Inhalt des Dokuments
AbsolventenSeminar • Numerische Mathematik
Verantwortliche Dozenten: 
Prof. Dr. Christian Mehl [1], Prof. Dr. Volker Mehrmann
[2] 

Koordination: 
AnnKristin Baum 
Termine:  Do 10:0012:00 in MA
376 
Inhalt:  Vorträge von
Diplomanden, Doktoranden, Postdocs und manchmal auch Gästen zu
aktuellen
Forschungsthemen 
Datum  Zeit  Raum  Vortragende(r)  Titel 

Do 18.10.  10:15 Uhr  MA
376  Antje Relitz  Eigenvalue problems for Hamiltonian matrices  The curse
of Van Loan Vorbesprechung 
Do 25.10.  10:15
Uhr  MA 376  Robert
Altmann  Moving Dirichlet Boundary
Conditions 
Do
01.11.  10:15 Uhr  MA
376  AnnKristin Baum  Positivity preserving simulation of DAEs with
variable coefficients 
Do 08.11.  10:15 Uhr  MA 376  Volker Mehrmann Linghui Zeng  Matrix functions
that commute with their derivative Equivalence Transformations for Quadratic Eigenvalue Problems 
Do 15.11.  10:15 Uhr  MA 376   kein Seminar   
Do 22.11.  10:15 Uhr  MA 376  Heiko K. Weichelt Vinh Tho Ma  RiccatiBased Boundary Feedback Stabilization of Flow
Problems Numerical Computation of the Polar Decomposition 
Do
29.11.  10:15 Uhr  MA 376  Lena Scholz Christian Mehl  StructuralAlgebraic Remodeling of Coupled
Dynamical Systems Structured backward errors for eigenvalues of Hermitian pencils 
Do 06.12.  10:15
Uhr  MA 376  Phi Ha
Michal Wojtylak  Analysis and reformulation of linear delay
differentialalgebraic equations On three problems on Hselfadjoint matrices with one eigenvalue of nonpositive type 
Do
13.12.  10:15 Uhr  MA
376  Andreas Steinbrecher Randolf Altmeyer  Regularization via Overdetermined Formulations and
Numerical Simulation of Multibody Systems Modeled with Modelica (I) Regularization via Overdetermined Formulations and Numerical Simulation of Multibody Systems Modeled with Modelica (II) 
Do
20.12.  10:15 Uhr  MA
376  Duc Thuan Do  Stability and robust stability of linear
differentialalgebraic equations with delay 
Do 10.01.  10:15 Uhr  MA 376   kein Seminar   
Do 17.01.  10:15 Uhr  MA 376   kein Seminar   
Do 24.01.  10:15 Uhr  MA 376  Leonhard Batzke Ute Kandler  AntiTriangularizing Talternating
Matrix Polynomials Spectral Error Bounds for an Inexact Arnoldi Method 
Do 31.01.  10:15 Uhr  MA 376  Andre Gaul  Projections, angles and spectra in Hilbert spaces

Do
07.02.  10:15 Uhr  MA
376  Christoph Zimmer Sarosh Quraishi  Adaptive simulation of an elastic
pendulum A dictionary based adaptive finite element approach 
Do
14.02.  10:15 Uhr  MA
376  Jan Heiland Helia Niroomand Rad  Projector chains for
semiexplicit index2 DAEs and
ADAEs 
Jan Heiland (Tu Berlin)
Donnerstag, 14. Februar 2013
Projector chains for semiexplicit index2 DAEs and ADAEs
To start with, we consider a linear differentialalgebraic Equation (DAE) with a finitedimensional state. In a DAE, differential and algebraic equations are interlocked, what makes their analysis and numerical treatment involved. This interlocking is quantified by various index concepts. Simply put, the lower the index, the better the algebraic and differential parts are seperated. Via the projector chain approach by Griepentrog and März*, one can recursively define a projector that is used to decouple the algebraic and differential part of the linear DAE. This is achieved by a scaling of the equations from the right and variable substitutions, but without variable transformations. Having discussed, how this operator chain applies to semiexplicit partially nonlinear index2 DAEs, I will turn towards DAEs with states in Banach spaces, often referred to as abstract DAEs (ADAEs). I will introduce the functional analytical setting, the particular type of ADAEs considered here, and the difficulties that come with the infinite dimensional setting. Then I will show that, under additional regularity conditions, the ADAE has the same solutions as a decoupling of the ADAE via a projector. To assure you that I haven't lost touch with reality, I will illustrate why the chosen approach is very natural if considering the weak formulation of the NavierStokes equations. *Griepentrog, März: DAEs and their numerical treatment, 1986
Christoph Zimmer (TU Berlin)
Donnerstag, 07. Februar 2013
Adaptive simulation of an elastic pendulum
Mathematical models combining rigid and elastic bodies are developed and analysed in recent years. An easy example of a nonrigid body is the elastic pendulum. In this talk, I will present methods for modelling and simulation of an oscillating, flexible bar. I will give a short glimpse of a discretization of an elastic pendulum. After this I will show first numerical results and explain three ways for an adaptive >simulation. Finally I will compare these three methods.
