University, Diss., 2004--Tübingen.

As shown by Dahmen, Harbrecht and Schneider, the fully discrete wavelet Galerkin scheme for boundary integral equations scales linearly with the number of unknowns without compromising the accuracy of the underlying Galerkin scheme. The supposition is a wavelet basis with a sufficiently large number of vanishing moments. In this paper we present several constructions of appropriate wavelet bases on manifolds based on the biorthogonal spline wavelets of A. Cohen, I. Daubechies and J.-C. Feauveau. By numerical experiments we demonstrate that it is worthwhile to spent effort on their construction to increase the performance of the wavelet Galerkin scheme considerably.

This paper is intended to present wavelet Galerkin schemes for the boundary element method. Wavelet Galerkin schemes employ appropriate wavelet bases for the discretization of boundary integral operators. This yields quasisparse system matrices which can be compressed to O(N_J) relevant matrix entries without compromising the accuracy of the underlying Galerkin scheme. Herein, O(N_J) denotes the number of unknowns. The assembly of the compressed system matrix can be performed in O(N_J) operations. Therefore, we arrive at an algorithm which solves boundary integral equations within optimal complexity. By numerical experiments we provide results which corroborate the theory.

We focus on the programming aspect of the treatment of digitized geometries for subsequent use in mesh-free and mesh-based numerical solvers. That perspective includes the description of our C/C++ implementations which use OpenGL for the visualization and MFC classes for the user interface. We report on our experience about implementing with the IGES interface which serves as input for storage of geometric information. For mesh-free numerical solvers, it is helpful to decompose the boundary of a given solid into a set of four-sided surfaces. Additionally, we will describe the treatment of diffeomorphisms on four-sided domains by using transfinite interpolations. In particular, Coons and Gordon patches are appropriate for dealing with such mappings when the equations of the delineating curves are explicitly known. On the other hand, we show the implementation of the mesh generation algorithms which invoke the Laplace-Beltrami operator. We start from coarse meshes which one refine according to generalized Delaunay techniques. Our software is also featured by its ability of treating assembly of solids in B-Rep scheme.

Functions of bounded variation are most important in many fields of mathematics. This thesis investigates spaces of functions of bounded variation with one variable of various types, compares them to other classical function spaces and reveals natural “habitats” of BV-functions. New and almost comprehensive results concerning mapping properties like surjectivity and injectivity, several kinds of continuity and compactness of both linear and nonlinear operators between such spaces are given. A new theory about different types of convergence of sequences of such operators is presented in full detail and applied to a new proof for the continuity of the composition operator in the classical BV-space. The abstract results serve as ingredients to solve Hammerstein and Volterra integral equations using fixed point theory. Many criteria guaranteeing the existence and uniqueness of solutions in BV-type spaces are given and later applied to solve boundary and initial value problems in a nonclassical setting. A big emphasis is put on a clear and detailed discussion. Many pictures and synoptic tables help to visualize and summarize the most important ideas. Over 160 examples and counterexamples illustrate the many abstract results and how delicate some of them are. / Funktionen beschränkter Variation sind in vielen Bereichen der Mathematik besonders wichtig. Diese Dissertation untersucht Räume von Funktionen einer Variable von beschränkter Variation unterschiedlichen Typs, vergleicht sie mit klassischen Funktionenräumen und enthüllt natürliche „Lebensräume“ von BV-Funktionen. Neue und umfassende Ergebnisse über Abbildungseigenschaften wie Surjektivität und Injektivität, verschiedene Arten von Stetigkeit und Kompaktheit von linearen und nichtlinearen Operatoren zwischen solchen Räumen werden präsentiert. Eine neue Theorie über verschiedene Konvergenzarten von solchen Operatoren wird entwickelt und schließlich auf einen neuen Beweis für die Stetigkeit des Kompositionsoperators im klassischen BV-Raum angewendet. Diese abstrakten Ergebnisse dienen als Zutat für die Lösung von Hammerstein- und Volterra-Integralgleichungen mithilfe von Fixpunktsätzen. Diese liefern viele Kriterien, welche die Existenz und Eindeutigkeit von Lösungen garantieren, die sodann auf Anfangs- und Randwertprobleme in einem nichtklassischen Setting angewendet werden. Besonders Augenmerk liegt auf einer klaren und detaillierte Darstellung. Viele Abbildungen und Tabellen helfen, die wichtigsten Ideen zu visualisieren und zusammenzufassen. Über 160 Beispiele und Gegenbeispiele illustrieren die abstrakten Ergebnisse und zeigen deren Grenzen.

