Pierre-Antoine Absil:   Stiefel manifolds and their applications

The Stiefel manifold St(p,n) is the set of all orthonormal p-frames in $R^n$. We will give a basic introduction to the geometry of St(p,n), viewed as an embedded submanifold of the set of nxp matrices, as a quotient of orthogonal groups, and also as the total space of a fiber bundle over the Grassmann manifold. Some geometric questions on Stiefel manifolds will be reviewed, such as the choice of a Riemannian metric and the initial-value and two-point boundary-value geodesic problems. An important part of the talk will be dedicated to presenting applications of St(p,n) in component extraction algorithms.

Costas Bekas:   Massively Parallel Mixed Precision Iterative Refinement Solution of Linear Systems with Applications

We present a quadratic complexity Iterative Refinement framework for the solution of very large dense SPD linear systems. The method relies on iterative internal solvers rather than matrix factorizations. We apply this framework in the context of uncertainty quantification for the estimation of the diagonal of inverse covariance matrices. Thanks to the quadratic complexity of the overall scheme and its excellent scaling and performance characteristics, that reach hundreds of TFLOPS, we are able to address unprecedented problem sizes.

Matthias Bolten:   Preconditioning systems arising from the KKR Green function method

The Korringa-Kohn-Rostoker Green function method is another aproach for the calcuation of electron structures. While these calculations are known to be computationally exepensive, recently a linear scaling method has been proposed. This method uses QMR to solve a linear system with multiple right hand sides. The convergence rate depends on the physical properties of the discretized system, resulting in slow convergence for interesting configurations. To overcome this limitation, we are currently investigating preconditioning techniques. In this talk we will introduce the systems at hand and provide some preliminary results we achieved for the preconditioned QMR method.

Christos Boutsidis:   Unsupervised Feature Selection for the k-means Clustering Problem

We discuss the topic of feature selection for the $k$-means clustering problem. We present a novel unsupervised feature selection algorithm that, given $n$ points described with respect to $d$ features (modeled in the $n \times d$ matrix A), and the number of clusters $k$, randomly selects and returns $O(k \log(k))$ features. Our method carefully explores the Singular Value Decomposition of $A$ in order to identify the most ``influential'' and ``discriminative'' features. We argue that running a $k$-means approximation algorithm on the selected features we can find a partition of the rows of $A$ that is a constant factor approximation to the optimal partition. To our knowledge, this is the first feature selection method for $k$-means clustering that comes with provable quality-of-approximation guarantees.

Tobias Breiten:   Krylov Subspace Methods for Model Order Reduction of Bilinear Control Systems

In this talk we will discuss the use of Krylov subspace methods with regard to the problem of model order reduction. We will focus on bilinear time-invariant control systems of the following form \begin{equation*} \Sigma:\mbox{\quad} \left\{ \begin{aligned} \dot{x}(t) & = Ax(t) + \sum_{j=1}^m{N_jx(t)u_j(t)} + Bu(t), \\ y(t) & = Cx(t), \quad x(0)=x_0 \end{aligned}\right. \end{equation*} with $A,N_j \in \mathbb R^{n \times n},B \in \mathbb R^{n \times m}, C \in \mathbb R^{p \times n}, x(t) \in \mathbb R^n, u(t)=[u_1(t),\dots,u_m(t)]^T \in \mathbb R^{m}.$ There already exist some approaches to this topic. However, all of them base on the matching of so-called low frequency multimoments which result from multidimensional Taylor series expansions at zero. We will show that there are other possible choices of the expansion points. It will turn out that these approaches can be seen as bilinear generalizations of the problems known as rational interpolation and partial relization, respectively. We want to compare these new ideas with some existent approaches and show that they indeed yield satisfying results and should be preferred in some cases.

Tobias Brüll:   Checking dissipativity of linear systems in behavior form

Dissipativity and systems in behavior form have been studied throughoutly by J.C.Willems. However, a computationally feasible method to check dissipativity of linear systems given in behavior form seems to be not known so far. In this talk it will be shown that dissipativity of controllable linear systems in behavior form is closely related to the existence of purely imaginary eigenvalues of a certain even matrix pencil. In particular, the results can be used to check passivity and contractivity of arbitrary (i.e., potentially rectangular and thus singular) controllable descriptor systems.

