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• Absolventen Seminar WS 12/13 [3]
• Absolventen Seminar SS 12 [4]
• Absolventen Seminar WS 11/12 [5]

Solvability analysis and reformulation of general Linear Delay Differential-Algebraic Equations

Delay differential equations (DDEs) arise in a variety of applications, including physical systems, biological systems and electronic networks. If the states of the physical system are constrained, e.g., by conservation laws or interface conditions, or some economical interest are involved in the biological model, then algebraic equations have to be included and one has to analyze delay differential-algebraic equations (Delay-DAEs).

In this talk, we present our recent result on the solvability analysis of general linear Delay-DAEs, i.e., system with linear time variable coefficients. We propose the method to reformulate a Delay-DAE into its underlying delay system, which can be used to address structural properties of the system like existence and uniqueness of a solution, consistency and smoothness requirements.

In the second part of the talk, if time permits, we shall consider some spectral properties of Linear Delay-DAEs and their relations with the solvability and stability analysis of the system.

This is the joint work with Volker Mehrmann and Andreas Steinbrecher.

Numerical Methods for Complex T-even Eigenvalue Problems

We investigate when T-even matrix polynomials allow a structure preserving linearization and develop an appropriate condensed form for T-even matrix polynomials.  Then a structure preserving numerical method for the complex T-even polynomial eigenvalue problems is presented.

Low-Rank Perturbations of Alternating Matrix Pencils

Many applications give rise to alternating matrix pencils. In this talk, structure-preserving low-rank perturbations of alternating (i.e. symmetric / skew-symmetric) matrix pencils will be investigated. While unstructured perturbations are well-understood, in the structured case not all canonical forms are 'allowed' for the perturbed pencil. Hence, the generic result deviates substantially from the unstructured case. We will derive a generic canonical form for alternating matrix pencils under symmetric rank-1 and skew-symmetric rank-2 perturbations.

Low rank perturbations of hermitian pencils.

Let A+zE be a liner pencil with matrices A,E being hermitian-symmetric. The following family of linear pencils  A+tB+zE where t is a real parameter and B is a low rank perturbation of A, will be considered. In particular, the parameter dependence of the eigenvalues and of the canonical forms will be discussed.
Special attention will be put to the case when the pencil is silngular for some values of t. In addition, a few facts infinite-dimensional counterpart of the problem will be recalled.

Stopping Criteria for Flow Equations

One of the key issues in efficient computations is to balance the various error components. This may include discretization, linearization, algebraic, or quadrature errors. In this talk, I give a short overview of a recent paper by A. Ern and M. Vohralik, where stopping criteria are derived for nonlinear diffusion problems. Therein, the technique of flux reconstruction is used to distinguish several error components. In the second part of the talk, I present a first attempt to adopt this method for the unsteady Stokes equation.

Modeling the Crosstalk Phenomenon within Electromagnetic Systems

In this talk, after introducing the crosstalk phenomenon in electromagnetic systems, in particular for electrical circuits, we will model the crosstalk for two neighboring close electrical circuits via presenting a coupled bilateral system. This system is obtained by coupling two unilateral subsystems each describing partially the crosstalk in the field theory and in the circuit theory. We will present Maxwell’s equations and modified nodal analysis formalism as the tools to describe the crosstalk in the field and in the circuit theory, respectively.

Eigenvalue problems arising from FE modeling of squealing disk brake Sqealing in a car brake is a very common problem, the origin of which is unstable friction induced self-excited vibration. The simplified governing equation is a quadratic eigenvalue problem with a parameter dependence. The eigenvalues with positive real part correspond to unstable modes which result in brake squeal. In this talk I will present the traditional approach of dimension reduction and a new proper orthogonal decomposition based model reduction which takes into account the parametric dependence of matrices and is more accurate than traditionally used approach, but at the same time computationally more expensive.

Structure And Sign Preserving Transformations Between Real T-even Quadratics And Real Hamiltonian Matrices

In this talk, we will give a sufficient and necessary condition for structure and sign preserving transformations between real T-even quadratics and real Hamiltonian matrices.

Localisation techniques for eigenvalue problems via diagonal dominance

From the simple observation of Geršgorin that eigenvalues of a given matrix lie in the union of disks centred in its diagonal entries, many eigenvalue localisation techniques have been developed through the years. In this talk, we will consider a certain unifying approach to this subject via classes of diagonally dominant matrices, and present few general principles that cover large range of singular results published by many authors. In addition, Geršgorin-type localisation technique for generalised eigenvalue problem will be defined and its underlying principles presented. To conclude the talk, some issues concerning stability will be discussed.

Hierarchically enriched adaptive finite element method for PDE eigenvalue/eigenvector approximations

In this talk we present an adaptive finite element method for PDE eigenvalue
problems which exploits hierarchical basis functions to improve the quality of
the approximated eigenpairs. Starting from the results of Grubiv{s}i'{c} and Ovall
on the reliable and efficient asymptotically exact a posteriori hierarchical error estimators in the self-adjoint case,
we explore the possibility to use the enhanced Ritz values and vectors to restart the iterative algebraic procedures
within the adaptive algorithm. Using higher order hierarchical polynomial finite element bases, as
indicated by Bank and by Ovall and Le Borne, our method yields a structure which is particularly suitable
for designing computational algorithms with low complexity. Some preliminary numerical results will
present the potential of this approach.

Automated Multilevel Substructuring: Introduction and Properties [6]

In this talk we will present Automated Multilevel Substructuring (AMLS), a method to compute /many/ eigenvalues for large, sparse generalized eigenvalue problems.

We will introduce you to the method using an important engineering problem and then we will walk you through all of the steps of the method. We will also talk about the useful properties of AMLS and its applications.

Afterwards, you will know about the suffering of the engineers, a new generalized eigenvalue solver, when you should consider using AMLS, and what it is able to accomplish. The method will be illustrated by some practical examples obtained with our implementation of AMLS.

Enforcing Solvability of a certain Nonlinear Matrix Equation

I will talk about current work with Tobias, Federico and his friend Giacomo about perturbing the coefficients of the matrix equation
 X+AX^{-1}A^T=B
in order to make sure that a solution exists. I will also show you an application in econometrics (i.e., the art of predicting Wall Street).