Talks

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.

In this talk, we present new explicit bounds on the Gaussian approximation of Poisson functionals, based on novel estimates of moments of Skorohod integrals. Combining these with the Malliavin-Stein method, we derived bounds in the Wasserstein and Kolmogorov distances whose application requires minimal moment assumptions on add-one cost operators — thereby extending the results from (Last, Peccati and Schulte, 2016). Our main application is a CLT for the Online Nearest Neighbour graph, whose validity was conjectured in (Wade, 2009; Penrose and Wade, 2009). We also applied our techniques to derive quantitative CLTs for edge functionals of the Gilbert graph, of the k-Nearest Neighbour graph and of the Radial Spanning Tree, both in cases where qualitative CLTs are known and unknown.