Angelika Bunse-Gerstner:   Balanced truncation based methods for unstable linear systems

Balanced truncation based methods for unstable linear systems For large scale linear dynamical systems it is often desirable to find reduced order systems whose short to medium term input-output behaviour is close to the one of the original system. In some applications the large scale systems are asymptotically unstable, and for this case only few automatic numerical model reduction methods have been developed so far. We recently proposed a balanced truncation based method approximating the large scale dynamical system with respect to the so-called $H_{\infty,\alpha}$ norm. Here we give a motivation for this specific approach and we compare it with balanced truncation based methods used so far for such problems. joint work with Nancy K. Nichols, University of Reading, GB

Paul Constantine:   Spectral Methods for Parameterized Matrix Equations

Katrijn Frederix:   Large eigenvalue problems and spectral clustering

For an overview of spectral clustering methods, we refer the interested reader to [1]. When performing spectral clustering on large data sets, one has to look for the eigenvectors corresponding to the largest eigenvalues of a matrix whose size depends on the number of data points. This number can be huge. A possibility to avoid solving this large eigenvalue problem is to approximate the similarity matrix by a low rank matrix and then solve the reduced eigenvalue problem. In this talk we will investigate how we can adaptively look for a suitable rank for this approximation. The study will be illustrated by numerical examples.\\{} [1] U. von Luxburg. A tutorial on spectral clustering. \emph{Statistics and Computing}, 17:395--416, 2007.

Chen Greif:   Augmentation Preconditioners for Saddle Point Linear Systems

Saddle point linear systems arise in a variety of constrained PDE and optimization problems. When these systems are very large and sparse, iterative methods must be used to compute a solution. A challenge here is to derive and apply preconditioners that exploit the properties and the structure of the underlying operators, and yield fast convergence while imposing reasonable storage requirements. In this talk we focus on augmentation preconditioners, which are based on incorporating the primal Schur complement of a related, regularized saddle point matrix. This preconditioning approach seems to work well in situations where the (1,1) block is singular or ill-conditioned, but in order for it to be computationally efficient an effective approximation of the Schur complement is required. We discuss spectral properties, bounds on convergence, and computational qualities, and illustrate our finding on a semidefinite programming problem and other applications.

Ralf Hiptmair:   Operator Preconditioning

Operator preconditioning offers a general recipe for constructing preconditioners for discrete linear operators that have arisen from the Galerkin approach of operators on function spaces. The key idea is to employ matching Galerkin discretizations of operators of complementary mapping properties. If these can be found, the resulting preconditioners will be robust with respect to the choice of the bases for trial and test spaces. I am going to survey the application of operator preconditioning to finite elements and boundary elements.

