Talks

Inhomogeneous random graphs are a prominent tool for modeling real-world complex networks as they manage to capture key concepts such as the scale-free property. In this talk we will focus on two particular inhomogeneous random graph models, the Norros-Reittu model and the random connection model. The Norros-Reittu model uses a deterministic vertex set and can be seen as a generalisation of the famous Erdős–Rényi graph. The random connection model on the other hand yields a spatial random graph, which leads to natural clustering effects. Our main goal is to determine the asymptotic behaviour of the size of the largest component as the number of vertices or the size of the observation window, respectively, goes to infinity. For the Norros-Reittu model we also study asymptotics of other counting statistics.

This talk gives an overview of the results obtained in my PhD under the supervision of Prof. Dr. Matthias Schulte.

Lower bounds for variances are often needed to derive central limit theorems. In this talk, a generalised reverse Poincaré inequality is established, which provides a lower variance bound for Poisson functionals that depends on the difference operator of some fixed order.
Poisson functionals, i.e. random variables that depend on a Poisson process, have many applications in stochastic geometry. In this talk we apply the introduced lower variance bound to statistics of spatial random graphs, $L^p$ surface areas of random polytopes and geometric functionals of excursion sets of Poisson shot noise processes.

This talk gives an overview of the results obtained in my PhD under the supervision of Prof. Dr. Matthias Schulte.

Accurate flow simulations remain a challenging task. In this talk we discuss the use of deep neural networks for augmenting classical finite element simulations in fluid-dynamics.
We first investigate the classical DFG-benchmark in 3D. We extend these settings to higher Reynolds numbers. At high Reynolds numbers, accurate simulations in 3D settings become increasingly difficult, and the classical methods reach their limits. To address this issue, we discuss approaches to connect the finite element method with neural networks. We propose the Deep Neural Network Multigrid Solver, which combines a geometric multigrid solver with a deep neural network to overcome limitations of classical methods. This approach uses classical simulation techniques where their strengths are eminent, such as the efficient representation of a coarse, large-scale flow field. Neural networks are used when a full resolution of the effects does not seem possible or efficient. While our method is tightly embedded in a geometric multigrid framework, it remains flexible, allowing for the coarse grid problem to be addressed with various solvers and enabling local enrichment without a global fine grid.
We demonstrate the efficiency, generalizability, and scalability of our proposed approach using 3D simulations. Our focus is on issues of stability, generalizability.
In the second part of the talk we give an outlook on future developments towards an extension to space-time multigrid methods within the framework of space-time finite element methods. We discuss the efficient implementation of space-time multigrid methods using the matrix-free framework provided by the deal.ii finite element library and demonstrate its effectiveness for the heat and acoustic wave equation.

Zoomlink:
https://tuhh.zoom.us/j/81920578609?pwd=TjBmYldRdXVDT1VkamZmc1BOajREZz09

Radial basis function finite difference (RBF-FD) discretization has recently emerged as an al-
ternative to classical finite difference or finite element discretization of (systems) of partial
differential equations. We focus on the construction of preconditioners for the iterative solution
of the resulting linear systems of equa- tions. In RBF-FD, a higher discretization accuracy may
be obtained by increasing the stencil size. This, however, leads to a less sparse and often also
worse conditioned stiffness matrix which are both challenges for subsequent iterative solvers. We
propose to construct preconditioners based on stiffness matrices resulting from RBF-FD dis-
cretization with smaller stencil sizes compared to the one for the actual system to be solved. In
our numerical results, we focus on RBF-FD discretizations based on polyharmonic splines (PHS)
with polynomial augmentation. We illustrate the performance of smaller stencil preconditioners
in the solution of the three-dimensional convection-diffusion equation.

Zoomlink:
https://tuhh.zoom.us/j/81920578609?pwd=TjBmYldRdXVDT1VkamZmc1BOajREZz09

I will be introducing myself formally and presenting my master's thesis, which concerns the study of numerical schemes for solving singularly perturbed differential equations, focusing on the Haar wavelet method.

Zoomlink:
https://tuhh.zoom.us/j/81920578609?pwd=TjBmYldRdXVDT1VkamZmc1BOajREZz09

This talk focuses on the mathematical models underlying reinforcement learning in artificial intelligence, particularly the reward functions in Markov Decision Processes. I argue that ethical principles related to well-being, safety, and equality are inherently reflected in these mathematical models. Building on this foundation, I then demonstrate how ethics can inform computational mathematics, while also addressing the challenges one encounters in this domain. Specifically, I discuss how the mathematical models behind reinforcement learning may rely on a distorted representation of ethics with respect to the determinacy and commensurability of ethical values.

Zoomlink:
https://tuhh.zoom.us/j/81920578609?pwd=TjBmYldRdXVDT1VkamZmc1BOajREZz09