Vorträge

Tsunami modeling is - to first (and very accurate) approximation - performed with the help of shallow water theory and equations. This is still the method of choice for many applications, including forecasting, hazard assessment and inundation modeling. However, for long propagation distances as well as highly nonuniform topographies dispersive effects become important. While truly dispersive model equations are fully three-dimensional and therefore expensive with respect to computational requirements, a common approach to dispersive modeling comprises a non-hydrostatic correction of shallow water equations. In order to derive this correction term, a linear system of equations needs to be solved in each time step - even when the time-stepping scheme is explicit.

In the presentation we will introduce the basic modeling concepts for tsunami simulation, will show the derivation of non-hydrostatic correction terms and motivate further research on solvers for linear systems of equations.

We give new conditions for strong convergence of positive operator
semigroups as time tends to infinity. This is achieved by a new approach
that combines the splitting theorem by Jacobs, de Leeuw and Glicksberg
with a purely algebraic result about positive group representations.
Thus, we obtain convergence theorems not only for one-parameter
semigroups but also for a much larger class of semigroup representations
without any continuity or regularity assumption in time.
In particular, this generalizes results from the literature that, under
technical assumptions, a bounded positive strongly continuous semigroup
that contains or dominates a kernel operator converges strongly as time
tends to infinity. One can also derive a generalization of a famous
theorem by Doob for operator semigroups on the space of measures.

Ich werde in einem ca. 45 minütigen Vortrag meine Masterarbeit, die ich im Juni 2016 an der Uni Lund verteidigt habe, vorstellen.
Die Arbeit mit dem Titel ''Evaluation of a Gradient Free and a Gradient Based Optimization Algorithm for Industrial Beverage Pasteurisation Described by Different Modeling Variants'' entstand in Zusammenarbeit mit der Firma Krones AG in Kopenhagen.
Ziel ist die Optimierung thermaler Behandlung von Getränken und flüssigen Dosenkonserven. Dazu wurden Pasteurisierungsprozesse mathematisch formuliert, simuliert und die Optimierung mit Python durchgeführt.

Consider the usual function space L^2(D) on the unit disk D and
the (closed) subspace of holomorphic functions A^2(D). An important class of bounded
linear operators arises by restricting multiplication operators M_f on L^2(D) to A^2(D). More
precisely, if P denotes the orthogonal projection onto A^2(D), one considers operators of
the form PM_f in A^2(D), so-called Toeplitz operators. In this talk we are going to study the
essential spectrum of these Toeplitz operators. It is a classical result that if the defining
symbol f is continuous up to the boundary, the essential spectrum can be obtained by
evaluating f at the boundary. As it turns out, this statement can be generalized to more
general symbols by using techniques that were developed to solve a similar problem on
the sequence space \ell^2(\Z).

Im Vortrag wird es darum gehen, wie man eine vektorwertige Gleichung löst, wenn man die entsprechende Gleichung schon einmal skalarwertig gelöst hat. Typische Beispiele hierfür sind elliptische Differentialgleichungen. Hierbei geht es dann weniger darum, den Differentialoperator selbst zu untersuchen, sondern die Eigenschaften der Räume, auf denen er lebt.

Im wesentlichen ein 60 bis 90 minütiger Arbeitsstandbericht. Es werden grundlagen der Theorie fraktionaler Operatoren erläutert und danach auf die Problematik der Einführung durch harmonische Erweiterung eingegangen.

Vier konkrete Anwendungen von Toeplitzoperatoren

Es werden vier konkrete und sehr unterschiedliche Anwendungen von Toeplitzoperatoren vorgestellt. Diese sind (1) ein Problem aus der optimalen ell-eins-Kontrolle, (2) Spektralfaktorisierung von Polynomen vom Grad 20000, (3) Berechnung des Volumens der Fundamentalgebiete gewisser hochdimensionaler Gitter, und (4) Bestimmung der Grenzmenge der Nullstellen von Polynomen vom Fibonacci-Typ in der Hausdorffmetrik. Der Vortrag erlaubt es, viermal abzuschalten und ebenso oft wieder einzusteigen.

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Four concrete applications of Toeplitz operators

I present four concrete and very different applications of Toeplitz operators. These applications are (1) a problem in optimal ell-one control, (2) spectral factorization of polynomials of degree 20000, (3) computation of the volume of the fundamental domains of some high-dimensional lattices, and (4) the determination of the Hausdorff limit of the zero set of polynomials of the Fibonacci type. The talk allows you to switch off four times and to re-enter the same number of times.