# Diplomanden- und Doktorandenseminar Numerische Mathematik SS 2007

 Dozenten: Jörg Liesen, Christian Mehl, Volker Mehrmann, Reinhard Nabben Koordination: Falk Ebert LV-Termine: Do 10-12 in MA 376 Inhalt: Vorträge von Diplomanden, Doktoranden, Postdocs und manchmal auch Gästen zu aktuellen Forschungsthemen

Datum Uhrzeit Raum Vortragende(r) Titel Vorläufige Terminplanung: Do 19.4.2007 10:15 MA 376 --------------------- Vorbesprechung Do 3.5.2007 10:15 MA 376 Daniel Kressner Eigenvalue Computation on the PlayStation 3 (Abstract) Do 10.5.2007 10:15 MA 376 Christian Schröder Optimal execution of transactions and the nearest AFR matrix (Abstract) Do 24.5.2007 10:15 MA 376 Elena Virnik Stability of positive descriptor systems - A toolbox of (simple but nice) tricks (Abstract) Do 31.5.2007 10:15 MA 376 Kathrin Schreiber Nonlinear Rayleigh functionals (Abstract) Do 7.6.2007 10:15 MA 376 Timo Reis Circuit Synthesis - An MNA Approach (Abstract) im An-schluss Rakporn Dokchan Numerical Integration of DAEs with critical points (Abstract) Do 14.6.2007 10:00 MA 376 Eva Abram Index analysis of electro-mechanical systems (Abstract) im An-schluss Tobias Brüll Linear Discrete-Time Descriptor Systems (Abstract) Do 21.6.2007 10:15 MA 376 ---no speakers--- Do 28.6.2007 10:15 MA 376 --------------------- Do 5.7.2007 10:00! MA 376 Lena Wunderlich Trimmed First Order Formulations for Linear Higher Order DAEs (Abstract) im An-schluss Sadegh Jokar Compressed Sensing and Partial Differential Equations (Abstract) Do 12.7.2007 10:00! MA 376 Sander Wahls A Minimum Norm Solution to the Operator Corona Problem (Abstract) im An-schluss Lisa Poppe H∞ Control of Differential-Algebraic Equations (Abstract)

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Abstracts zu den Vorträgen:

Daniel Kressner (TU Berlin)
Eigenvalue Computation on the PlayStation 3
Thu 03.05.2007, 10:15 h in MA 376

Abstract:
The Cell architecture, upon which the recently released PlayStation 3 is based, has been demonstrated to have tremendous potential for scientific computations in terms of both raw performance and power efficiency. Realizing this potential in praxis is challenging, mainly due to the fact that the design of Cell radically differs from conventional multiprocessor or multicore architectures. For example, while PlayStation 3's Cell CPU achieves a peak performance of 204 Gflops in single precision, it only achieves 15 Gflops in double precision. To obtain both high performance and accuracy, it is therefore desirable to do a large part of the computation in single precision before using the considerably slower 64-bit unit. The well-known iterative refinment, which will be reviewed in this talk, provides a convenient framework to achieve this goal. For computing eigenvalues and eigenvectors, however, standard iterative refinment schemes have some limitations, e.g., encountering convergence problems when the eigenvalues become too ill-conditioned. These limitations can be completely avoided if instead of individual eigenvalues the complete Schur decomposition is refined. A novel algorithm will be sketched, which can be seen as a mixture between the Jacobi method and Newton methods for refining invariant subspaces. This is joint work at an early stage with Jack Dongarra.

Christian Schröder (TU Berlin)
Optimal execution of transactions and the nearest AFR matrix
Thu 10.05.2007, 10:15 h in MA 376

Abstract:
We will consider the problem of selling a large amount of shares at the stock market. After introducing the setting we will discuss a model covering the mayor effects. The model will then be solved using techniques from optimal control theory.
In the last part a method to approximate model parameters is presented. This involves finding the rank-1 matrix $B$ that can be written as $B=uv^T$ with $v_i=\frac{1}{u_i}$ that is closest to a given matrix $A$.

Elena Virnik (TU Berlin)
Stability of positive descriptor systems - A toolbox of (simple but nice) tricks
Thu 24.05.2007, 10:15 h in MA 376

Abstract:
We consider differential-algebraic linear homogeneous continuous-time systems. Positivity implies that the solution trajectory is non-negative for all times $t$. In the case of standard positive systems, most classical stability criteria take a simpler form. In this talk we establish a complete correspondence of stability criteria in the case of standard positive systems and in the positive descriptor case.

