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

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.