Vorträge

Hierarchical Matrices are dense but data-sparse matrices that use low-rank factorisations of suitable submatrices to allow for storage with linear-polylogarithmic complexity. Furthermore, efficient approximations of matrix operations like matrix-vector and matrix-matrix multiplication, matrix inversion and LU decomposition are available. There are several approaches for the computation of QR factorisations in the hierarchical matrix format, however, they suffer from numerical drawbacks that limit their use in many applications. In this talk, I will present a new approach based on block Householder transformations that improves upon some of those problems. To prevent unnecessary high ranks in the resulting factors and increase speed as well as accuracy the algorithm meticulously tracks for which intermediate results low-rank factorisations are available.

I will try to keep things as simple as possible and give a short introduction to hierarchical matrices as well. Previous knowledge of them is not necessary to understand the basic ideas and main obstacles of the new algorithm. I will focus on aspects, that I haven't talked about yet in similar talks in the past, mainly on how a cost estimate is possible although the hierarchical structure of the resulting QR decomposition is step-wise created during the algorithm and not defined beforehand.

We present a novel local pressure correction method for incompressible fluid flows. Pressure correction methods
decouple the velocity and pressure components of the time-dependent Navier-Stokes equations and lead to a sequence of elliptic partial differential equations for both components instead of a saddle point problem. In some situations, the equations
for the velocity components are solved explicitly (with time step restrictions) and thus the elliptic pressure problem remains to be the most expensive step. Here, we employ a multiscale procedure for the solution of the Poisson problem related to pressure. The procedure replaces the global Poisson problem by local Poisson problems on subregions.We propose a new Robin-type boundary condition design for the
local Poisson problems, which contains a coarse approximation of the global Poisson problem. Accordingly, no further communication between subregions is necessary and the method is perfectly adapted for parallel computations. Numerical experiments regarding a known analytical solution and flow around cylinder benchmarks show the effectivity of this new local pressure correction method.

An increased use of renewable energy could significantly contribute to decelerate climate change but cannot be realized easily since most renewable energy sources underlie volatile availability. Using of storage devices and scheduling consumers to times when energy is available can increase the amount of renewable energy that is used. For this purpose, adequate models that forecast the energy generation and consumption as well as the behavior of storage devices are essential. We present a computationally efficient modeling approach based on a least-squares problem that is extended by a hybrid model approach based on kmeans clustering and evaluate it on real-world data at the examples of modeling the state of charge of a battery storage and the temperature inside a milk cooling tank. The experiments indicate that the hybrid approach leads to better forecasting results, especially if the devices show a more complicated behavior. Furthermore, we investigate whether the behavior of the models is qualitatively realistic and find that the battery model fulfills this requirement and is thus suitable for the application in a smart energy management system. Even though forecasts for the hybrid milk cooling model have low error values, further steps need to be taken to avoid undesired effects when using this model in such a sophisticated system.

I will introduce myself and present the topic of my master thesis.

Computed Tomography (CT) is a standard procedure in clinical imaging. In dynamic CT, several CT scans are made to make a process inside the patient visible. Therefore, the X-ray exposure to the patient is relatively high during such a survey. Thus, it is desirable to lower the X-ray exposure to the patient.

In my thesis I investigated an approach which requires only sparse angular sampling for every scan. In order to be able to reconstruct the image anyway, I used a shearlet system combined with an $\ell^1$-regularization. I compared different shearlet systems and checked for different parameters the impact on the results. I used both simulated data as well as real CT data for the tests.

Back in 2009, both Stein's method - a probabilistic technique to derive quantitative limit theorems - and Malliavin calculus - a stochastic version of the calculus of variations - had already established themselves as standard tools in their respective domain, even though both were discovered quite recently in 1972 and 1978, respectively. Then they started an innocent liaison in Paris which quickly developed into a very strong bond (despite numerous affairs), leading to fame and success both in- and outside the probabilistic community. This bond is today known as the Malliavin-Stein approach.

I will highlight some exciting parts of this story, also attributing a fair share of time to yet unwritten chapters (i.e. open problems). Mathematically, this will feature non-linear approximation, limit theorems (central and non-central), stochastic processes, chaos, Markov generators, non-commutative probability theory and the first four moments. Catering to the fact that probabilists are in the minority in our department, things will also be presented from a functional analytic point of view.

The talk will mostly be informal and understandable by non-specialists.

Coupling means the joint construction of two or more random variables, processes, or any random objects. The aim of the construction could be to deduce properties of the individual objects, or to gain insight into distributional relations between them, or to simulate a particular object. It has been called The Probabilistic Method since it is not based on methods from other fields of mathematics.

In this talk we shall consider some basic examples such as the Poisson approximation, stochastic domination, Markov chains and Brownian motion, and perfect simulation

Solving evolution equations with random coefficients numerically requires discretizing in space, time and of random parameters. As numerical methods for all three discretisations are well-known, it is natural to ask under which conditions they can be combined. In this talk, we discuss this question with a special emphasis on preservation of strong convergence rates.

A common approach to spatial discretization consists of solving the weak formulation on finite-dimensional approximating spaces. We present a novel quantified version of the Trotter-Kato theorem in this setting, yielding rates of strong convergence under a joint condition on properties of the corresponding form and the approximating spaces.

This is joint work with Christian Seifert.