Ph.D. student will investigate the discrete-to-continuum limits of graph gradient flows, with the goal of rigorously proving convergence for various different gradient flow models on various different classes on graphs. This will require a strong background in rigorous mathematics, in particular in the analysis of (ordinary and partial) differential equations and related fields such as variational methods, functional analysis and measure theory. Some familiarity with graph theory is also welcome, but the main techniques and models will come from the area of differential equations. In particular, this will not be a graph theory project.

Even more so than specific mathematical background knowledge, the project requires the skills, abilities, and motivation to work on detailed and technical mathematical problems that require rigorous proofs.

The main focus of this project lies on the theoretical aspects of graph gradient flows and their convergence properties. Applications of such flows, such as image segmentation and data clustering, will be important motivators, but will not be central to the project.

The student will be supervised on a daily basis by dr. Yves van Gennip and have regular meetings with dr. Johan Dubbeldam, both at the Delft Institute of Applied Mathematics (DIAM) at Delft University of Technology. There will be opportunities to collaborate with other researchers, both nationally and internationally.

Early applications are welcome. If a suitable candidate is found before the end date of the vacancy, the position may close earlier than listed.

## Functie-eisen

We are looking for excellent candidates that satisfy the following requirements:

• B.Sc. and M.Sc. degrees in Mathematics or a related field with a strong focus on rigorous mathematics;
• a strong background (at MSc level) in the analysis of ordinary and partial differential equations and in one or more related fields such as variational methods, functional analysis, or measure theory;
• a willingness to learn more about the fields mentioned above and any other topics that may prove relevant to the project;
• proficiency in written and oral English, coupled with clear communication and presentation skills;
• good technical writing skills (as can be evidenced by, for example, B.Sc. and M.Sc. theses) and a willingness to continually and critically assess one's own writing and improve it;
• intrinsic motivation to work on the project and the abilities both to work independently and to collaborate with others;
• a willingness to travel for conferences and research visits (especially in later years of the project).

Nice-to-haves:

• knowledge of graphs and graph theory;
• programming skills, for example in Python or MATLAB, to be able to quickly implement models and test hypotheses.

## Conditions

TU Delft offers PhD-candidates a 4-year contract, with an official go/no go progress assessment after one year. Salary and benefits are in accordance with the Collective Labour Agreement for Dutch Universities, increasing from € 2434 per month in the first year to € 3111 in the fourth year. As a PhD candidate you will be enrolled in the TU Delft Graduate School. The TU Delft Graduate School provides an inspiring research environment with an excellent team of supervisors, academic staff and a mentor. The Doctoral Education Programme is aimed at developing your transferable, discipline-related and research skills.

The TU Delft offers a customisable compensation package, discounts on health insurance and sport memberships, and a monthly work costs contribution. Flexible work schedules can be arranged. For international applicants we offer the Coming to Delft Service and Partner Career Advice to assist you with your relocation.