Alfred Inselberg:   Visual Linear Algebra and Multidimensional Geometry

${\cal I}$n parallel coordinates (abbr. $\|$-cs) a point in $\mathbb{R}^N$ is represented by a polygonal line (polyline). Polylines representing collinear points \begin{equation} \nonumber \label{eq:ij} \ell_{i,j} : x_i = m_{i,j} x_j + b_{i,j} \quad \longleftrightarrow \quad \bar{\ell}_{i,j} : (\frac{j-i}{1-m_{i,j}} + (i-1), \frac{b_{j-i}}{1-m_{i,j}}) \end{equation} intersect at a single point $\bar{\ell}_{i,j}$ given above in Cartesian coords. A {\bf line $\ell \subset \mathbb{R}^N$ is represented by $(N-1)$ independent points} $\bar{\ell}_{i,j}$. For $i \not = j \not = k$, the points $\bar{\ell}_{ij}, \bar{\ell}_{ik}, \bar{\ell}_{jk}$ are on a line $\bar{L}$. This is an instance where {\bf linear dependence} between linear relations is manifested as {\bf collinearity} of points. {\bf Coplanarity} is recognized from the {\bf {\em lines} $\ell$ on a plane} $\pi \subset \mathbb{R}^3$ by the $\bar{L}$ intersecting at a point $\bar{\pi}_{123}$. A second intersection $\bar{\pi}_{1'23}$, with the same $y$-coord, is obtained via an axis translation. These {\bf two points represent the plane}. Their 3 indices distinguish them from points with two indices representing lines. There is a family of 2-flats, the {\bf super-planes}, whose points appear in $\|$-cs as straight rather than polygonal lines. From their nice properties the {\bf representation of hyperpanes in $\mathbb{R}^N$ by $(N-1)$ points with $(N + 1)$ indices} is efficiently obtained. {\em Linear dependence} (containment) between flats is recognized from the {\em collinearity} of their indexed points providing pleasing intersection algorithms and {\em translation} $\leftrightarrow$ {\em rotation} dualities. Smooth hypersurfaces in $\mathbb{R}^N$ are represented by their tangent hyperplanes, or equivalently by their normal vectors, generating $(N-1)$ linked planar regions. Each point must be correctly matched with the $(N-2)$ others altogether representing a proper tangent hyperplane. From the resulting patterns surface features, usually hidden in other representations, like folds, cusps, bumps and {\em {\bf non-orientability}} are revealed. {\bf \em Convexity} can be recognized in any dimension. Intuition gained from the $\mathbb{R}^3$ representations leads to the to $\mathbb{R}^N$ generalizations with beautiful new {\bf dualities like: {\em cusp} in $\mathbb{R}^N \; \leftrightarrow \; (N-1)$ {\em "swirls"} in $\mathbb{R}^2$~, \, {\em twist} in $\mathbb{R}^N \; \leftrightarrow \; (N-1)$ {\em cusps} in~$\mathbb{R}^2$.} Several interactive demonstrations are included in the presentation.

Additional material

Elias Jarlebring:   Computing all pairs $(\lambda,\mu)$ such that $\lambda$ is a double eigenvalue of $A+\mu B$

Consider the structured matrix distance problem of finding pairs $(\lambda,\mu)\in\mathbb{C}^2$ such that $\lambda$ is a double eigenvalue of $A+\mu B$ for given matrices $A,B\in\mathbb{C}^{n\times n}$. Many problems related to matrices with double non-semisimple eigenvalues are badly conditioned because of the typically infinite sensitivity of the eigenvalues. As a first result we show that the problem is structured in such a way that $(\lambda,\mu)$ is generically well conditioned with respect to perturbations in $A$ and $B$. The set of solutions is (under mild conditions) finite and we construct a method which converges \emph{globally} and simultaneously to \emph{all} solutions. The method is based on the observation that the problem of finding $\mu$ such that $\lambda$ and $\lambda(1+\varepsilon)$ are eigenvalues of $A+\mu B$ can be stated as a \emph{two-parameter eigenvalue problem}, \begin{eqnarray*} Av_1&=&(\lambda I- \mu B)v_1\\ Av_2&=&(\lambda(1+\varepsilon) I-\mu B)v_2 \end{eqnarray*} where $\varepsilon$ is a small fixed non-zero constant. The available methods for two-parameter eigenvalue problems allow us to solve the problem reliably. We also show that the method convergences to the solution of the original problem and that the convergence speed with respect to $\varepsilon$ is generically of R-order two. The regularization parameter $\varepsilon$ is non-zero. This implies that the pairs $(\lambda,\mu)$ do not have full accuracy. We propose a Newton type method which we use to correct non-semisimple eigenvalues as well as semisimple eigenvalues to high accuracy. This is joint work with Simen Kvaal (U. Oslo) and Wim Michiels (K.U. Leuven).

