Diplomanden- und Doktorandenseminar
Numerische Mathematik WS 2006/07

Interessenten sind herzlich eingeladen!

Weitere Vorträge siehe auch:

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

• 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 investigate an integrated circuit model coupling the modified nodal analysis (MNA) equations the non-stationary drift-diffusion equations. By considering perturbations only in the MNA equations we could obtain the first perturbation estimate for this partial differential algebraic equation (PDAE). It is assumed that the linear circuit without the semiconductor branch is of index 1, that the diode contacts are connected by a capacitive path and that the semiconductor region is a one-dimensional.

Abstract:
It will be motivated that differential algebraic equations DAEs with piecewise analytical coefficients are a nice setup for considering initial value problems as well as switched systems. On the other hand it is well known that for a complete solution theory of DAEs generalized functions (i.e. distributions) are needed. The aim of this talk is to give an overview how DAEs with piecewise analytical coefficients can be embedded in the distributional setup.

Abstract:
Simulating bubbly flows is a very popular topic in CFD and it receives a lot of attention in various research projects in the last decades. The bubbly flows are governed by the Navier-Stokes equations. In many popular operator splitting formulations for these equations, solving the singular SPSD linear system Ax=b coming from the Poisson equation with discontinuous coefficients takes the most computational time. Standard Krylov solvers like conjugate gradient (cg) method with any preconditioner appear to converge very slowly.

We present a method based on cg by adding a so-called deflation technique to accelerate the iteration process significantly. Using some numerical linear algebra, we give theoretical results to illustrate that the deflation technique can indeed be applied on bubbly flow problems. Finally, some 3-D numerical experiments with bubbles an droplets will be presented in order to show that the deflation method works very well by considering both the number of iterations and the computational time.

Abstract:
Proper orthogonal decomposition aka. POD aka. principal component analysis aka. Karhuenen-Loewe expansion has been used in data representation and compression for a long time. By means of a Gelerkin projection they can be employed in the model order reduction of nonlinear ODEs. The simple approach, however, fails when dealing with differential-algebraic equations. There the state-reduction must be applied to the inherent dynamic system only. We will present a general introduction to the POD method and show why it fails for DAEs. A POD method adapted to DAEs will be presented together with related issues such as computation of the relevant projection subspaces.

Abstract:
We will give a definition of passivity of LTI systems and explain how to determine, if a given system is passive. Furthermore, we will discuss an algorithm to perturb a given system to make it passive. Finally, numerical examples are presented.

Abstract:
Many complex dynamical systems in different application areas are switched or hybrid systems, i.e. the mathematical model itself may change with time, depending on certain indicators. In this talk we consider the modeling and numerical simulation of hybrid differential-algebraic systems. We deal with certain difficulties as index changes after mode switching events or the computation of consistent initial values after mode changes.

Abstract:
In this talk an introduction to H∞-control is given. First the control problem for ODEs is introduced and it is shown how the conditions for the existence of a controller can be obtained. In the second part we show how the ODE techniques can be used to approach the H∞-control problem for descriptor systems and discuss different methods for this class of problems.

Abstract:
We will discuss a new concept for linearization of matrix polynomials and first order formulations of DAEs. This new approach allows to reduce the lengths of Jordan chains associated with singular parts or infinte eigenvalues and is in line with the approach used for first order formulations in multiboy dynamics.

This is joint work with Ralph Byers and Hongguo Xu.

Abstract:
We will present a method for the model reduction of passive descriptor system such that the reduced order model remains to be passive. As a concrete application, we discuss the equations from the modified nodal analysis of electrical circuits.

Abstract:
We present a new extension of the well-known Perron-Frobenius theorem to regular matrix pairs (E,A) that has a number of advantages over previous extensions. The new extension is based on projector chains and is motivated from the solution of positive descriptor systems. We give an extensive introduction to the underlying topics and present examples where the new condition holds, whereas conditions in previous literature are not satisfied.