Talks

For two Banach spaces $X,Y$ let $u:\mathbb{R}_{\geq 0}\rightarrow \mathcal{L}(X;Y)$ be an operator valued function and $\mathtt{P}$ a regularity property. Assume that each orbit $t\mapsto u(t)x$ has the regularity property $\mathtt{P}$ on some interval $(t_x,\infty)$ in general depending on $x\in X$. In this paper we prove a Baire-type theorem, which allows to remove the dependency of $x$ in certain situations. Afterwards, we provide some applications which are of interest in semigroup theory. In particular, we generalize and explain the result obtained by Bárta in his article ``\emph{Two notes on eventually differentiable families of operators}'' (Comment. Math. Univ. Carolin. 51,1 (2010), 19-24).

In this presentation we will talk about the application of hierarchical matrices to solve the least squares problem arising in the RBF Approximation of scattered data.

We will shortly introduce hierarchical matrices as well as some central aspects of the RBF approach to scattered data. The main part will be about different ideas regarding the QR decomposition of hierarchical matrices.

I will present an introduction of the RBF-FD method and properties of the arising linear systems.

In this talk we present results of a comparative study, where we analyze and benchmark the error and the convergence order of finite difference, finite-element as well as Voronoi finite-volume discretization schemes for the drift-diffusion equations describing charge transport in bulk semiconductor devices, i.e., the van Roosbroeck system.

The relation between the quasi-Fermi levels and the densities of electrons and holes is given by the equation of state. Three common challenges, that can corrupt the precision of numerical solutions of the van Roosbroeck system, will be discussed: boundary layers of the quasi-Fermi potentials at Ohmic contacts, discontinuties in the doping profile, and corner singularities in L-shaped domains. The influence on the order of convergence is assessed for each computational challenge and the different discretization schemes. Additionally, we provide an analysis of the inner boundary layer asymptotics near Ohmic contacts to support our observations.

The Hot Spots Conjecture of J. Rauch asserts that the hottest and coldest points of an insulated body should move towards its boundary for large times, if the insulation is perfect. Via the semigroup associated with the Neumann Laplacian this reduces to proving that maximum and minimum of the eigenfunction(s) associated with the smallest positive eigenvalue are located on the boundary. This conjecture is not true in full generality but is currently open, for example, for convex domains.

In this talk we will examine the corresponding question on metric graphs: for the Laplacian on a finite metric graph with standard (continuity and Kirchhoff) vertex conditions we consider the possible distribution of maxima and minima of eigenfunctions associated with the smallest nonzero eigenvalue. Among other things, we give examples to show that the usual notion of “boundary” of a metric graph, namely the set of vertices of degree one, has limited relevance for determining the “hottest” and “coldest” parts of a graph.

This is joint work with James Kennedy (Lisbon).

In this talk we study approximation techniques for input-output systems, which appear in the modeling process of mechanical systems. So, the focus will be on linear dynamical systems with a second derivative term.
These system can become very large in practice and therefore, expensive to be used for simulations and controller design.
Since this frequently happens to all control systems coming from real-live application, model order reduction became a major field in control theory over the last decades.
Here however, beside approximating the input-output behavior of the original system, the special structure should be preserved in the reduced-order model.
So far, reduction techniques designed for the linearized model fail in this aspect. On the other hand, there is a wide variety of methods that directly treat the second order control system. However, up to this point none of those methods deliver reasonable error-bounds for the approximation.
In this talk an approximation method is presented for the special class of passive mechanical systems. Roughly speaking passivity for control systems means that the system itself cannot produce energy. For this class the special canonical structure, given by so called Jordan triples for matrix polynomials, can be exploited.
In the end an error bound in the gap metric will be derived. The gap metric is used as a measure for the distance of two linear systems. It is defined via the distance of the closed subspaces of stable trajectories corresponding to zero initial conditions of the systems. Hence, the gap metric error-bound ensures the quality of the approximation of the state/signal system.

Abstract
We study two standard biased (1 : b) Maker-Breaker positional games
| the Perfect Matching game and the Hamilton Cycle game, played on
the edge set of the complete graph on n vertices, Kn. Given Breaker's bias
b, possibly depending on n, our goal is to determine the minimal number
of moves in which Maker can win in each of these two graph games.

This is joint work with Miloš Stojakovic.

In this talk we outline classical connections between such mathematical objects as operator semigroups, evolution equations and Markov processes. Further, we present a method to approximate operator semigroups with the help of the Chernoff theorem. Many \emph{Chernoff approximations} lead to representations of solutions of (corresponding) evolution equations in the form of limits of $n$-fold iterated integrals of elementary functions when $n$ tends to infinity. Such representations are called \emph{Feynman formulae}. They can be used for direct computations, modelling of the related dynamics, simulation of underlying stochastic processes.
In some cases, Chernoff approximations can be understood as a version of the operator splitting method (known in the numerics of PDEs); some Feynman formulae provide Euler--Maruyama schemes for SDEs. Moreover, the limits in Feynman formulae sometimes coincide with path integrals with respect to probability measures (\emph{Feynman-Kac formulae}) or with respect to Feynman type pseudomeasures (\emph{Feynman path integrals}). It is planned to discuss different Chernoff approximations for semigroups corresponding to some Markov processes (e.g., subordinate Feller diffusions on star graphs and Riemannian manifolds) and for Schr\''{o}dinger groups.
Furthermore, the constructed Chernoff approximations for operator semigroups can be used to approximate solutions of some time-fractional evolution equations describing anomalous diffusion (solutions of such equations do not posess the semigroup property).