Kathrin Schreiber (TU Berlin)
Nonlinear Rayleigh functionals
Thu 07.06.2007, 10:15 h in MA 376

Abstract:
After a short review of Rayleigh quotients for Hermitian and general matrices we introduce appropriate Rayleigh functionals $p(u)$ and $p(u,v)$  defined by $(T(p(u))u,u)=0$, $T(p(u,v))u,v)=0$ resp. for nonlinear eigenvalue problems $T(\lambda)x=0$, where $u$, $v$ are approximations for right and left eigenvectors. Local existence and uniqueness of $p$ is shown as well as "stationarity" (technically $p$ is not differentiable). Bounds for the distance of $p$ and the exact eigenvalue are provided, which are of the same order as in the linear case. We give a numerical example of Rayleigh functional iteration and related methods applied to a quadratic problem.

Timo Reis (TU Berlin)
Circuit Synthesis - An MNA Approach
Thu 07.06.2007, 10:15 h in MA 376, first talk

Abstract:
Given is a descriptor system. We consider the following question: Can we perform state space transformations such that the descriptor system has the form of the equations of modified nodal analysis arising in theory of electrical circuits? For the class of passive and reciprocal systems, we will answer this question by giving an algorithm.

Rakporn Dokchan (HU Berlin)
Numerical Integration of DAEs with critical points
Thu 07.06.2007, 10:15 h in MA 376, second talk

Abstract:
We consider linear, time-varying differential-algebraic equations (DAEs) with critical points. By means of projector-based analysis critical points shall be characterized. Assuming the existence of continuous extensions of certain projectors and the density of the regular points, one can introduce harmless critical points that allow us to apply Radau IIA method or BDF method like in the case of regular DAEs.

Eva Abram (TU Berlin)
Index analysis of electro-mechanical systems
Thu 14.06.2007, 10:00 h in MA 376, first talk

Abstract:
We consider two examples of electro-mechanical systems and analyse them. We compare the index of the subsystems with the index of the whole system and check when it differs. Later, an outlook on how to analyse general electro-mechanical systems is given.

Tobias Brüll (TU Berlin)
Linear Discrete-Time Descriptor Systems
Thu 14.06.2007, 10:00 h in MA 376, second talk

Abstract:
We consider linear discrete-time descriptor systems, i.e. systems of linear equations of the form $E_{k+1} x_{k+1} = A_k x_k + f_k$, where $E_k$ and $A_k$ are matrices, $f_k$ are vectors and $x_k$ are the vectors of the solution we are looking for. Analogously to the book "Differential-Algebraic Equations - Analysis and Numerical Solution" by V.Mehrmann and P.Kunkel the existence and uniqueness of solutions is first studied for the constant coefficient case, i.e. where $E_k = E$ and $A_k = A$ and then for the variable coefficient case. A strangeness index is defined for such systems.

Lena Wunderlich (TU Berlin)
Trimmed First Order Formulations for Linear Higher Order DAEs
Thu 05.07.2007, 10:00 h in MA 376, first talk

Abstract:
We consider linear higher order differential-algebraic equations. The classical reduction to linear first order systems leads to different solvability results and higher smoothness requirements. We present trimmed first order formulations for higher order DAEs based on a derivative array approach that allow order reduction without introducing higher smoothness requirements. Further, these trimmed first order formulations allow an explicit representation of solutions for higher order DAEs.

Compressed Sensing and Partial Differential Equations
Thu 05.07.2007, 10:00 h in MA 376, second talk

Abstract:
In signal processing we are often interested in a substitute representation of a signal and seeking some simplification for an obvious gain. This is the rationale behind the so many transforms proposed over the past several centuries, such as the Fourier, cosine, wavelets, and many others. In this talk we are interested in finding sparse solutions of underdetermined linear systems. We will show that sparsity and redundancy can be used to design new/renewed and powerful signal/image processing tools. In this way we review some known results on compressed sensing and some open problems in this area. Then we give some ideas and theoretical results on the possible relation between compressed sensing and partial differential equations. Finally we will give some experimental results on finding the sparse solutions of partial differential equations. Computational experience looks promising.

Sander Wahls (TU Berlin)
A Minimum Norm Solution to the Operator Corona Problem
Thu 12.07.2007, 10:00 h in MA 376, first talk

Abstract:
We consider the Hardy space $H^\infty$ of bounded and analytic operator-valued functions on the complex unit disc. The question whether there exists a right inverse function G in $H^\infty$ for some given F in $H^\infty$ is often termed "Operator Corona Problem". As right inverse functions do not need to be unique we are interested in a right inverse having minimum norm. The talk outlines the construction of such a minimum norm right inverse using the so-called "lurking isometry" method and gives results on the approximation of the minimum norm whose a-priori knowledge is required in the construction.

Lisa Poppe (TU Berlin)
H Control of Differential-Algebraic Equations
Thu 12.07.2007, 10:00 h in MA 376, second talk
 Impressum Falk Ebert 07.06.2007