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

Koordination: 
AnnKristin Baum, Benjamin Unger 
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 16.10.  10:15 Uhr  MA 376  Vorbesprechung  
Namita Behera  Fiedler
Linearizations for LTI StateSpace Systems and for Rational Eigenvalue Problems  
AnnKristin
Baum  A flowonmanifold formulation
ofdifferentialalgebraic equations.Application to positive
systems.  
Do 23.10.  10:15 Uhr  MA
376  Benjamin Unger  Analysis of the Structure of the POD Basis

Pratibhamony Das  A priori and a posteriori uniform error estimates based on
moving meshes for singularly perturbed problems  
Do
30.10.  10:15 Uhr  MA 376   kein Seminar
  
Do 06.11.  10:15 Uhr  MA
376  Christian Schröder  A special topic in Krylov subspaces 
 
Do
13.11.  10:15 Uhr  MA 376  Cornelia
Gamst  Maxwell's eigenvalue problem for
photonic crystals 
Phi
Ha  Analysis and Numerical solutions of
DelayDAEs  
Do 20.11.  10:15 Uhr  MA
376   kein Seminar   
Do 27.11.  10:15 Uhr  MA
376   kein Seminar   
Do 04.12.  10:15 Uhr  MA
376  Michael Overton  Investigation of Crouzeix's Conjecture via
Optimization 
Volker
Mehrmann  When is a linear control system
equivalent to a portHamiltonian system  
Do
11.12.  10:15 Uhr  MA 376  Michał Wojtylak
 On deformations of classical Jacobi
matrices 
Do 18.12.  10:15 Uhr  MA
376  Christoph Conrads  The OffDiagonal Block Method 
Helia Niroomand Rad  TimeHarmonic
Scattering Problem Considering The Pole
Condition  
Do 08.01.  10:15 Uhr  MA
376   kein Seminar   
Do 15.01.  10:15 Uhr  MA
376  Judith Simon  Structure Preserving Discretization Methods for
SelfAdjoint DifferentialAlgebraic Equations 
Matthias Voigt  The
KalmanYakubovichPopov Inequality for DifferentialAlgebraic
Equations  
Do 22.01.  10:15 Uhr  MA
376  Jeroen Stolwijk  Error Analysis for the Euler Equations in Purely Algebraic
Form 
Do 29.01.  10:15 Uhr  MA
376  Philipp Schulze  StructurePreserving Model
Reduction 
Robert
Altmann  Optimal Control for Problems with
Servo Constraints  
Do 05.02.  10:15 Uhr  MA
376  Leo Batzke  Rankk Perturbations of Hamiltonian Matrices 
Sarosh Quraishi  Parametric study of a brake squeal using machine learning
 
Do 12.02.  10:15 Uhr  MA
376  Ute Kandler  Computation of extreme eigenvalues of a largescale
symmetric eigenvalue problem 
Deepika Gill  MultiMode Control
of Tall Building using Distributed Multiple Tuned Mass
Dampers 
Rückblick
 Absolventen Seminar SS 14 [3]
 Absolventen Seminar WS 13/14 [4]
 Absolventen Seminar SS 13 [5]
 Absolventen Seminar WS 12/13 [6]
 Absolventen Seminar SS 12 [7]
 Absolventen Seminar WS 11/12 [8]
Namita Behera (TU Berlin)
Donnerstag, 16. Oktober 2014
Fiedler Linearizations for
LTI StateSpace Systems and for
Rational Eigenvalue
Problems
In this seminar, we introduce a family of linearizations, which we
refer to as
Fiedler linearizations, of the Rosenbrock system
matrix of an LTI system in state space form for computation of
transmission and invariant zeros of the system. We define
linearizations for the transfer function of the LTI system and show
that under appropriate assumptions a Fiedler linearization of the
system matrix is also a linearization of the transfer function. Thus,
given a rational eigenvalue problem, we reformulate the problem of
computing eigenvalues of a rational matrix function to that of
computation of transmission zeros of an LTI state space system. Hence
we compute the eigenvalues by solving a generalized eigenvalue problem
for Fiedler pencil of the system matrix.
