Vorträge

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.

The trend towards massively parallel high-performance computers requires the development of parallel algorithms to employ their computational power.
The Parareal algorithm computes the solution of time-dependent problems parallel in time, meaning that approximations to the solution at different times are computed simultaneously. In this talk, we will focus on hyperbolic one-dimensional problems, where a combination of Parareal and a discontinuous Galerkin method will be used. The practical use and challenges of this method will be illustrated by means of a Python implementation for shallow water equations and corresponding numerical results.

How will the next generation of Artificial Intelligence (AI) look like? Comparing today's AI algorithms with biological intelligence, one of the most remarkable differences is the ability of the human brain to somehow understand the 'essence' of things: A small child can easily identify any type of object after having seen only a few examples or recognize a song even when played on different instruments or in a different key. In other words: Brains are able to create abstract concepts of real-world entities - and today's algorithms are not.

With today's AI largely being based on neuron models already invented by the mid of last century, I will argue that we should take a new look at the brain to find inspiration for the next generation of machine learning algorithms. Even though there is still only a very limited understanding of how the brain works computationally, I'll explain why there is hope that we can reverse-engineer some of its algorithmic principles and implement them in a computer. I'll explain why a highly interdisciplinary approach is required from neuroscience, computer science, mathematics and physics to make progress in this question.

The talk will be held on Zoom:
https://tuhh.zoom.us/j/86836210324?pwd=ajJURGY2T3pFNWMvUzVQTkduSTNCQT09
Meeting-ID: 868 3621 0324
Kenncode: 521014

For many processes in sciences, the coefficients of the partial differential equation describing a dynamical system as well as the boundary conditions of it may vary with time. In such cases one speaks of non-autonomous (or time-varying) evolution equations. From an operator theoretical point of view one considers families of Banach space operators which depend on the time parameter and studies the associated non-autonomous abstract Cauchy problem. We consider time-dependent Desch-Schappacher perturbations of non-autonomous abstract Cauchy problems and apply our result to non-autonomous uniformly strongly elliptic differential operators on Lp -spaces. This is joint work with Christian Seifert (TUHH).

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

A fundamental principle in developing the Theory of Quantum Mechanics is to take
well-studied concepts from the Theory of Classical Mechanics and to define
analogues in the quantum mechanical setting.
One such important tool in Classical Mechanics is the theory of optimal
transport and in particular the Wasserstein distance.

In my thesis I studied the mathematical objects needed to translate
the concepts of the optimal transport problem to the realm of Quantum
Mechanics. In particular,
one wants to establish a relation between density matrices (trace-class operators
of trace one) and
probability measures. This can be done by the so-called
(generalized) Toeplitz operators and the (generalized) Husimi
transform.

After I give a brief introduction into both the Optimal Transport and Quantum
Mechanics I will introduce both
the Toeplitz operators and the Husimi transform and discuss some of their
properties.