Talks

In this talk I will present some recent results on polynomial decay rates for C0-semigroups, assuming that the resolvent grows polynomially at infinity in the complex right half-plane. Unlike many of the recent developments in the literature our results do not require the semigroup to be uniformly bounded. The talk is based on joint work with Chenxi Deng and Jan Rozendaal.

Entropy stability is the foundation of numerical methods for hyperbolic conservation laws, thereby ensuring the stability and reliability of the resulting numerical solutions. Summation-by-parts (SBP) operators provide a general framework to systematically develop entropy-stable schemes by mimicking continuous properties on a discrete level. They have proven to be a powerful tool to provide stable and high-order accurate numerical solutions. Classically, they are developed in order to differentiate polynomials up to a certain degree exactly. However, in many cases alternative function spaces are more appropriate to approximate the underlying solution space. Especially in multidimensional problems with potentially complex domains radial basis functions are known to possess very good approximation properties. The theory of radial basis function approximation provides us with stability and convergence results for scattered data approximation in a meshfree setting.
This talk discusses properties and efficient construction algorithms for multidimensional function space SBP (MFSBP) operators based on scattered data. I focus on radial basis function spaces and show some preliminary results for using MFSBP operators to solve conservation laws. I give an outlook on how convergence results of radial basis functions can be used to prove long-time error behavior of SBP discretizations for linear advection problems.

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

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.