Diplomanden- und Doktorandenseminar
Numerische Mathematik SS 2008

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Weitere Vorträge siehe auch:

• Kolloquium der Arbeitsgruppe Numerik, Modellierung, Differentialgleichungen
• Seminar des Fachgebiets Optimierung bei partiellen Differentialgleichungen (Gruppe Tröltzsch)

• Diplomanden- und Doktorandenseminar WS 2007/08
• Diplomanden- und Doktorandenseminar SS 2007
• Diplomanden- und Doktorandenseminar WS 2006/07
• Diplomanden- und Doktorandenseminar SS 2006
• Diplomanden- und Doktorandenseminar WS 2005/06
• Diplomanden- und Doktorandenseminar SS 2005
• Diplomanden- und Doktorandenseminar WS 2004/05
• Diplomanden- und Doktorandenseminar SS 2004
• Diplomanden- und Doktorandenseminar WS 2003/04
• Diplomanden- und Doktorandenseminar SS 2003
• Diplomanden- und Doktorandenseminar WS 2002/03

Abstracts zu den Vorträgen:

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.

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.

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.

Abstract (pdf)

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.

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