Talks

Ich präsentiere einen allgemeinen Ansatz für das Jacobi-Davidson-Verfahren basierend auf einem beliebigen iterativen Verfahren zum Lösen eines linearen oder nichtlinearen Eigenwertproblems. Die Auswirkung eines inexakten Lösens der Korrekturgleichung wird betrachtet und hieraus kann lineare Konvergenz für drei Fälle von verschiedene Vorbedingungen bewiesen werden.

We give an overview or recent developments on Deep Learning, its relations to wavelet
theory and applications to image analysis with interactions with persistent homology.

We consider differential-algebraic equations on (possibly) infinite dimensional Hilbert spaces, that is, we consider equations of the form
\begin{align*}
(Eu)'+Au & =0\quad(\text{on }\mathbb{R}_{\geq0}),\\
u(0) & =u_{0},
\end{align*}
where $E,A$ are linear operators on a Hilbert space $H$ and $E$ is bounded and allowed to have a non-trivial kernel. These equations cannot have a unique solution for each $u_{0}\in H$ (just look at the case $E=0$). Thus, finding the ``right'' space of initial values arises as a natural question. Imposing growth conditions on the operator pencil
\[
z\mapsto(zE+A)^{-1}
\]
we determine the maximal space of admissible initial values. First, we treat the case of bounded $A$ and then generalise the results to the case of unbounded $A$. In particular, we discuss whether we can find a $C_{0}$-semigroup yielding the mild solutions of the above problem.