Sarosh Quraishi (TU Berlin)
Donnerstag, 07. Februar 2013
A dictionary based adaptive finite element approach
In this talk we present a dictionary based approach for finite element discretization. We construct a basic dictionary using tensor product of simple functions like multilevel Bspline functions. The special features of solution (like singularities and shape of domain which is available from the problem) are subsequently incorporated into basic dictionary. In this approach the geometric model of the domain and inherent singularities does not influence the mesh generation or mesh refinement. Given the high accuracy of dictionary based discretization, the system size is relatively small and can be solved by a variety of techniques. Finally, we present simple examples to illustrate our approach.
Andre Gaul (Tu Berlin)
Donnerstag, 31. Januar 2013
Projections, angles and spectra in Hilbert spaces
In this talk I will present some fundamental yet very interesting results (old and new ones) from the wonderful world of projections, angles between subspaces and spectral perturbation theory in Hilbert spaces.
Leonhard Batzke
Donnerstag, 24. Januar 2013
AntiTriangularizing Talternating Matrix Polynomials
Talternating matrix polynomials can occur in different applications. While it has recently been shown that any square matrix polynomial is triangularizable, the structurepreserving triangularization of alternating matrix polynomials has not yet been investigated. In this talk, we present an algorithm to create a triangular form for alternating matrix polynomials of the same structure and degree. Unfortunately, this procedure is restricted to the case that all elementary divisors/ eigenvalues are finite, the consequences of allowing infinite elementary divisors and a possible solution to the arising problems are presented as well.
Ute Kandler (TU Berlin)
Donnerstag, 24. Januar 2013
Spectral Error Bounds for an Inexact Arnoldi Method
We investigate the behavior of an inexact Arnoldi method to
approximate eigenvalues, eigenvectors and/or invariant
subspaces corresponding to k<<n eigenvalues of a
Hermitian, complex n by n matrix A.
The dominant operations of
Arnoldi's method are matrixvector products and weighted vector sums
(for the sake of orthogonalization).
Also scalar products and
vector scalings are necessary.
In practice neither
matrixvector multiplication nor vector sums nor vector scaling can
be done exactly. Instead only approximations of the intended
quantities are available. However it is asumed that scalar products
can be evaluated exactly.
Our main goal is to give bounds
on the approximation quality of the approximate eigenvalues,
vectors, and invariant subspaces obtained using an inexact Arnoldi
method. The presented error bounds generalize the known results for
the unperturbed Arnoldi method. Furthermore, as a
prerequisite we also analyze the distance to orthonormality of the
nownotanymore orthogonal basis vectors.
Joint work with
Christian Schroeder
Duc Thuan Do (TU Berlin)
Donnerstag, 20. Dezember 2012
Stability and robust
stability of linear differentialalgebraic equations with
delay
In this talk, we deal with the exponential stability and the stability radius of linear differentialalgebraic equations with delay when the equation is subjected to the structured perturbations. Necessary and sufficient conditions for the exponential stability are studied. Computable formulas for the stability radius are given. Examples are derived to illustrate results.
This is joint work with Volker Mehrmann, Vu Hoang Linh and Nguyen Huu Du
Andreas Steinbrecher (TU Berlin)
Donnerstag, 13. Dezember 2012
Regularization via
Overdetermined Formulations and Numerical
Simulation of
Multibody Systems Modeled with Modelica (I)
In
this talks we will discuss the efficient and robust
numerical
simulation of dynamical systems modeled with Modelica.
