Diplomanden- und Doktorandenseminar
Numerische Mathematik WS 2007/08

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 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:
On the basis of a small slope approximation, dimension-reduced models for the faceting of a growing nano-crystal shape that take into account surface diffusion, anisotropic surface tension and normal flux onto the growing surface are presented. We analyze the possible steady state solutions of the resulting Cahn-Hilliard type equations using phase-space analysis and asymptotic methods.

Abstract:
The subject of the talk is the so-called even eigenvalue problem (1)  Mx=\lambda Nx,   where M is symmetric, and N is skew symmetric. The matrices M and N can be transformed into the so-called even URV form   involving two transformation matrices U and V. This even URV form makes it   possible to read off the eigenvalues and to a certain extend also the   eigenvectors of (1). But what one really wants is to transform the matrices M and N into the   so-called even Schur form form involving only one transformation matrix Q. In the talk, we will present a method that takes an even URV form and   produces an even Schur form. This is done by equalizing, step by step, the   two matrices U and V as they really should be equal ... to the matrix Q.

Abstract:
This is an introductory talk on pseudospectra and $\mu$-functions for structured matrix perturbations. We discuss the relationship between these quantities and  give explicit formulas for several perturbation structures. In particular, we consider pseudospectra for small paerturbations and obtain formulas for structured eigenvalue condition numbers.

Abstract:
Linear differential-algebraic equation motivate the study of pairs of matrix-valued functions. In many applications from mechanics or control theory the underlying matrices are structured, e.g., they are symmetric or Hermitian or have a self-adjoint structure. Structure preserving condensed forms for pairs of Hermitian matrix functions as well as pairs of self-adjoint matrix functions are studied. Furthermore, a structure preserving strangeness-free system for differential-algebraic equations is derived using the derivative array approach.

Abstract:
Possible ways of reducing matrix polynomials of higher order to second order are discussed. A certain class of quadratic matrix polynomials constitutes a vector space of potential canonical reductions. The infinite spectrum of the original matrix polynomial is in general not preserved during the reduction. Therefore it is investigated, in which way the infinite spectrum changes. Moreover the preservation of some structural properties of the original polynomial during the reduction process are examined. At last an alternative class of reductions is introduced, which preserves the infinite spectrum.

Abstract:
In this talk we will discuss the H∞-control problem for descriptor systems. We want to find a dynamic controller such that the transfer matrix T(s) of the closed loop system is stable and has minimal H∞ norm. Conditions for the existence of such a (sub)optimal dynamic controller are often given in terms of the solvability of two Riccati Equations, but this method leads to several numerical difficulties. The use of a matrix pencil approach instead, can avoid some of these difficulities and will be introduced in this talk for index one systems as well as for higher index systems. The functionality of this approach is demonstrated with the help of a numerical example.

Abstract:
I will present a talk that is based on the reseaerch that had been carried out during my last visit to TU Berlin (November 2006). This is a joint work with Reinhard Nabben, Kees Vuik and Yogi Erlangga. For various applications, it is well-known that a two-level conjugate gradient method is an efficient method for solving large and sparse linear systems. A combination of traditional and projection-type preconditioners is used to get rid of the effect of both small and large eigenvalues of the coefficient matrix. The resulting projection methods are known in literature, coming from the fields of deflation, domain decomposition and multigrid. At first glance, these methods seem to be different. However, from an abstract point of view, it can be shown that they are closely related to each other and some of them are even equivalent. The aim of this talk is to compare projection methods both theoretically and numerically. We investigate their equivalences, robustness, spectral and convergence properties. We end up with a suggestion of a two-level preconditioner, that is as robust as the abstract balancing preconditioner and as cheap and fast as the deflation preconditioner.

Abstract:
In an industry cooperation with SFE the GMRES algorithm with subspace recycling and preconditioning has been implemented to solve large sparse system of linear equations. First, the physical background will be introduced. After the mathematical statement of the problem, we will discuss the method employed and some implementational issues. Finally, some results will be shown.

Abstract:
In the simplex algorithm for linear programming, sequences of large, unstructured, sparse linear systemes of equations have to be solved. The most commonly used linear algebra kernel in simplex algorithms employs a direct solution scheme (LU) based on the Markowitz rule, which was published first in 1957. We analyze the effectiveness of the existing kernel and explain why other techniques do not offer a competitive alternative.

Abstract:
This talk gives an introduction to my research project in the SFB 557 "Control of complex turbulent shear flows". First, I will present the mathematical model for flow control of a high-lift configuration and give an overview of different solution approaches. Afterwards, I will talk about system identification. Finally, the implementation of a high-lift configuration in the CFD code FEATFLOW will be considered and some first results will be presented.

Abstract:
Starting with the nonlinear Jacobi--Davidson algorithm we derive a version which uses the special structure of complex symmetric nonlinear eigenvalue problems. We will show, why this improves convergence on the basis of nonlinear Rayleigh functionals, in particular, the complex symmetric Rayleigh functional. The latter is stationary in contrast to the standard functional, which is the one-dimensional equivalent to the Rayleigh-Ritz step eigensolution in the nonlinear Jacobi--Davidson algorithm. Numerical examples will be shown.

Abstract:
In this talk, we will present a topological approach for the index analysis of electro-meachanical systems. The first step will be to analyse the structure of the coupled system. Afterwards, an algorithm to get an upper bound for the index of the electro-mechanical system is presented. Based on that, I will give an algorithm which determines the index of the coupled system and present an example of how the algorithm works.

Abstract:
In this talk we will give an introduction to the idee of combining adaptive finite element method (AFEM) with fast iterative solvers for general algebraic eigenvalue problems, in terms of solving parameter dependent eigenvalue problems for PDEs. To achieve that we have to concentrate on equilibrating the errors and computational work between the discretization and the approximation errors and the errors in the solution of the resulting finite dimensional linear and nonlinear eigenvalue problems. This talk will give an overview of the problem with some first research results.

Abstract:
This talk is about 2-level-methods to solve linear equationsystems, particulary the method of Ruge and Stüben. This method was developed in 1987 by J.W. Ruge and K. Stüben and they showed, that it is convergent for nonsingular symmetric M-matrices and nonsingular symmetric positiv definite matrices. In the talk I will demonstrate that splitting methods are a good approach to find general convergence results and I will show the convergence for special nonsingular nonsymmetric M-matrices. Before I give a few numerical examples I will demonstrate a modification of the method of Ruge and Stüben which is convergent for all nonsymmetric nonsingular M-matrices.

Abstract:
Numerical simulation of electrical circuits has been and will be a challenging task. The difficulties arise from the huge problem sizes, nonsmooth data and possibly high index of the involved differential algebraic equations. Recently, some of the circuit elements are even described by PDEs, adding even more problems to the solution of the coupled system equations.
We will give a short introduction on dynamic iteration methods and show how these can be applied to electrical circuits while maintaining the network structure. We will discuss common problems for dynamic iteration methods and how they can be overcome. Finally, we will show how the theory can be applied to solve coupled DAE-PDE systems while using only existing solvers.

Abstract:
We will present several open questions concerning the Sendov conjecture. Since Blagovest Sendov posed it in 1958, it remains unsolved. Though there are many partial results, the problem itself seems still far from being solved completely. We will mention several of them and also some results that are related to the conjecture and discuss difficulties that are hidden behind this problem that can be stated so easily, but is so difficult. We will also present a matrix approach to the conjecture.