Diplomanden- und Doktorandenseminar
Numerische Mathematik WS 2005/06

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 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:
The talk consists of two parts. In the first part a special URV decomposition of a square matrix is introduced. An algorithm to to compute this form is also derived. The second part deals with the symmetric/anti-symmetric eigenvalue problem. The just introduced URV decomposition and some postprocessing is shown to make up an efficient algorithm.

Abstract:
In this talk, the simulation of drop size distributions in stirred liquid-liquid systems will be considered. This simulation requires, on the one hand, the computation of the flow field in the stirred tank and, on the other hand, the simulation of the population dynamical phenomena of the drops. In the application considered here, the flow simulation is done with the CFD code FEATFLOW, whereas the drop size distributions are calculated with the package PARSIVAL. In this talk, we will consider one way how the coupling of the two solvers can be realized.

Abstract:
In this talk the interaction between sparsity and l_1 minimization is explained. Also recovery of sparse signals from incomplete and noisy measurements is briefly explained. In this way a beautiful connection between sparse recovery and uncertainty principles is given. Finally some applications in Computerized Tomography(CT) are given.

Abstract:
The equations ariasing in circuit simulation are usually very large differential-algebraic equations (DAEs). In practical applications, these equations almost always have a differentiation index larger than one. (cf. almost everything by D. Estevez-Schwarz and C. Tischendorf) It is a well known fact that in general DAEs of index higher than 1 show bad numerical behaviour with respect to stability and accuracy. It is therefor recommendable to reduce the index of such equations. (cf. many works by P. Kunkel and V. Mehrmann) Many known methods rely on the algebraic transformation of the DAE. In industrial simulation however, this is rarely possible as alterations to the used solvers are nearly impossible. We will present alternative methods that change the circuit topology instead of the resulting equations to generate analytically equivalent systems of index one. We will give examples of the procedure using the free circuit simulator SPICE as reference simulator. We will show that even without changes to the simulator, just by the topologial preprocessing step, we will be able to improve accuracy and computation time of small test examples.

Abstract:
In this talk, we show how we can transform a linear second order differential-algebraic system by equivalent transformations to a condensed form and how we can use this condensed form to derive a strangeness-free system. This approach is very useful for the analysis of higher order DAEs but it is not suitable for numerical methods. Therefore, we use the derivative array approach for higher order differential-algebraic systems to derive an equivalent strangeness-free system.

Abstract:
This talk is about explicit two-step methods with peer-variables for the solution of ordinary differential equations. We would like to analyze consistency and convergence and optimize the stability regions of the methods. Finally, we compare several methods with ode45.

Abstract:
In this talk we want to introduce an algorithm for the solution of  nonlinear constrained optimization problems. This quasi Newton method uses a new kind of low-rank corrections computed by the SR1- and TR1-update  formulas. Dealing with large dense approximations of derivatives it is necessary to use a decomposed representation of the KKT system. This leads to the task of  efficiently updating a factorized system after a low-rank modification. Here, we want to show two different algorithms for updating a  rectangular LU factorization. Further, we will introduce a new hybrid method which is well suited in the optimization context.

Abstract:
We sketch how the perturbation index for the MNA-equations coupled with the stationary drift-diffusion equations can be determined. Methods and results from DAEs in finite dimensional spaces are generalised in an expected straight-forward way.
In the second part the instationary drift-diffusion equations are seen as an abstract DAE with a dynamic part living in an infinite dimensional subspace. In this case the unique solvability of the inherent ODE is not evident. We present two different appraoches to obtain this abstract ODE and suggest how the solvability can be checked.

Abstract:
With regard to the purpose of this seminar, a brief overview of my current research work is given:

• Comfortable testing of control configurations with the CFD-Code Featflow by means of a Matlab-interface (with Lars Henning, Sonja Schlauch, Andrey Sokolov)
• Adaptive time- and space discretizations using the FEM-package deal.ii (with Christian Kamm)
• A framework for the low-dimensional input-output modeling of linear infinite-dimensional systems

Abstract:
Domain decomposition methods are widly used for solving partial differential equations. Strongly connected with domain decompostion methods are the multiplicative and additive Schwarz-type methods for solving the related linear systems. Here we establish some new variants of Schwarz methods. We give some convergence results for these methods and we consider the effect on convergence of algorithmic parameters such as the number of subdomains, the amount of overlap, and the result of inexact local solves. Moreover a numerical example is considered.

Abstract:
The 'Algebraic Multilevel Iteration' (AMLI) developed by Axelsson and Vassilevski is an algebraic multigrid method, which uses an approximation of the Schur complement as a coarse grid operator.
In its basic form the AMLI method can be rewritten as an additive Schwarz method. By taking a look at the corresponding multiplicative Schwarz method we get a new iteration scheme. We will show convergence of the additive and multiplicative form for nonsymmetric M-matrices. Beside new comparison results a bound for the asymptotic convergence rate will be presented.

Abstract:
In this talk we continue the previous talk about sparse signal recovery using l_1 minimization. The miracle about l_1 minimization and sparsity is briefly explained. The Mathematics behind this problem will be discussed. Finally, stable signal recovery in the presence of noise will be given.