We
will present an approach which mainly consists in two steps.
The first step consists in an automatic regularization of the
model
equations provided in Modelica source code. The obtained
overdetermined
formulation satisfies the requirements for a
regularization and, in
particular, is equivalent to the original
DAE in the sense that both
have the same solution set.
The second step in the approach is the subsequent robust and
efficient
numerical integration of the regularized
overdetermined formulation with
adapted discretization methods.
Within the Modelica framework, there currently exist no
numerical
integrator which is suited for overdetermined systems.
This is joint work with Volker Mehrmann and Andreas
Steinbrecher.
Randolf Altmeyer (TU Berlin)
Donnerstag, 13. Dezember 2012
Regularization via
Overdetermined Formulations and Numerical
Simulation of
Multibody Systems Modeled with Modelica (II)
In
this talks we will discuss the efficient and robust
numerical
simulation of dynamical systems modeled with Modelica.
We
will present an approach which mainly consists in two steps.
The first step consists in an automatic regularization of the
model
equations provided in Modelica source code. The obtained
overdetermined
formulation satisfies the requirements for a
regularization and, in
particular, is equivalent to the original
DAE in the sense that both
have the same solution set.
The second step in the approach is the subsequent robust and
efficient
numerical integration of the regularized
overdetermined formulation with
adapted discretization methods.
Within the Modelica framework, there currently exist no
numerical
integrator which is suited for overdetermined systems.
This is joint work with Volker Mehrmann and Andreas
Steinbrecher.
Phi Ha (TU Berlin)
Donnerstag, 06. Dezember 2012
Analysis and reformulation of linear delay
differentialalgebraic 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
differentialalgebraic equations (DelayDAEs).
In this
talk, we study the analysis of linear DelayDAEs from both
theoretical and numerical viewpoints. First, we generalize some
wellknown results in DAEs theory to DelayDAEs.
Then, we
propose algorithms to reformulate a DelayDAE into its underlying
delay system. Moreover, the constructed forms are also used to
address structural properties of the system like solvability,
regularity, consistency and smoothness requirements.
This
is joint work with Volker Mehrmann and Andreas Steinbrecher.
Michal Wojtylak (Jagiellonian University)
Donnerstag, 06. Dezember 2012
On three problems on Hselfadjoint matrices with one eigenvalue of nonpositive type.
Let H be a hermitiansymmetric, invertible matrix with one
negative eigenvalue and let A be Hsymmetric, that is HA=A^*H.
It is well known, that the spectrum of A is real, except the
eigenvalue of nonpositive type, which may be real or complex (in the
latter case its cojugate is an eigenvalue as well). The main
object of the presented research is the study the behavior of this
eigenvalue under a certain change of the matrix A. We consider three
instances
 A is a one dimensional perturbation of a
given matrix (or operator)
 A is a finite dimensional
truncation of a given infinite Jacobi matrix
 A is a
large random matrix.
Apparently, these three
problems can be treated with the same method of Weyl functions
and Nevanlinna functions with one negative square.
The results
are obtained in cooperation with H.S.V. de Snoo (RUG Groningen), H.
Winkler (TU Ilmenau), M. Derevyagin (TU Berlin) and P. Pagacz
(Jagiellonian University).
Lena Scholz (TU Berlin)
Donnerstag, 29. November 2012
StructuralAlgebraic Remodeling of Coupled Dynamical Systems
The automated modeling of multiphysical dynamical systems is usually realized by coupling different subsystems together via certain interface or coupling conditions. This approach results in largescale highindex differentialalgebraic equations (DAEs). Since the direct numerical simulation of these kind of systems leads to instabilities and possibly nonconvergence of the numerical methods a regularization or remodeling of the system is required. In many simulation environments a kind of structural analysis based on the sparsity pattern of the system is used to determine the index and a reduced system model. However, this approach is not reliable for certain problem classes, in particular not for coupled systems of DAEs. We will present a new approach for the remodeling of coupled dynamical systems that combines the structural analysis, in particular the Signature Method, with classical algebraic regularization techniques and thus allows to handle socalled structurally singular systems and also enables a proper treatment of redundancies or inconsistencies in the system.