Effrosyni Kokiopoulou:   Graph-based dimensionality reduction with repulsion Laplaceans

Graph-based methods for linear dimensionality reduction have recently attracted much attention and research efforts. The main goal of these methods is to preserve, after reduction, the properties of a graph representing the affinity between data points in the input high-dimensional space. In the context of classification, each data point is associated with a class label and the problem is to predict the class label of a new data point based on previously seen ones and their corresponding labels. It has been observed that, in general, supervised graph-methods outperform their unsupervised peers in various classification tasks. Supervised graphs are typically constructed by allowing two data points to be adjacent only if they are of the same class. However, such graphs are oblivious to the proximity of data from different classes. We propose a methodology which builds on ``repulsion graphs'', i.e., graphs that model undesirable proximity between points. The main idea is to repel points from different classes that are close by in the input high-dimensional space. The proposed methodology is generic and can be combined with any graph-based method for linear dimensionality reduction. We provide ample experimental evidence, which shows that the proposed methodology offers significant performance gains to different graph-based methods. This is joint work with Prof. Y. Saad (University of Minnesota, USA).

Piotr Krzyzanowski:   On block preconditioners for generalized saddle point problems with singular/indefinite (1,1) block

We will consider a class of block preconditioners (including block diagonal, triangular and symmetric indefinite preconditioners) for symmetric algebraic saddle point problems, where the (1,1) block matrix may be indefinite or singular. We will show conditions under which the convergence rate of the Preconditioned Conjugate Residuals method is independent of the discretization mesh parameter.

Manoj Kumar:   Computer Simulation of a class of Singular Boundary Value Problems Arising in Science and Engineering

Here, we present an efficient numerical algorithm for solving singular two-point linear and non-linear problems, which is based on the Modified Adomian Decomposition Method (MADM). Also, we proposed a new operator for solving singular boundary value problems, which gives less error comparative to MADM and other existing techniques in the literature at neighborhood of the right boundary. To illustrate the effectiveness, the algorithm is tested on two linear and two non-linear examples and obtained results using Mathematica 6.0 demonstrate the reliability and efficiency of the proposed algorithm.

Ofer Levi:   Perfect diagonal preconditioners for non-uniform Discrete Fourier Transforms

Non uniform Discrete Fourier Transforms (DFT) play very important role in variety of practical inverse problems. Unlike the uniform version where the operator is orthogonal, non uniform DFT's are known to be highly ill conditioned. In this work we show how the special structure of the operator's matrix can be facilitated to build a left diagonal preconditioner that pefectly orthogonalizes the product columns. We define the required conditions for which such a preconditioner exists, prove its existance and provide an efficient iterative method for computing the precondioner diagonal elements. We treat both 1D and 2D cases. We also review some important applications of non uniform DFT's such as reconstruction problems in medical and biological imaging, rigid body transformation and more.

Thomas Mach:   LR-Cholesky Transformations for Hierarchical Matrices and other Algorithms to Compute the Eigenvalues of Hierarchical Matrices

We investigate the use of Rutishauser's LR-Cholesky transformation to compute all eigenvalues of symmetric hierarchical ($\mathcal{H}$-) matrices. Historically, the LR transformation was the first algorithm of the class of GR algorithms. \begin{align*} &L_{m+1} L^T_{m+1} = M_m - \mu_m I\\ &M_{m+1} = L^T_{m+1} L_{m+1} = L_{m+1}^{-1} M_m L_{m+1}. \end{align*} The hierarchical matrix format allows storing a variety of dense matrices from certain applications in a special data-sparse way with linear-polylogarithmic complexity. Most $\mathcal{H}$-arithmetic operations, including the $\mathcal{H}$-Cholesky decomposition, have linear-polylogarithmic complexity, too. We substitute the Cholesky decomposition in the LR-Cholesky transformation by the $\mathcal{H}$-Cholesky decomposition. In principle this works, since the limit of the iterative process is a diagonal matrix and hence an $\mathcal{H}$-matrix, too. But some of the intermediate iterates may not have good $\mathcal{H}$-matrix approximations. We discuss ideas to overcome this difficulty. We further discuss alternative approaches to solve the eigenvalue problem for hierarchical matrices. This is a joint work Peter Benner. The work is supported by a grant of the Free State of Saxony.

Volker Mehrmann:   On the perturbation of a system to a nearby passive system

Perturbations of Hamiltonian matrices are studied that move eigenvalues off the imaginary axis. This problem is important in the passivation of control systems. We give explicit formulas for the minimal perturbations to move purely imaginary eigenvalues of different sign characteristics together and also to move multiple purely imaginary eigenvalues to specific values in the complex plane. We also present a numerical method to compute upper bounds for the minimal perturbations and discuss how to employ these ideas in the context of passivation.

