Vorträge

It is a conjecture going back to J. Rauch (1974) that the hottest and coldest spots in an insulated homogeneous medium such as an insulated plate of metal should converge to the boundary, for "most" initial heat distributions, as time tends to infinity. This so-called hot spots conjecture can be phrased alternatively as follows: the eigenfunction(s) corresponding to the first non-zero eigenvalue of the Neumann Laplacian on a Euclidean domain should take its maximum and minimum on the boundary only. This has been proven to be false for certain domains with holes, but it was shown to hold for several classes of simply connected or convex planar domains. One of the most recent advances is the proof for all triangles given by Judge and Mondal (Annals of Math. 2020). The conjecture remains open in general for simply connected or at least convex domains. In this talk we provide a new approach to the conjecture. It is based on a non-standard variational principle for the eigenvalues of the Neumann and Dirichlet Laplacians.

Jonathan Rohleder is an associate professor at Stockholm University, Sweden. His work focusses on spectral theory.

Specific solutions of the nonlinear Schrödinger equation, such as the Peregrine breather, are considered to be prototypes of extreme or freak waves in the oceans. An important question is whether these solutions also exist in the presence of gusty wind. Using the method of multiple scales, a nonlinear Schrödinger equation is obtained for the case of wind-forced weakly nonlinear deep water waves. Thereby, the wind forcing is modeled as a stochastic process. This leads to a stochastic nonlinear Schrödinger equation, which is calculated for different wind regimes. For the case of wind forcing which is either random in time or random in space, it is shown that breather-type solutions such as the Peregrine breather occur even in strong gusty wind conditions.

Atmospheric dynamics can be described by the Boussinesq approximation which models bouyancy-driven fluid flows. Its simulation involves the repeated solution of the Navier-Stokes equations. This requires numerical solution methods for the dense Schur complement. In this talk, we will be concerned with Schur complement preconditioners. Furthermore, we will discuss a low-rank update for the Schur complement preconditioners. The update method is based on the error between the preconditioned Schur complement and the identity. It will be illustrated with some numerical results.

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.