Christian Mehl (TU Berlin)
Donnerstag, 29. November 2012
Structured backward errors
for eigenvalues of Hermitian pencils
In this
talk, we consider the structured backward errors for eigenvalues of
Hermitian pencils or, in other words, the problem of finding the
smallest Hermitian perturbation so that a given value is an
eigenvalue
of the perturbed Hermitian pencil.
The
answer is well known for the case that the eigenvalue real, but
in the case of nonreal eigenvalues, only the structured backward
error for eigenpairs has been considered so far, i.e., the problem of
finding the smallest
Hermitian perturbation so that a given pair
is an eigenpair of the perturbed Hermitian pencil.
In
this talk, we give a complete answer to the question by reducing the
problem to an eigenvalue minimization problem of Hermitian matrices
depending on two real parameters. We will see that
the
structured backward error of complex nonreal eigenvalues may be
significatly different from the corresponding unstructured backward
error  which is in conrast to the case of real eigenvalues where
the structured and unstructured backward errors coincide.
Heiko K. Weichelt (MPI Magdeburg)
Donnerstag, 22. November 2012
RiccatiBased Boundary Feedback Stabilization of Flow Problems
In order to explore boundary feedback stabilization of flow problems, we consider the(Navier) Stokes equations that describe instationary, incompressible flows for moderate Reynolds (in case of NavierStokes) or viscosity (in case of Stokes) numbers. Following the analytic approach by Raymond [Raymond ’06], we have to find a numerical treatment of the Leray projector. After a finite element discretization we get a differentialalgebraic system of differential index two. We show how to reduce this index with a projection method based on [Heinkenschloss/Sorensen/Sun ’08] and point out the connection to the Leray projector. This leads to generalized state space systems where a linear quadratic control approach can be applied. Avoiding the explicit projection,we end up with largescale saddle point systems which have to be solved. We will show numerical results regarding the arising nested iteration and how the different parameters influence its convergence. Furthermore, we show some examples where we want to apply these ideas in multifield flow problems. Finally, we point out recent problems and tasks related to our approach.The work is developed within the project Optimal ControlBased Feedback Stabilization in MultiField Flow Problems with Eberhard B ̈nsch and Peter Benner as principal investigators. This project is part of the DFG priority program 1253: Optimization With Partial Differential Equations. Furthermore, this is a joined work with Jens Saak(general state space systems), Martin Stoll (preconditioning of iterative solvers) and Friedhelm Schieweck, Piotr Skrzypacz (finite element techniques).
Vinh Tho Ma
Donnerstag, 22. November 2012
Numerical Computation of the Polar Decomposition
The factorization A=UH of a complex matrix A into an isometric matrix U and a Hermitian positive semidefinite matrix H is called polar decomposition.. I am going to summarize some basic properties of this factorization like existence and uniqueness and afterwards I will discuss some numerical methods for computing the polar factor U, e.g. the socalled scaled Newton iteration.
Volker Mehrmann (TU Berlin)
Donnerstag, 08. November 2012
Matrix functions that commute with their derivative
Joint work with Olga Holtz and Hans Schneider
We examine when a matrix whose elements are differentiable functions in one variable commutes with its derivative. This problem was discussed in a letter from Issai Schur to Helmut Wielandt written in 1934, which we found in Wielandt's Nachlass. The topic was rediscovered later and partial results were proved. However, there are many subtle observations in Schur's letter which were not obtained in later years. Using an algebraic setting, we put these into perspective and extend them in several directions. We present in detail the relationship between several conditions mentioned in Schur's letter. We also present several examples that demonstrate Schur's observations.
Linghui Zeng
Donnerstag, 08. November 2012
Equivalence Transformations for Quadratic Eigenvalue Problems
Since there does not exist a simple equivalence transformation which can triangularize or diagonalize a quadratic matrix polynomial, we study more general equivalence transformations for quadratic eigenvalue problems in this paper. In order to well understand the equivalence of quadratic matrix polynomials, we first analyze properties of their decomposable pairs, which are closely related to the equivalence. Furthermore, based on the theory about the equivalence of linear matrix polynomials, we construct equivalence transformations for quadratic matrix polynomials. Specifically, first, linearize a regular quadratic matrix polynomial as a linear matrix polynomial; second, apply a structure preserving transformation to the linear matrix polynomial to get another one; finally, recover a new regular quadratic matrix polynomial from the transformed linear matrix polynomial. In particular, we propose our definitions of structure preserving transformations and analyze their properties. We find that those transformations satisfying some constraints can transform a symmetric linearization for a regular quadratic $Q(lambda)$ into another symmetric linearization for a new regular quadratic $wQ(lambda)$. Meanwhile, we discover that they can preserve the structures of the original structured quadratic matrix polynomial and its structured linearizations.
