Diese Monographie spannt einen Bogen rund um die aktuelle Thematik Wavelets, um neueste Entwicklungen anhand aufeinander aufbauender Probleme darzustellen und das konzeptuelle Potenzial von Waveletmethoden für Partielle Differentialgleichungen zu demonstrieren.

This thesis is concerned with the numerical solution of boundary value problems (BVPs) governed by nonlinear elliptic partial differential equations (PDEs). To iteratively solve such BVPs, it is of primal importance to develop efficient schemes that guarantee convergence of the numerically approximated PDE solutions towards the exact solution. The new adaptive wavelet theory guarantees convergence of adaptive schemes with fixed approximation rates. Furthermore, optimal, i.e., linear, complexity estimates of such adaptive solution methods have been established. These achievements are possible since wavelets allow for a completely new perspective to attack BVPs: namely, to represent PDEs in their original infinite dimensional realm. Wavelets in this context represent function bases with special analytical properties, e.g., the wavelets considered herein are piecewise polynomials, have compact support and norm equivalences between certain function spaces and the $ell_2$ sequence spaces of expansion coefficients exist. This theoretical framework is implemented in the course of this thesis in a truly dimensionally unrestricted adaptive wavelet program code, which allows one to harness the proven theoretical results for the first time when numerically solving the above mentioned BVPs. Numerical studies of 2D and 3D PDEs and BVPs demonstrate the feasibility and performance of the developed schemes. The BVPs are solved using an adaptive Uzawa algorithm, which requires repeated solution of nonlinear PDE sub-problems. This thesis presents for the first time a numerically competitive implementation of a new theoretical paradigm to solve nonlinear elliptic PDEs in arbitrary space dimensions with a complete convergence and complexity theory.

This book takes readers on a multi-perspective tour through state-of-the-art mathematical developments related to the numerical treatment of PDEs based on splines, and in particular isogeometric methods. A wide variety of research topics are covered, ranging from approximation theory to structured numerical linear algebra. More precisely, the book provides (i) a self-contained introduction to B-splines, with special focus on approximation and hierarchical refinement, (ii) a broad survey of numerical schemes for control problems based on B-splines and B-spline-type wavelets, (iii) an exhaustive description of methods for computing and analyzing the spectral distribution of discretization matrices, and (iv) a detailed overview of the mathematical and implementational aspects of isogeometric analysis. The text is the outcome of a C.I.M.E. summer school held in Cetraro (Italy), July 2017, featuring four prominent lecturers with different theoretical and application perspectives. The book may serve both as a reference and an entry point into further research.

The book of invited articles offers a collection of high-quality papers in selected and highly topical areas of Applied and Numerical Mathematics and Approximation Theory which have some connection to Wolfgang Dahmen's scientific work. On the occasion of his 60th birthday, leading experts have contributed survey and research papers in the areas of Nonlinear Approximation Theory, Numerical Analysis of Partial Differential and Integral Equations, Computer-Aided Geometric Design, and Learning Theory. The main focus and common theme of all the articles in this volume is the mathematics building the foundation for most efficient numerical algorithms for simulating complex phenomena.

Wavelet methods are by now a well-known tool in image processing (jpeg2000). These functions have been used successfully in other areas, however. Elliptic Partial Differential Equations which model several processes in, for example, science and engineering, is one such field. This book, based on the author's course, gives an introduction to wavelet methods in general and then describes their application for the numerical solution of elliptic partial differential equations. Recently developed adaptive methods are also covered and each scheme is complemented with numerical results , exercises, and corresponding software.

Contains lecture notes on four topics at the forefront of research in computational mathematics. This book presents a self-contained guide to a research area, an extensive bibliography, and proofs of the key results. It is suitable for professional mathematicians who require an accurate account of research in areas parallel to their own.

Like FEM, the Boundary Element Method (BEM) provides a general numerical tool for the solution of complex engineering problems. In the last decades, the range of its applications has remarkably been enlarged. Therefore dynamic and nonlinear problems can be tackled. However they still demand an explicit expression of a fundamental solution, which is only known in simple cases. In this respect, the present book proposes an alternative BEM-formulation based on the Fourier transform, which can be applied to almost all cases relevant in engineering mechanics. The basic principle is presented for the heat equation. Applications are taken from solid mechanics (e.g. poroelasticity, thermoelasticity). Transient and stationary examples are given as well as linear and nonlinear. Completed with a mathematical and mechanical glossary, the book will serve as a comprehensive text book linking applied mathematics to real world engineering problems.

Isogeometric Analysis is a groundbreaking computational approach that promises the possibility of integrating the finite element method into conventional spline-based CAD design tools. It thus bridges the gap between numerical analysis and geometry, and moreover it allows to tackle new cutting edge applications at the frontiers of research in science and engineering. This proceedings volume contains a selection of outstanding research papers presented at the second International Workshop on Isogeometric Analysis and Applications, held at Annweiler, Germany, in April 2014.

Sparse grids have gained increasing interest in recent years for the numerical treatment of high-dimensional problems. Whereas classical numerical discretization schemes fail in more than three or four dimensions, sparse grids make it possible to overcome the “curse” of dimensionality to some degree, extending the number of dimensions that can be dealt with. This volume of LNCSE collects the papers from the proceedings of the second workshop on sparse grids and applications, demonstrating once again the importance of this numerical discretization scheme. The selected articles present recent advances on the numerical analysis of sparse grids as well as efficient data structures, and the range of applications extends to uncertainty quantification settings and clustering, to name but a few examples.