Agnieszka Miedlar:   A homotopy approach for the adaptive solution of PDE-eigenvalue problems

We introduce a new adaptive finite element algorithm for PDE-eigenvalue problems combined with a homotopy approach to determine a few eigenpairs of non-selfadjoint eigenvalue problems for PDEs. In order to reduce the complexity while retaining good accuracy for symmetric problems the adaptive algorithms with guaranteed computable bounds in the algebraic part are used. However, large, sparse non-symmetric real matrices may have complex eigenvalues, occurring as complex conjugate pairs. Unfortunately, in this case the min-max theorem does not hold and we are not able to use standard eigenvalue solvers without additional information about the spectrum. The homotopy method allow us to control the convergence process and detect whether a conjugate pair appeared. Our new algorithm combines those two techniques to solve non-selfadjoint eigenvalue problems with desired accuracy and reasonable complexity. \begin{flushright}This is a joint work with J. Gedicke.\end{flushright}

Keiichi Morikuni:   Preconditioning Krylov subspace methods using inner iterations for least squares problems

Consider solving large sparse least squares problems, where the coefficient matrix may be rank deficient and the problem may be inconsistent. We consider performing inner iterations in order to precondition the normal equation before applying Krylov subspace methods for solving such problems. The advantage of using inner iteration is that we can avoid computing, constructing, and storing the preconditioners. Hence we can save memory and accelerate the convergence of the outer iteration. We propose using stationary iterative methods such as the SOR and Kaczmarz(Cimmino) methods based on the normal equation. We show by numerical experiments on a variety of problems that the proposed approach is competitive compared to previous methods especially for highly ill-conditioned problems. Joint work with Xiaoke Cui and Ken Hayami.

Reinhard Nabben:   Multilevel Krylov methods

We develop a new type of multilevel methods to solve linear systems of equations. These methods are called multilevel Krylov methods (MK methods). The basic idea of this type of methods is to shift small eigenvalues that are responsible for slow convergence to an a priori fixed constant. The shifting of the eigenvalues is similar to projection type or deflation type methods and uses the solution of subspace or coarse level systems. Numerical results show that these MK methods work very well for convection-diffusion equations and the Helmholtz equation. The convergence can be made almost independent of grid size h and also only mildly dependent of the wavenumber k for the Helmholtz equation. We also combine the multilevel Krylov idea with the algebraic way of choosing the coarse-grid system and the restriction and prolongation operators. The resulting method is called algebraic multilevel Krylov method or AMK method. This is a joint work with Yogi A. Erlangga.

Gerhard Opfer:   The computation of zeros of quaternionic polynomials

Quaternionic polynomials may have two types of zeros. Isolated zeros and so-called spherical zeros. The spherical zeros consist of an equivalence class of infinitely many zeros, which can be described by just one representative. We shall show, how to compute, numerically, all zeros, regardless of the type. We will also give an overview on various attempts to find these zeros. Some of the results are based on work by Pogorui and Shapiro, 2004. The paper is written jointly with Drahoslava Janovsk\'a, Prague.

Juan Manuel Pena:   Totally nonnegative matrices and iterative methods

A matrix is called totally nonnegative if all its minors are nonnegative. These matrices present important applications to many fields, and the interest of iterative methods for these matrices is motivated by problems of Computer Aided Geometric Design. We present results on the convergence of iterative methods for totally nonnegative matrices. Preconditioning is also considered.

