Vorträge

We study the interplay between moment and tail inequalities in the concentration of measure phenomenon. A motivating example are so-called higher order concentration bounds, where functions are addressed which have unbounded first order derivatives (or differences) but whose derivatives of some higher order are bounded. A variety of different situations is considered like (classical) Euclidean spaces, discrete situations, functions of independent random variables and the Poisson space. A special emphasis is put on pointing out the parallels and common ground throughout all these cases.

Since we now have implemented an efficient solver for the Maxey-Riley equation, we can generate a lot of trajectory data. This data could be used to train a neural network, which can predict the trajectories given a starting postion. Because the dynamics are governed by an integro-differential equation, the future path of a trajectory depends on the whole past. This characteristic could be handled using recurrent neural networks.
I will show how the network performs on different velocity fields. There are cases where the model does a great job in the prediction, but there are still many problems to discuss and it will be interesting to hear some thoughts.
Finally I will show a network architecture that combines a Verlet integrator with a recurrent neural network.

Tomographic imaging is an essential tool in medical diagnostics, allowing diseases to be detected much earlier
than would be possible from external observations alone. The aim is to determine a function representing the inner
of the human body from external measurements only so that the procedure is non-invasive and not harmful.
Determining this function, or in practice an appropriately discretized form, involves solving an
inverse problem, which is often ill-posed and must be solved for noisy measurements. In this talk, an
overview of different image reconstruction challenges and ways to address them algorithmically is given.
We also sketch possibilities that arise and allow for multi-contrast image reconstruction from only single
measurements.

Fluid flow problems can be modelled by the Navier-Stokes, or Oseen equations. Their discretization results in saddle point problems. These systems of equations are typically very large and need to be solved iteratively. Standard (block-) preconditioning techniques for saddle point problems rely on an approximation of the Schur complement. Such an approximation can be obtained by a hierarchical matrix (H-Matrix) LU-decomposition for which the Schur complement is computed explicitly. The computational complexity of this computation depends, among other things, on the hierarchical block structure of the involved matrices. However, widely used techniques do not consider the connection between the discretization grids for the velocity field and the pressure, respectively. Thus, a problem dependent hierarchical block structure for the FEM discretization of the gradient operator is presented. The block structure of the corresponding saddle point matrix block is improved by considering the connection between the two involved grids.Numerical results will show that the improved block structure allows for a faster computation of the Schur complement, the bottleneck for the set-up of the H-Matrix LU-decomposition.

The first part of the presentation provides an introduction to quantum mechanics and quantum algorithms. In the second part, I will present an overview of the research at the new TUHH institute on the topic (see www.tuhh.de/quantum) and explain the mathematical aspects of it.