Numerik-Oberseminar WS 2003/04

Interessenten sind herzlich eingeladen!

Weitere Vorträge siehe auch:

• Diplomanden- und Doktorandenseminar des Fachgebiets Numerische Mathematik (Gruppe Mehrmann)
• Seminar des Fachgebiets Optimierung bei partiellen Differentialgleichungen (Gruppe Tröltzsch)

Rückblick:

• Numerik-Oberseminar SS 2003
• Numerik-Oberseminar WS 2002/03
• Numerik-Oberseminar SS 2002
• Numerik-Oberseminar WS 2001/02
• Numerik-Oberseminar SS 2001

Abstracts zu den Vorträgen:

Abstract:
A general framework for analyzing convergence of proximal-like methods for variational inequalities with set-valued maximal monotone operators is developed, including discretization of the space and data approximation (operators and feasible sets).
This approach is devoted to methods coupling successive approximation of the variational inequality with the proximal point algorithm as well as to related methods using regularization on a subspace and/or weak regularization.
The convergence results are proved under mild assumptions with respect to the original variational inequality and admit, in particular, the use of the ε-enlargement of the operator and the use of non-quadratic regularization functionals. The latter permit us to deal with methods having an interior point effect.
Taking into account the specific structure of non-differentiable terms in energy functionals of several problems in mathematical physics, we analyze the construction of ε-enlargements for some special operators.
The connection between proximal point method and the Auxiliary problem principle will be issued, which leads to several splitting algorithms.

Abstract:
Given a matrix A of size n, and a fixed perturbation matrix E, the effect of linear perturbations A+tE as t varies over the complex numbers, on certain spectral properties of A is analyzed. The special effect of the matrix E on these properties is exhibited. A geometric framework is developed for spectral analysis of A+tE to achieve this goal. It is shown that this framework leads to a better understanding of the sensitivity of eigenvalues and spectral decompositions of A. Finally, a set of necessary and sufficient conditions for the spectrum of A to be equal to the spectrum of A+tE for all complex numbers t is provided.

Abstract:
Local defect correction (LDC) is an iterative method for solving elliptic boundary value problems on composite grids based on simple data structures and simple discretization stencils on uniform or tensor-product grids. In the LDC method, the discretization on the composite grid is based on a combination of standard discretizations on several uniform grids with different grid sizes that cover different parts of the domain.
LDC converges very fast; in practice, one or two iterations are sufficient to reach the fixed point. The convergence behavior of the method has been analyzed for a model problem, Poisson's equation on the unit square with standard five-point finite difference discretizations on uniform grids.
The standard LDC method has been combined with multi-level adaptive gridding and domain decomposition. The domain decomposition algorithm provides a natural way for parallelization and enables the usage of many small tensor-product grids rather than a single large unstructured grid. It has been shown that this may greatly reduce memory usage.
The properties above will be illustrated by applying the adaptive multi-level LDC algorithm with domain decomposition to a combustion problem. The mathematical model is a system of nonlinear partial differential equations with strongly nonlinear chemical source terms. The solutions of the system have large gradients in a very small part of the domain and are smooth elsewhere.

Abstract:
Modernes Halbleiterdesign stellt im wesentlichen drei Ansprüche an den angewandten Mathematiker: Die Entwicklung von geeigneten Modellen zur Beschreibung der physikalischen Effekte, die Bereitstellung von numerischen Verfahren zur Simulation der Modellgleichungen und neuerdings die Umsetzung von Optimierungsstrategien zur schnellen Berechnung von optimalen Designvorschlägen. In diesem Vortrag sollen die ersten beiden Punkte anhand des Quanten Drift Diffusionsmodells erläutert werden. Es werden insbesondere geeignete Zeit- und Ortsdiskretisierungen vorgestellt. Den dritten Punkt betreffend wird ein neuer Optimierungsalgorithmus präsentiert, der es z.B. erlaubt das optimale Design eines MESFET Bauteils mit äußerst geringem Aufwand zu berechnen.

Abstract:
Recent research on switched and hybrid systems has resulted in a renewed interest in determining conditions for the existence of a common quadratic Lyapunov function for a finite number of stable LTI systems. While efficient numerical solutions to this problem have existed for some time, compact analytical conditions for determining whether or not such a function exists for a finite number of matrices have yet to be obtained. In this talk we present a geometric approach to this problem. By making one simplifying assumption we obtain a compact time-domain condition for the existence of such a function for a pair of matrices.
Our conditions also relate the existence of such a function to the stability boundary of the underlying switched linear system (thereby indicating that requiring the existence of a such a function does not, in a certain sense, lead to overly conservative stability conditions). We show that classical Lyapunov results can be obtained using our framework. In particular, we obtain simple time-domain versions of the SISO Kalman-Yacubovich-Popov lemma, the Circle Criterion, and stability multiplier criteria. Finally, we indicate how our approach can be used to analyse n-tuples of LTI systems and present preliminary results on the existence of common non-quadratic Lyapunov functions of a certain form.

Abstract:
For the solution of sparse linear systems from circuit simulation, parallel flexible iterative methods with distributed Schur complement preconditioning are presented. The parallel efficiency of the solvers is increased by exploitation of parallel graph partitioning methods. The costs of local, incomplete LU decompositions are decreased by fill-in reducing reordering methods of the matrix. The efficiency of the parallel solvers is demonstrated for real circuit simulation runs with NEC's circuit simulator MUSASI.

Abstract:
Aggressive early deflation is a QR algorithm deflation strategy that takes advantage of matrix perturbations outside of the subdiagonal entries of the Hessenberg QR iterate. It identifies and deflates converged eigenvalues long before the classic small-subdiagonal strategy would. The small-bulge multi-shift QR sweep with aggressive early deflation maintains a high rate of execution of floating point operations while significantly reducing the number of operations required. We will discuss variations on aggressive early deflation and revisit the question of how best to choose shifts and where to expect deflations.

Abstract:
Let A be a simple matrix and d(A) be the distance of A from the set of defective matrices. The determination of d(A) and a defective matrix A' such that ||A-A'|| = d(A) is widely known as Wilkinson's problem. We characterize the nearest defective matrices, analyze their structure and describe a simple guaranteed way to compute d(A).

Abstract:
The talk is a synopsis of my PhD thesis on spectral value sets. These sets, also called structured pseudospectra, are unions of spectra of perturbed matrices of the form A -> A+BDC, where A,B and C are fixed matrices and the matrix D an element of a given class of matrices (perturbation class). The content of the talk is the following.

1. The general framework (the connection between spectral value sets, stability radii and μ-functions)
2. Real perturbations, real perturbation values and hermitian-symmetric inequalities
3. Real pseudospectra of normal matrices
4. Pseudospectra of upper triangular matrices and an open problem
5. Root sets of uncertain polynomials
6. Spectral value sets for composite systems and the eigenvalue inclusion theorem of Brualdi
7. Eigenvalue condition numbers for structured perturbations
8. An almost trivial remark on the distance of a matrix pair (A,B) to the set of uncontrollable pairs

Abstract:
Rational Krylov is a development of the shifted and inverted Arnoldi algorithm where several shifts (poles) are used in one run. Two variants will be described. The first one transforms the matrix pencil into a pencil of two Hessenberg matrices, the second gives a new Arnoldi factorization each time the shift is moved. The first is a natural alternative for Model reduction, while the second is used to solve an eigenvalue problem that is nonlinear in the eigenvalue parameter.
Results, taken from the thesis of Patrik Hager, are reported for test examples coming from finite element approximations modelling viscously damped vibrating structures.