Miro Rozloznik:   Rounding error analysis of partitioned triangular tridiagonalization

We consider a partitioned algorithm for reducing the symmetric matrix $A$ to tridiagonal form, which computes a factorization $PA P^T = L T L^T$ where $P$ is a permutation matrix, $L$ is lower triangular with a unit diagonal and bounded off-diagonal elements, and $T$ is symmetric tridiagonal. We show that such a partitioned factorization is backward stable provided that the corresponding growth factor is not too large (the entries can grow in the factor $T$). The only slight change with respect to the basic (nonpartitioned ) algorithm is in the constant that includes the the size of partition which, on the other hand, allows to exploit modern computer architectures through the use of the level-3 BLAS. Experimental results demonstrate that such algorithm achieves approximately the same level of performance as the blocked Bunch-Kaufman code implemented in Lapack. The Bunch-Kaufman method is also conditionally backward stable (assuming no or moderate growth in triangular factors) making these two main approaches comparable also from the numerical stability point of view.

Christian Schroeder:   A software package to antitriangularize a matrix

In some applications, e.g. the palindromic eigenvalue problem, for a given square matrix $A$ one has to find a unitary matrix $U$ such that $U^*AU=R$ is anti triangular, that is $r_{ij}$ vanishes whenever $i+j<=n$. While this is not possible in general, every matrix can be transformed to block anti triangular form. One algorithm for this task, the so-called palindromic Laub trick, has been implemented in the software package PLAUBPACK. My talk gives an overview on the palindromic eigenvalue problem and the palindromic Laub trick and focusses on the features of the software (like support for mixed precision arithmetic).

Hubert Schwetlick:   Parametrized Perturbation Results for Nonlinear Eigenvalue Problem

Consider the nonlinear eigenvalue problem $T(\lambda,\varepsilon)=0$ with $T:\,\mathbb{C}\times\mathbb{R}^p\mapsto \mathbb{C}^{n\times n}$ where $\varepsilon$ is a vector of real perturbation parameters. Under the assumption that $\lambda_*$ is a simple eigenvalue of $T(\lambda,0)$, first order perturbation results for both the eigenvalue $\lambda=\lambda(\varepsilon)$ and the corresponding right and left eigenvectors $x=x(\varepsilon)$ and $y=y(\varepsilon)$ of $T(\lambda,\varepsilon)$ are derived by using appropriate primal and dual singularity systems.

Meiyue Shao:   A supernode approach to incomplete LU factorization

We present a new supernode-based incomplete LU factorization method to construct a preconditioner for solving sparse linear systems with iterative methods. The new algorithm is primarily based on the ILUTP approach by Saad, and we incorporate a number of techniques to improve the robustness and performance of the traditional ILUTP method. These include new dropping strategies that accommodate the use of supernodal structures in the factored matrix. Numerical experiments demonstrate that our new method is competitive with the other ILU approaches and is well suited for today's high performance architectures.

Valeria Simoncini:   Spectral properties of preconditioned saddle point linear systems

In the past few decades there has been significant interest in the algebraic analysis of matrices in the form $$ {\cal A} = \left [ \begin{array}{cc} A & B^\top \\ B & -C \end{array}\right ], $$ and of their {\it preconditioned} version ${\cal P}^{-1}{\cal A}$, with the nonsingular matrix ${\cal P}$ specifically chosen. These structured matrices arise in the solution of algebraic linear systems stemming from saddle point problems in a variety of applications. By means of various characterizations of the involved matrices $A$, $B$ and $C$, most results aim at analyzing the spectrum of ${\cal P}^{-1}{\cal A}$ when ${\cal P}$ has a block structure similar to that of ${\cal A}$. Main requirements are that ${\cal P}^{-1}$ be cheap to apply and that ${\cal P}^{-1}{\cal A}$ have {\it favorable} spectral properties. In this talk we discuss some recent results in the spectral algebraic analysis of matrices in the form ${\cal P}^{-1}{\cal A}$, under various hypotheses on the blocks of $\cal A$.

Martin Stoll:   Preconditioning for PDE-constrained optimization

Solving problems from the optimal control of partial differential equations typically leads to the solution of linear systems in saddle point form. We will present methods for solving the all-at-once formulation of the PDE-constrained optimization problem. In particular, our focus will be on short-term recurrence methods such as MINRES or the Bramble-Pasciak CG and the construction of suitable preconditioners. In addition, we will show that these techniques carry over to the case when control constraints are introduced. Numerical results will illustrate the competitiveness of this approach.