AnnKristin Baum (TU Berlin)
Donnerstag, 16. Oktober 2014
A flowonmanifold formulation ofdifferentialalgebraic equations.Application to positive systems.
Differentialalgebraic equations (DAEs) are coupled systems of
differential and algebraic equations that model dynamical processes
constrained by auxiliary algebraic conditions, like e.g. connected
joints in multibody systems, connections or loops in networks or
balance equations and conservation laws in advectiondiffusion
equations. Considering applications in economy, social sciences,
biology or chemistry, the analyzed values typically represent
nonnegative quantities like the amount of goods, individuals or the
density of a chemical or biological species. This leads to positive
systems, i.e., systems for which every solution that starts with a
componentwise nonnegative initial value remains nonnegative for its
lifetime.
In this talk, we derive a systematic description of
positive systems that allows to validate if a given model possesses
the property of positivity. We generalize the notion of a flow from
ordinary differential equations to DAEs and give an explicit solution
formula using a projection approach. We characterize flow invariant
sets for DAEs and derive a uniform framework of flow invariant sets
for implicit and explicit differential equations. Considering the
nonnegative orthant, in particular, we thus give a uniform description
of positive systems, constrained or unconstrained.
Benjamin Unger (TU Berlin)
Donnerstag, 23. Oktober 2014
Analysis of the Structure of the POD Basis
Over the past two decades model reduction of very high dimensional systems, arising from physical measurements or generated by partial differential equations, has become increasingly important to the CFD and optimal control community. In this talk, we outline the general idea of reduced order modeling (ROM) and focus on Proper Orthogonal Decomposition (POD) as a particular technique. A analysis of the structure of the POD basis is performed and conclusions concerning the efficiency of the method are drawn. Moreover, the nonlinear ROM technique Discrete Empirical Interpolation Method (DEIM) is analyzed with respect to different discretisation techniques.
Pratibhamony Das (TU Berlin)
Donnerstag, 23. Oktober 2014
A priori and a posteriori uniform error estimates based on moving meshes for singularly perturbed problems
In this talk, I will give a small introduction of singularly perturbation and moving mesh methods. Then, a nonlinear parametrized problem will be considered to show the graphical differences between a priori and a posteriori generated meshes. Thereafter, a posteriori errror estimate based on equidistribution principle will be considered for system of reaction diffusion problems. If time permits, a two parametric parabolic convection diffusion problem will be considered for convergence analysis on moving meshes.
Christian Schröder (TU Berlin)
Donnerstag, 06. November 2014
A special topic in Krylov subspaces
I will talk about a recent project dealing with
numerical linear algebra, Krylov subspaces and Arnoldi's recurrence.
Cornelia Gamst (TU Berlin)
Donnerstag, 13. November 2014
Maxwell's eigenvalue problem for photonic crystals
Photonic crystals are periodic optical nanostructures that affect the motion of photons through their geometry. They can be modeled as infintely periodic geometric structures in order to calculate their characteristic resonant wavelengths for the electromagnetic fields. The model can be reduced to the eigenvalue problem of Maxwell's equations with a form depending on the parameters of the material and geometry. In this talk I will give a description of the mathematical model for photonic crystals and illustrate some results on the existence of solutions for different examples of material and geometric parameters. Afterwards, I will consider the finite element method for solving the resulting eigenvalue problem and present some results on error estimation for simple cases of material and geometric parameters and indicate directions for further work regarding more complicated cases of parameters.
Phi Ha (TU Berlin)
Donnerstag, 13. November 2014
Analysis and Numerical solutions of
DelayDAEs
Differentialalgebraic equations
(DAEs) have an important role in modeling practical systems, wherever
the system needs to satisfy some algebraic
constraints due to
conservation laws or surface conditions. On the other hand,
timedelays occur naturally in various dynamical systems, both
physically, when the transfer phenomena (energy, signal, material) is
not instantaneous, and artificially, when a timedelay is used in the
controller. The combination of differentialalgebraic equations and
timedelays leads to a new mathematical object: "delay
differentialalgebraic equations (DelayDAEs)", which is a source
of many complex behavior.