AnnKristin Baum (TU Berlin)
Donnerstag, 01. November 2012
Positivity preserving simulation of DAEs with variable coefficients
Positive dynamical systems arise in every application in which the
considered variables represent a material quantity that does not
take
negative values, like e.g. the concentration of chemical
and biological
species or the amount of goods and individuals in
economic and social
sciences.
Beside positivity, the
dynamics are often subject to constraints resulting
from
limitation of resources, conservation or balance laws, which extend
the differential system by additional algebraic equations.
In order to obtain a physically meaningful simulation of such
processes,
both properties, the positivity and the constraints,
should be reflected
in the numerical solution.
In this
talk, we discuss these issues for linear timevarying systems, as
they arise for example in the linearization of nonlinear systems in
chemical reaction kinetics or process engineering.
As for linear timeinvariant systems [1], we pursue a projection
approach
based on generalized inverses that admits to separate
the differential and
algebraic components without changing
coordinates.
We first consider index1 problems, in which
the differential and
algebraic equations are explicitly given
and explain under which
conditions we can expect a positive
numerical approximation that meets the
algebraic constraints.
We then extend these results to higher index problems,
i.e., problems in
which some of the algebraic equations are
hidden in the system, using
derivative arrays and the index
reduction developed by Kunkel and Mehrmann
[2].
[1]
Numerical Integration of Positive Linear DifferentialAlgebraic
Systems. A.K. Baum and V. Mehrmann, Preprint TU Berlin, 2012.
www3.math.tuberlin.de/multiphysics/Publications/Articles/BauM12_ppt.pdf
[2] DifferentialAlgebraic Equations. Analysis and
Numerical Solution,
P. Kunkel and V. Mehrmann, EMS Publishing
House, Zuerich, CH, 2006.
Robert Altmann (TU Berlin)
Donnerstag, 25. Oktober 2012
Moving Dirichlet Boundary Conditions
In my last talk we analysed the dynamics of elastic media involving the Dirichlet boundary conditions as a weak constraint. Therefore, we introduced suitable spaces such that the resulting operator DAE has a unique solution. In this talk we derive a suitable model for moving Dirichlet boundary conditions. This means that we demand Dirichlet conditions on a part of the boundary which changes with time. In order to avoid ansatz spaces which depend on time, we need an additional coordinate transformation. In the second part of the talk, a stable discretization scheme is presented.
Antje Relitz (TU Berlin)
Donnerstag, 18. Oktober 2012
Eigenvalue problems for Hamiltonian matrices  The curse of Van Loan
I will present Van Loan's approach to computing all the eigenvalues of a Hamiltonian matrix. For this purpose I will give a short summary of the properties of Hamiltonian and skewHamiltonian matrices and matrices wich preserve their structure under similarity transformations. Afterwards I will explain the steps of the algorithm developed by Van Loan. I will compare the computational cost of Van Loan's algorithm to the QR algorithm and present the results of the error analysis of Van Loan's algorithm.
Rückblick
 Absolventen Seminar WS 2013
 Absolventen Seminar SS 2012 [3]
 Absolventen Seminar WS 2011/2012 [4]
 Diplomanden und Doktorandenseminar SS 2011 [5]
 Diplomanden und Doktorandenseminar WS 2010/11 [6]
 Diplomanden und Doktorandenseminar SS 2010 [7]
 Diplomanden und Doktorandenseminar WS 2009/10 [8]
 Diplomanden und Doktorandenseminar SS 2009 [9]
 Diplomanden und Doktorandenseminar WS 2008/09 [10]
 Diplomanden und Doktorandenseminar SS 2008 [11]
 Diplomanden und Doktorandenseminar WS 2007/08 [12]
 Diplomanden und Doktorandenseminar SS 2007 [13]
 Diplomanden und Doktorandenseminar WS 2006/07 [14]
 Diplomanden und Doktorandenseminar SS 2006 [15]
 Diplomanden und Doktorandenseminar WS 2005/06 [16]
 Diplomanden und Doktorandenseminar SS 2005 [17]
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