Diplomanden- und Doktorandenseminar
Numerische Mathematik SS 2010

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 WS 2009/2010
• Diplomanden- und Doktorandenseminar SS 2009
• Diplomanden- und Doktorandenseminar WS 2008/09
• Diplomanden- und Doktorandenseminar SS 2008
• 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:

In this talk, we analyze the use of inexact solves in interpolatory model reduction setting and present the resulting structured perturbation effects on the underlying model reduction problem. We prove that when inexact solves are performed within a Petrov-Galerkin framework, the resulting reduced order models are backward stable with respect to the approximating transfer function. General bounds on the system error associated with an inexact reduced order model are introduced that provide a new tool to understand the structured backward error and stopping criteria. We also show that for a selection of interpolation points that satisfy first-order necessary H2-optimality conditions, a primitive basis remains remarkably well-conditioned and errors due to inexact solves do not tend to degrade the reduced order models. Conversely, for poorly selected interpolation points, errors can be greatly magnified through the model reduction process. Several numerical examples will be presented to support the theoretical discussion.

Abstract:

I will try to give an introduction to several kinds of quadratic matrix equations encountered in applied probability problems (queuing theory, Markov chains), presenting the iterative algorithms used to solve them. Many of these equations are very similar to the problems encountered in control theory (CARE, QEP); thus the focus of the talk will be on highlighting the similarities and differences between these two classes of problems.
It is hoped that knowledge from the more mature field of structured eigenvalue problems can provide better understanding also for the probability setting.

Abstract:

Since the early work of Cullum and Donath (1974) and Underwood and Golub (1975, 1977), there has been, from time to time, a renewed interest in conjugate gradient methods for systems with multiple-right hand sides and in Lanczos methods with several starting vectors for solving linear systems, computing eigenvalues, or, more recently, system identification and model reduction.

While the formal definition of block versions of the Lanczos algorithm is quite easy --- at least when there are equally many right and left starting vectors (i.e., inputs and outputs) --- the true difficulties come when a dimension reduction of block Krylov space (so-called deflation) has to be treated properly, or, in the nonsymmetric case, when the block Lanczos process breaks down. Deflation for symmetric systems of equations was treated by Nikishin and Yeremin (1995) (and, to some extent, by others before). For the nonsymmetric case, the answer to both difficulties lies in an adjustment of the look-ahead Lanczos algorithm for SISO systems. The resulting nonsymmetric look-ahead (or cluster) band Lanczos algorithm has been developed in 1994--1996 in partly independent and partly cooperative efforts by Aliaga, Boley, Feldmann, Freund, Hern\'andez, Malhotra, and even others. The work culminates in the paper by Aliaga, Boley, Freund, and Hern\'andez in Math. Comp. (1999/2000, submitted 1996).

However, the detailed specification of a nonsymmetric block Lanczos algorithm (for MIMO systems) requires to choose among many options, and the solution chosen by the mentioned authors is far from optimal regarding its stability. There are trade-offs between cost, complexity, and stability. The purpose of this talk is to discuss these issues. We will in particular present the solution chosen in the dissertation of my student Damian Loher, which proved far more stable than previous ones.

Abstract:

Let X be a simple invariant subspace of the square matrix A. If the matrix E is sufficiently small then the matrix A+E has a simple invariant subspace X_E which is close to X. We provide a (hopefully) new bound for the maximum canonical angle between X and X_E. Furthermore, we compare our estimate with the classical results by Davis and Kahan, Demmel and Stewart.

Abstract:

The H∞ control problem is studied for linear constant coefficient descriptor systems. Necessary and sufficient optimality conditions are derived in terms of deflatign subspaces of even matrix pencils for problems of arbitrary index. A simple numerical example is given to compare them to previous methods.

Abstract:

In this talk, we consider the linear quadratic optimal regulator for descriptor systems. The main ansatz will be, to transform the system into an SVD coordinate system. Using this transformed system, an algebraic Riccati equation is derived, whose solution can be used to compute the optimal regulator.

Abstract:

In this talk we will consider first-order linear systems with quadratic cost or energy functionals. For such systems we will see that (under some controllability assumptions) dissipativity is equivalent to the solvability of a certain linear matrix inequality.

Abstract:

To simulate the mixing of two or more immiscible fluids one can call on the multiphase Navier-Stokes Equations. For the setup considered in my work, where one phase is easily dominated by the other, it is common practice to approximate the flow by one fluid and to track the second by means of a population balance equation. To simulate turbulent regimes the use of turbulence models is inevitable. This talk addresses the modelling of multiphase flows and how the simplified, averaged and reduced equations used in my simulations can be derived. Doing the reduction in one framework delivers a closed representation for the approximation error.

Abstract:

We discuss a functional perturbation results, i.e., a functional normwise backward error for PDE eigenvalue problems. Following the work of M. Arioli et al. for boundary value problems we will extend the ideas of functional compatibility and condition number to eigenvalue problems. At the end some first ideas about stopping criteria for iterative eigenvalue solvers will be introduced.

Abstract:

The results of my diploma-thesis are presented, a Newton-like algorithm that solves the singular Lur'e-equations in low-rank arithmetics and its advantages in comparison to another method for solving this problem are shown.

Abstract:

Deflation is a technique to improve convergence of Krylov subspace methods for the solution of linear systems. We will consider the case of symmetric (but not necessarily positive definite) matrices in this talk. Eric de Sturler proposed a deflated MinRes method but our investigation revealed that breakdowns can occur. Using our results for MinRes applied to singular but consistent systems, we construct a method not suffering from breakdowns.

Abstract:

In this talk we will continue the discussion of linear, time-invariant DAEs with the property of positivity. We recall the positivity conditions for ODEs and their generalization to the differential part of linear, time-invariant DAEs and point out how the assumption of absolute monotonicity can be avoided. For the algebraic part of a DAE discretization, we will demonstrate how the consistency conditions of the applied method can be used to derive assumptions for positivity.

Abstract:

The aim of this talk is the strangeness-index of a linear time invariant, delay differential-algebraic equation of retarded type $$ E\dot{x}(t)=A0 x(t) + A1 x(t-h) + f(t),$$ based on the structure of matrix triple $(E,A0,A1)$. We will compare the strangeness-index of this system with the strangeness-index of its non-delay part, which is often used as the index of regular delay differential-algebraic equations. In the second part of the talk, we will discuss another strangeness-index concept introduced by the behavior approach and the relation between these strangeness-indices.