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Date Time Venue Talk
09/06/18 02:00 pm Am Schwarzenberg-Campus 3 (E), Room 3.074 Utilizing Geometry of Smoothness-Increasing-Accuracy-Conserving (SIAC) filters for reduced errors
Prof. Dr. Jennifer Ryan, Mathematics, University of East Anglia

Smoothness-Increasing Accuracy-Conserving (SIAC) filters for Discontinuous Galerkin (DG) methods are designed to increase the smoothness and improve the convergence rate of the DG solution form p+1 to 2p+1 through post-processing. However, introducing these filters can be challenging for multi-dimensional data since a tensor product filter grows in support size as the field dimension increases [(3p+2)*h]^d, where p + the polynomial order and d is the dimension. This becomes computationally prohibitive as the dimension increases. An alternative approach is to utilize a one-dimensional univariate filter. In this talk we introduce the Line SIAC filter and explore how the orientation, structure and filter size affect the order of accuracy and global errors. We show how line filtering preserves the properties of traditional tensor product filtering, including smoothness and improvement in the convergence rate, given an appropriate rotation. Furthermore, numerical experiments are included, exhibiting how these filters achieve the same accuracy at significantly lower computational costs.
08/09/18 03:45 pm H0.09 A glimpse on interpolation theory and interpolation with mixed boundary conditions*
Sebastian Bechtel, Arbeitsgruppe Analysis, TU Darmstadt, Schlossgartenstrasse 7, 64289 Darmstadt

First, we give a short introduction to abstraction interpolation theory and
relate it to the well-known interpolation results from Riesz--Thorin and
Marcinkiewicz. Then we apply the abstract methods to concrete spaces
incorporating (mixed) boundary conditions and give an overview on arising
challenges and ways to resolve them.
07/25/18 11:00 am Am Schwarzenberg-Campus 3 (E), Room 3.074 Iterative Löser für RBF Kollokation zur Lösung von partiellen Differentialgleichungen (Bachelorarbeit)
Felix Kieckhäfer, Mathematik
07/19/18 03:45 pm Am Schwarzenberg-Campus 3 (E), Room 3.074 Quantitative unique continuation principles and application to control theory for the heat equation
Martin Tautenhahn, TU Chemnitz, Fakultät für Mathematik

This talk is divided into two. In the first part we discuss a so-called scale-free and quantitative unique continuation principle for spectral projectors of Schr\''odinger operators.
Let $\Omega = \Lambda_L = (-L,L)^d$ or $\Omega = \mathbb{R}^d$, and $H = -\Delta + V$ be a Schr\''odinger operator on $L^2 (\Omega)$ with a bounded potential $V$. If $\Omega = \Lambda_L$ we impose Dirichlet, Neumann, or periodic boundary conditions. The unique continuation principle states that for any $E \geq 0$, and any $\phi \in \operatorname{Ran} \chi_{(-\infty , E]} (H)$ we have
\begin{equation} \label{quc}
\lVert \phi \rVert_{L^2 (\Omega)}^2 \leq C_{\rm sfuc} \lVert \chi_{S_\delta \cap \Omega} \phi \rVert_{L^2 (\Omega)}^2,
\end{equation}
where $S_\delta \subset \mathbb{R}^d$ is a union of equidistributed $\delta$-balls, and $C_{\rm sfuc} = C_{\rm sfuc} (d , E ,\allowbreak \delta , \lVert V \rVert)$ some explicitly given constant.
\par
In the second part of the talk we will discuss an applications thereof to control theory. On the time interval $[0,T]$ we consider the controlled heat equation
\begin{equation} \label{eq:parabolic}
\partial_t u + H u = f\chi_{S_\delta \cap \Omega} ,
\end{equation}
where $u,f \in L^2([0,T] \times \Omega)$, and $u (0,\cdot) \in L^2 (\Omega)$.
Note that the control function $f$ acts on the set $S_\delta$ only. Our aim is to study null-controllability in time $T > 0$, i.e.\ there is a control function $f$ such that $u(T,\cdot) = 0$. We provide explicit estimates on the costs of the form $\lVert f \rVert_{L^2([0,T]\times \Omega )} \leq C \lVert u_0 \rVert_{L^2 (\Omega)}$.
07/17/18 11:00 am H - SBC5 / H0.06 Maximum number of clique-free edge coloring in graphs
Hiep Han, Universidad de Santiago de Chile
07/17/18 10:00 am H - SBC5 / H0.06 Gallai's Conjecture for regular graphs and planar graphs
Andrea Jimenez, Universidad de Valparaíso
07/12/18 03:45 pm tba Sparse Frequency Estimation*
Benedikt Diederichs, Fachbereich Mathematik, Universität Hamburg

Prony's problem - estimating the frequencies of an exponential sum - and its higher dimensional
analogs have attracted a lot of attention in recent years. A somewhat neglected question is whether
this problem is well-posed. In this talk, some results in this direction will be presented.
We start by giving a brief introduction to stability in compressed sensing. Compressed sensing is
concerned with solving nite dimensional linear systems under a priori sparsity assumptions. Stability
follows from the so-called restricted isometric property (RIP) of the system matrix.
We then discuss sparse frequency estimation. Due to the continuous nature, proving an analogue of
the RIP is more dicult. To this end, we briey introduce specic functions, which are well localized
in the spatial and frequency domain. Then we deduce stability results as well as a posteriori error
estimates.
This talk is based on joint work with Armin Iske.
07/04/18 01:30 pm Am Schwarzenberg-Campus 3 (E), Room 3.074 Image segmentation methods and an application to brain images.
Christoph Nicolai
06/28/18 03:45 pm Am Schwarzenberg-Campus (D), Room D1.021 A minimax principle in spectral gaps*
Albrecht Seelmann, Fakultät für Mathematik - Technische Universität Dortmund

In [Doc. Math. 4 (1999),275--283], Griesemer, Lewis, and Siedentop devised an abstract minimax principle for eigenvalues in spectral gaps of perturbed self-adjoint operators. We show that this minimax principle can be adapted to the particular situation of bounded additive perturbations with hypotheses that are easier to check in this case. The latter is demonstrated in the framework of the Davis-Kahan sin(2\Theta) theorem.

This talked is based on joint work with I. Nakic, M. Täufer, M. Tautenhahn, and I. Veselic.
06/27/18 01:00 pm Am Schwarzenberg-Campus 1 (A), Room A 0.14 Tiling edge-coloured complete graphs with few pieces
Jan Corsten, London School of Economics, Department of Mathematics

* Talk within the Colloquium on Applied Mathematics