In this talk, we address the
computational problem for numerical solutions to general linear
DelayDAEs.
First, we discuss the characteristic properties,
which have not been mentioned in prior studies of numerical solutions
to DelayDAEs.
Then, we propose an algorithm, which extends the
classical (Bellman) method of steps, to determine the solution of
general linear DelayDAEs.
Second, we examine the functionality
and the efficiency of our algorithm by some illustrative examples
Michael Overton (Courant Institute, NYU)
Donnerstag, 04. Dezember 2014
Investigation of Crouzeix's
Conjecture via Optimization
We investigate a
challenging problem in the theory of nonnormal matrices
called
Crouzeix's conjecture, which we will explain in some detail.
We
present experimental results using nonsmooth optimization and
CHEBFUN,
a very useful tool for computing with functions in
MATLAB.
Volker Mehrmann (TU Berlin)
Donnerstag, 04. Dezember 2014
When is a linear control system equivalent to a portHamiltonian system Joint work with Christopher Beattie and Hongguo Xu The synthesis of system models describing physical phenomena often follows a systemtheoretic network paradigm that formalizes the interconnection of naturally specified subsystems. When the models of dynamic subsystem components arise from variational principles, the aggregate system model typically has structural features that characterize it as a portHamiltonian system. We discuss the structure of linear portHamiltonian systems and then answer the question when there exists a state space transformation that turns a general linear inputoutput system into a portHamiltonian system.
Michał Wojtylak (Jagiellonian University, Cracow)
Donnerstag, 11. Dezember 2014
On deformations of classical Jacobi matrices
Let A be an infinite tridiagonal Hermitian matrix. We will try to reveal the spectrum of the product HA, where H=diag(1,1,1,...). Our main interest will lie in locating the (unique) nonpositive eigenvalue of HA, i.e. the unique eigenvalue with the eigenvector x with x'Hx less or equal zero. It appears that this eigenvalue can be computes as a limit of eigenvalues of nonpositive type of the finite sections of the matrix HA. The character of the convergence will be discussed in detail. The research is motivated by a problem in signal analysis: detecting damped oscillations in a highly noisy signal. Joint work with Maxim Derevyagin (University of Mississippi).
Christoph Conrads (TU Berlin)
Donnerstag, 18. Dezember 2014
The OffDiagonal Block Method
The OffDiagonal Block Method We present a new method for finding eigenspaces of large, sparse, symmetric positive semidefinite matrices. The method is well suited for computer implementation and can be used, f. e., to compute eigenpairs anywhere in the spectrum of the matrix or to calculate approximations to all matrix eigenvalues. We will explain how the method works, compare it to existing methods, and present numerical examples.
Joint work with Volker Mehrmann.
The slides can be found at
ftp://ftp.math.tuberlin.de/pub/numerik/conrads/ConradsODBM20141218.pdf [9]
Helia Niroomand Rad (TU Berlin)
Donnerstag, 18. Dezember 2014
TimeHarmonic Scattering Problem Considering The Pole Condition
Scattering problem in the field of wave propagation describes a large variety of problems in acoustic and electromagnetics. In such problems, in order to have a proper formulation for the propagation in an unbounded domain, one may consider the radiating condition. In addition, to deal with the problem numerically within a bounded computational domain, one should impose an artificial boundary with a suitable boundary condition illustrating the radiating condition, which is canonically described by the pole condition. In this talk, we mainly show that one can apply the Laplace technique leading to the pole condition, and it is followed by some existence and uniqueness results.
