Vorträge

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.

The numerical solution of saddle-point systems arising in computational fluid dynamics with iterative solvers
requires efficient preconditioners. We focus on preconditioners for the pressure Schur complement. Low-rank
corrections aim to enhance spectral properties of standard preconditioners to accelerate convergence. We discuss
a multiplicative low-rank update that exploits a (randomized) low-rank approximation to the error between the
identity and the preconditioned Schur complement. Relaxing the initial Schur complement preconditioner can
have a significant impact on the convergence behaviour of the iterative solver. We test the presented method
for the linearized incompressible Navier-Stokes equations. Numerical results illustrate the performance of the
relaxed update scheme.