# Diplomanden- und Doktorandenseminar Numerische Mathematik SS 2008

 Dozenten: Jörg Liesen, Volker Mehrmann, Reinhard Nabben Koordination: Falk Ebert LV-Termine: Do 10-12 in MA 376 Inhalt: Vorträge von Diplomanden, Doktoranden, Postdocs und manchmal auch Gästen zu aktuellen Forschungsthemen

Datum Uhrzeit Raum Vortragende(r) Titel Vorläufige Terminplanung: Do 17.04.2008 10:15 MA 376 Elena Virnik Model reduction of positive descriptor systems (Abstract) im An-schluss Vorbesprechung Do 24.04.2008 10:15 MA 376 Özlem Cavusoglu A characterization of Lyapunov operators (Abstract) Do 08.05.2008 10:15 MA 376 Hannes Gruschinski Recent investigations on optimal control and estimation of stochastic descriptor and singularly perturbed systems (Abstract) Do 15.05.2008 10:15 MA 376 no talk Do 22.05.2008 10:15 MA 376 Kai Wassmuss Ein algebraisches Mehrgitterverfahren zur Lösung der Helmholtz-Gleichung (in german) Do 12.06.2008 10:15 MA 376 Lisa Poppe H∞-Control of Discrete-Time Descriptor Systems (Abstract (pdf)) im Anschluss MA 376 Tobias Brüll A short introduction into dissipative systems theory (Abstract) Do 19.06.2008 10:15 MA 376 Maciek Korzec An introduction to pseudospectral methods with applications onto sixth order equations describing the faceting of growing surfaces (Abstract) Do 26.06.2008 10:15 MA 376 Elena Teidelt Quadratic Eigenvalue Problems Arising from Noise Control Problems (Abstract (pdf)) Do 03.07.2008 10:00! MA 376 Agnieszka Miedlar Adaptive solution of elliptic eigenvalue problems (Abstract) im Anschluss MA 376 Özlem Cavusoglu A new Tensor SVD (Abstract)

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Abstracts zu den Vorträgen:

Elena Virnik (TU Berlin)
Model reduction of positive descriptor systems
Thu 17.04.2008, 10:15 h in MA 376

Abstract:
We consider linear time-invariant control systems of the form
\begin{eqnarray*}
E\dot x&=&Ax+Bu, \ x(t_0)=x_0\\
y&=&Cx+Du,
\end{eqnarray*}
where $A,B,C,D$ are real constant coefficient matrices of appropriate size. The state $x$, input $u$ and output $y$ are real-valued vector functions. Positive systems arise, when economical, biological or chemical systems are modelled by descriptor systems, in which the state $x$ describes concentrations, populations of species, or numbers of cells. In positive systems it is required that the solution and the output have to be nonnegative vector functions for appropriately chosen initial state and input. We review some fundamental properties of positive systems and of the model reduction methods of balanced truncation and singular perturbation balanced truncation, which have the advantage of a guaranteed error bound. We present an adaptation of these methods that preserves the positivity of the system. First, we show the results for standard systems (joint work with Timo Reis) and then (if time is left) extend these to the descriptor case.

Özlem Cavusoglu (BMS)
A characterization of Lyapunov operators
Thu 24.04.2008, 10:15 h in MA 376

Abstract:
The aim of the talk is to discuss the problem of representing a linear matrix operator, in particular the inverse of a Lyapunov operator, as a sum of minimum number of elementary operators. Some concepts and properties related to Sylvester and Lyapunov operators as well as the concept of "Sylvester index" of a linear matrix operator and the concept of "Lyapunov index" will be presented. The characterization of real and complex Lyapunov operators will be given. The concept of "Lyapunov singular values" will be introduced. This talk is mainly based on the research paper: M. Konstantinov, V. Mehrmann, P. Petkov, On properties of Sylvester and Lyapunov operators, Linear Algebra Appl.312(2000) 35-71.

Hannes Gruschinski (Uni Magdeburg)
Recent investigations on optimal control and estimation of stochastic descriptor and singularly perturbed systems
Thu 08.05.2008, 10:15 h in MA 376

Abstract:
We introduce stochastic descriptor systems (SDS) in continuous-time with properly stated leading term and point out important special cases (singularly perturbed (SP) systems with time-varying and constant perturbation number and differential-algebraic systems). For many of these classes we propose novel families for the LQR/Kalman-Bucy/LQG/$H_\infty$/Lyapunov problems. We sketch derivations and a catalogization of the so-called "Obvious" as well as "Not-So-Obvious" solutions (Riccati transformations) including their solvability conditions and mathematical ramifications. Preliminary numerical results and future projects are discussed.

Lisa Poppe (TU Berlin)
H-Control of Discrete-Time Descriptor Systems
Thu 29.05.2008, 10:15 h in MA 376, first talk
Agnieszka Miedlar (TU Berlin)
Adaptive solution of elliptic eigenvalue problems
Thu 03.07.2008, 10:00 h in MA 376, first talk

Abstract:
As a first step we will introduce some existing results about a priori and a posteriori error estimation and global convergence of adaptive methods for elliptic eigenvalue problems. Because the goal of the project is to reduce the complexity while retaining good accuracy we concentrated our work on combining two currently used techniques: first spectralize then discretize and first discretize then spectralize. Our new approach uses Krylov subspace methods together with Galerkin approximations. The main idea is to use fine meshes only to decide which areas should be refined and iterative methods to solve finite dimensional problems on the coarse grids. At the end some first numerical results will be presented.

Özlem Cavusoglu (TU Berlin)
A new Tensor SVD
Thu 03.07.2008, 10:00 h in MA 376, second talk

Abstract:
The Singular Value Decomposition is an important tool in linear algebra. However, extension of the matrix SVD's to higher-order tensors is not straightforward. Some basic concepts such as "rank" or "best rank-k approximations" become more complicated. A new tensor-tensor multiplication, which constitutes an algebra, has been introduced recently by Carla Dee Martin et. all. In this talk I am going to present this new tensor-tensor multiplication and how this multiplication gives rise to an SVD-like decomposition of higher-order tensors. Finally, I will try to relate this algebra to Banach algebras, in order to relate the approximation problem of tensors to the approximation theory in Banach algebras, specifically algebras of Hilbert space operators.
 Impressum Falk Ebert 01.07.2008