Judith Simon (TU Berlin)
Donnerstag, 15. Januar 2015
Structure Preserving Discretization Methods for SelfAdjoint DifferentialAlgebraic Equations
We consider a fundamental problem in control theory of minimizing a
cost functional, subject to constraints that are modeled by a
differentialalgebraic equation with timevariant coefficient
matrices. This problem is called the linear quadratic optimal control
problem. Developing numerical methods for finding an optimal solution
is an important and difficult task in relation with
differentialalgebraic equations. In general, there are two different
approaches, which still form a big field in research. Firstly, one can
apply common optimization techniques to the problem and discretize the
resulting system afterwards. On the other hand, it is also possible to
discretize the initial problem first and then apply optimization
techniques. Here, we focus on the former approach.
It has been
shown that the necessary optimality conditions of the discretetime
optimal control problem underly certain properties,
i.e., the
optimality system is a selfconjugate operator associated with
selfadjoint triples of coefficient matrices. We analyze the
continuoustime setting and are partly able to show the same result
if we apply particular discretization methods to the continuoustime
optimality system. Finally, we examine the results with the help of an
example from the multibody mechanics.
Matthias Voigt (TU Berlin)
Donnerstag, 15. Januar 2015
The KalmanYakubovichPopov Inequality for
DifferentialAlgebraic Equations
Abstract: The
KalmanYakubovichPopov lemma is one of the most famous results in
systems and control theory. Loosely speaking, it states
equivalent conditions for the positive semidefiniteness of a
socalled Popov function on the imaginary axis in terms of the
solvability of a certain linear matrix inequality, namely the
KalmanYakubovichPopov (KYP) inequality. In applications, this lemma
plays an important role in assessing feasibility of linearquadratic
optimal control problems or characterizing dissipativity of linear
control systems.
In the literature, there exist manifold
attempts to generalize this lemma to differentialalgebraic
equations. However, most of these approaches make certain restrictive
assumptions such as a bounded index or impulse controllability. In
this talk we show how to drop these restrictions by considering the
KYP inequality on the system space, i.e., the subspace in which the
solution trajectories of the system evolve. Moreover, we present
results on the solution structure of this inequality. In particular,
we consider rankminimizing, stabilizing, and extremal solutions.
These results can then be interpreted as a generalization of the
algebraic Riccati equation to a very general class of
differentialalgebraic control systems.
Jeroen Stolwijk (TU Berlin)
Donnerstag, 22. Januar 2015
Error Analysis for the Euler Equations in
Purely Algebraic Form
Natural gas plays a
crucial role in the energy supply of Europe and the world. It is
sufficiently and readily available, is traded, and is storable. After
oil, natural gas is the second most used energy supplier in Germany.
The high and probably increasing demand for natural gas calls for a
mathematical modeling, simulation, and optimisation of the gas
transport through the existing pipeline network.
The gas
flow through a pipeline can be accurately modeled by the
onedimensional Euler equations. However, their numerical solution
requires much computational effort, such that several simplifications
are often performed. The Euler equations in purely algebraic form are
analysed in this presentation. More specifically, an error analysis is
performed for both the mass flux and the temperature (the pressure was
analysed in the previous talk). We ask ourselves: Are rounding and
measurement errors amplified in the solution? This question is
answered both theoretically and statistically.
Finally,
we will investigate whether the model can be further simplified by
assuming that the temperature is constant. Which condition(s) should
be satisfied such that this assumption can be made safely?
Philipp Schulze (TU Berlin)
Donnerstag, 29. Januar 2015
StructurePreserving Model Reduction
In the past decades, model order reduction has gained increasing attention in all fields where numerical simulations are performed. Especially, in largescale optimisation and control applications, reduced order models are needed to allow reasonable computation times and storage requirements while still maintaining an acceptable level of accuracy. When applying standard model reduction techniques, important properties of the original model (e.g. stability, passivity) are in general not included in the reduced model. Since these properties are often related to certain structures, this issue motivates for the field of structurepreserving model reduction. In this talk, we consider the Loewner framework, proposed by Mayo and Antoulas (2004), which is a technique for generalised realisation but may also be applied for the purpose of model reduction. The task is to extent this approach in order to preserve the socalled /portHamiltonian/ (pH) structure which very often arises by nature when modelling physical systems. Some first results regarding linear timeinvariant pH systems are presented.
