Diplomanden- und Doktorandenseminar
Numerische Mathematik SS 2007

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Weitere Vorträge siehe auch:

• Kolloquium der Arbeitsgruppe Numerik, Modellierung, Differentialgleichungen
• Seminar des Fachgebiets Optimierung bei partiellen Differentialgleichungen (Gruppe Tröltzsch)

• 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:
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.

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 $A$.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Abstract