Zdenek Strakos:   Approximation of the reflection coefficients in scatterometry using Krylov subspace methods

Diffraction of light on periodic media represents an important problem with numerous physical and engineering applications. The Rigorous Coupled Wave Analysis (RCWA) method assumes a specific form of gratings which enables a straightforward separation of space variables. Using Fourier expansions, the solutions of the resulting systems of ordinary differential equations for the Fourier amplitudes can after truncation be written in a form of matrix functions, with an elegant formulation of the linear algebraic problem for integrating constants. In our contribution we focus on subsequent estimation of reflection coefficients using the matching moments model reduction.

Zoran Tomljanovic:   Damping optimization of linear vibrating systems using dimension reduction

We will consider a mathematical model of a linear vibrational system \[ M\ddot{x}+D\dot{x}+Kx=0, \qquad x(0)=x_0, \qquad \dot x(0)=\dot x_0 \] where $M$ and $K$ are mass and stiffness, respectively, which are positive definite matrices of order $n$. Damping matrix is $D=C_u+C$, where $C_u$ represents internal damping and $C$ is external damping which is positive semidefinite matrix. Problem of determination of optimal damping is equivalent to minimization of the trace of the solution of the Lyapunov equation \[ A X + X A^T = - G G^T\,, \] where $A$ is matrix $2n \times 2n$ obtained from $M, D$ and $K$. Finding $C_0$ such that \[C_0=\displaystyle{\rm arg}\min\limits_{C}\, {\rm trace}\, X(C) \] is very demanding problem, caused by large number of trace calculations which can be required for bigger matrix dimensions. Thus we propose dimension reduction to accelerate optimization process. Our reduction is based on the construction of the matrix $Q_r$, of dimension $n\times r$, which contains $r$ columns of the matrix $\Phi$ which simultaneously diagonalize pair $(M,K)$. The trace approximation is obtained by solving Lyapunov equation which comes form reduced system $M_r\ddot{x}+D_r\dot{x}+K_rx=0$, where $M_r = Q_r^T M Q_r, D_r = Q_r^T D Q_r, Kr = Q_r^T K Q_r$. We will present a new result about a proper choice of $Q_r$, which allow us to control accuracy of trace approximation. \\ \\ Joint work with Peter Benner \footnote{Mathematics in Industry and Technology, Department of Mathematics, Chemnitz University of Technology, Chemnitz, Germany, {\tt benner@mathematik.tu-chemnitz.de}} and Ninoslav Truhar \footnote{Department of Mathematics, J.J. Strossmayer University of Osijek, Croatia, {\tt ntruhar@mathos.hr}}

Olivier Verdier:   The Kronecker Theorem and indices of DAEs

We will show a new proof of the Kronecker Theorem which allows a straightforward derivation of the popular index concepts used for differential algebraic equations.

Heinrich Voss:   Efficient Computation of the L-Curve in Large Scale Regularized Total Least Squares

The total least squares (TLS) method is a successful approach for linear problems if both the system matrix and the right hand side are contaminated by some noise. For ill-posed TLS problems regularization is necessary to stabilize the computed solution. In this paper we suggest the use of the L-curve for the determination of the regularization parameter. Setting up the L-curve is easy if the singular value decomposition can be obtained cheaply. We are interested in large scale problems where computing the SVD is too expensive. Values on the L-curve are obtained by solving a Regularized Total Least Squares (RTLS) problem each time. Within the sequence of RTLS problems it is highly favorable to reuse information from previously solved problems. The Nonlinear Arnoldi method with the use of thick starts turned out to be a powerful solver for this kind of problems. Numerical examples demonstrate the efficiency of the approach. This is joint work with J{\"o}rg Lampe.

Jens-Peter M Zemke:   IDR($s$)ORes and eigenvalue computations

We focus on the eigenvalue perspective of the IDR family, a recent family of Krylov subspace methods derived by Sonneveld and van Gijzen [SIAM J. Sci. Comput. Vol. 31, No. 2, pp. 1035-1062 (2008)]. We present five mathematically equivalent ways to extract information about eigenvalues using the quantities computed by IDR. This is joint work with Martin Gutknecht.