Robert Altmann (TU Berlin)
Donnerstag, 29. Januar 2015
Optimal Control for Problems with Servo Constraints
Consider a robot with an end effector which should follow a prescribed trajectory. The corresponding model equations lead to a DAE structure with a socalled servoconstraint. If the input and output variables do not 'fit well', we obtain a system which is of index 5. In this talk, we consider an alternative approach using an optimal control setting.
Leo Batzke (TU Berlin)
Donnerstag, 05. Februar 2015
Rankk Perturbations of Hamiltonian
Matrices
Many applications give rise to
Hamiltonian matrices, that
is, matrices H with H^TJ=JH, where J
is skewsymmetric and
invertible. It is our goal to determine
the change of the canonical
form of H when it is subjected to a
generic (`typical') perturbation
that preserves the Hamiltonian
structure.
Previously, generic Hamiltonian rank1 perturbations
were investigated
by Mehl, Mehrmann, Ran, and Rodman and it was
shown that they may
produce different results from
nonHamiltonian perturbations.
Even so, it is a nontrivial task
to extend these results to the case
of rankk perturbations. In
this talk, we will discuss the
difficulties of going from rank1
perturbations to ones of arbitrary
rank and then derive a
generic canonical form for Hamiltonian matrices
under
Hamiltonian rankk perturbations employing a new technique.
Sarosh Quraishi (TU Berlin)
Donnerstag, 05. Februar 2015
Parametric study of a brake squeal using machine learning
In this talk we review POD (proper orthogonal decomposition) based model reduction for parametric studies of brake squeal and address some shortcomings and areas that need further improvement. I also present some preliminary results on using machine learning (classification using Support Vector Machine (SVM)) for the task of classifying parameter values responsible for a brake squeal using a dataset generated from a minimal model for brake squeal [1].
Reference:
[1] Utz von Wagner, Daniel Hochlenert, and Peter
Hagedorn, Minimal models for disk brake squeal. Journal of Sound and
Vibration 302 (2007), 527539.
Ute Kandler (TU Berlin)
Donnerstag, 12. Februar 2015
Computation of extreme eigenvalues of a
largescale symmetric eigenvalue problem
We
consider a computation which approximates a few minimal eigenvalues of
a symmetric largescale eigenvalue problem. The application we have in
mind comes from strongly correlated quantum spin systems where we can
assume that the eigenvectors are representable in a lowrank tensor
format. We use the tensor train format (TT) for vectors and matrices
in order to overcome the curse of dimensionality and to make storage
and computation feasible. The eigenstates of interest are computed by
the minimization of a block Rayleigh quotient which is performed in an
alternating scheme for all dimensions.
Deepika Gill (TU Berlin)
Donnerstag, 12. Februar 2015
MultiMode Control of Tall Building using Distributed Multiple Tuned Mass Dampers
In the past decades, tuned mass damper (TMD) has gained increasing attention in the control of structural vibrations. The TMD has a mass, a spring and a dashpot connected to main system. The natural frequency of the TMD is tuned to natural frequency of the main system, the vibrations of the main system causes the damper to vibrate in resonance, and as a result, the vibration energy is dissipated through the damping in the TMD. The main disadvantage of a single TMD is its sensitivity of the effectiveness to the error in the natural frequency of the structure. This sensitivity can be reduced by multiple tuned mass dampers (MTMD). The basic configurations of MTMD consist of large number of dampers whose natural frequencies are distributed around the natural frequency of the controlled mode of structure and all dampers are placed at the topmost floor. The MTMD configurations result into placement intricacies. The problem of placement is solved by use of distributed multi tuned mass dampers (dMTMDs). The dampers of dMTMDs are distributed in the vertical direction along the height of building.
In this talk, we will review TMD, MTMD and dMTMDs. Moreover, we will see variation of different responses such as displacement, acceleration etc. for TMD, MTMD and dMTMDs. We will also discuss equation of motion for TMD, MTMD and dMTMDs and its solution using Newmark’s beta method.
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