Viele Probleme der Riss- und Bruchmechanik sowie der mathematischen Physik lassen sich auf Lösungen von singulären Integralgleichungen über einem Intervall zurückführen. Diese Gleichungen setzen sich im Wesentlichen aus dem Cauchy'schen singulären Integraloperator und zusätzlichen Integraloperatoren mit festen Singularitäten in den jeweiligen Kernen zusammen. Zur numerischen Lösung solcher Gleichungen werden polynomiale Kollokations-Quadraturverfahren betrachet. Als Ansatzfunktionen und Kollokationspunkte werden dabei gewichtete Polynome und Tschebyscheff-Knoten gewählt. Die Gewichte sind so gewählt, dass diese das asymptotische Verhalten der Lösung in den Randpunkten widerspiegeln. Mit Hilfe von C*-Algebra Techniken, werden in dieser Arbeit notwendige und hinreichende Bedingungen für die Stabilität der Kollokations-Quadraturverfahren angegeben. Die theoretischen Resultate werden dabei durch numerische Berechnungen anhand des Problems der angerissenen Halbebene und des angerissenen Loches überprüft.

In this thesis we deal with the degree of ill-posedness of linear operator equations in Hilbert spaces, where the operator may be decomposed into a compact linear integral operator with a well-known decay rate of singular values and a multiplication operator. This case occurs for example for nonlinear operator equations, where the local degree of ill-posedness is investigated via the Frechet derivative. If the multiplier function has got zeroes, the determination of the local degree of ill-posedness is not trivial. We are going to investigate this situation, provide analytical tools as well as their limitations. By using several numerical approaches for computing the singular values of the operator we find that the degree of ill-posedness does not change through those multiplication operators. We even provide a conjecture, verified by several numerical studies, how these multiplication operators influence the singular values of the operator equation. Finally we analyze the influence of those multiplication operators on the opportunities of Tikhonov regularization and corresponding convergence rates. In this context we also provide a short summary on the relationship between nonlinear problems and their linearizations. / Diese Arbeit beschaeftigt sich mit dem Grad der Inkorrektheit linearer Operatorgleichungen in Hilbertraeumen, die sich als Komposition eines vollstetigen linearen Integraloperators mit bekannter Abklingrate der Singulaerwerte und eines Multiplikationsoperators darstellen lassen. Dieser Fall tritt beispielsweise bei nichtlinearen Operatorgleichungen auf, wobei der lokale Inkorrektheitsgrad ueber die Frechetableitung bestimmt wird. Falls die Multiplikatorfunktion Nullstellen hat, so ist die Bestimmung des lokalen Grades der Inkorrektheit nicht einfach. Moeglichkeiten und Grenzen der Analysis fuer diese Situation werden betrachtet. Unterschiedliche numerische Ansaetze fuer die Bestimmung der Singulaerwerte liefern, dass der Grad der Inkorrektheit durch die Multiplikationsoperatoren nicht veraendert wird. Es wird sogar ein Zusammenhang angegeben, wie Multiplikationsoperatoren die Singulaerwerte beeinflussen. Schliesslich werden Moeglichkeiten der Tikhonov-Regularisierung unter Einfluss der Multiplikationsoperatoren untersucht. In diesem Zusammenhang wird auch eine kurze Zusammenfassung zur Beziehung von nichtlinearen Problemen und ihren Linearisierungen gegeben.

In this paper matrix compression techniques in the context of wavelet Galerkin schemes for boundary integral equations are developed and analyzed that exhibit optimal complexity in the following sense. The fully discrete scheme produces approximate solutions within discretization error accuracy offered by the underlying Galerkin method at a computational expense that is proven to stay proportional to the number of unknowns. Key issues are the second compression, that reduces the near field complexity significantly, and an additional a-posteriori compression. The latter one is based on a general result concerning an optimal work balance, that applies, in particular, to the quadrature used to compute the compressed stiffness matrix with sufficient accuracy in linear time. The theoretical results are illustrated by a 3D example on a nontrivial domain.

As shown by Dahmen, Harbrecht and Schneider, the fully discrete wavelet Galerkin scheme for boundary integral equations scales linearly with the number of unknowns without compromising the accuracy of the underlying Galerkin scheme. The supposition is a wavelet basis with a sufficiently large number of vanishing moments. In this paper we present several constructions of appropriate wavelet bases on manifolds based on the biorthogonal spline wavelets of A. Cohen, I. Daubechies and J.-C. Feauveau. By numerical experiments we demonstrate that it is worthwhile to spent effort on their construction to increase the performance of the wavelet Galerkin scheme considerably.

This document serves as a reference for the beta version of our evaluation library ITL. First, it describes a library which gives an easy way for programmers to evaluate the 3D image and the normal vector corresponding to a parameter value which belongs to the unit square. The API functions which are described in this document let programmers make those evaluations without the need to understand the underlying CAD complica- tions. As a consequence, programmers can concentrate on their own scien- tific interests. Our second objective is to describe the input which is a set of parametric four-sided surfaces that have the structure required by some integral equation solvers.