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While wavelets have since their discovery mainly been applied to problems in signal analysis and image compression, their analytic power has more and more also been recognized for problems in Numerical Analysis. Together with the functional analytic framework for different differential and integral quations, one has been able to conceptu ally discuss questions which are relevant for the fast numerical solution of such problems: preconditioning issues, derivation of stable discretizations, compression of fully popu lated matrices, evaluation of non-integer or negative norms, and adaptive refinements based on A-posteriori error estimators. This research monograph focusses on applying wavelet methods to elliptic differential equations. Particular emphasis is placed on the treatment of the boundary and the boundary conditions. Moreover, a control problem with an elliptic boundary problem as contraint serves as an example to show the conceptual strengths of wavelet techniques for some of the above mentioned issues. At this point, I would like to express my gratitude to several people before and during the process of writing this monograph. Most of all, I wish to thank Prof. Dr. Wolfgang Dahmen to whom I personally owe very much and with whom I have co-authored a large part of my work. He is responsible for the very stimulating and challenging scientific atmosphere at the Institut fiir Geometrie und Praktische Mathematik, RWTH Aachen. We also had an enjoyable collaboration with Prof. Dr. Reinhold Schneider from the Technical University of Chemnitz.
List of contents
1 Introduction.- 2 The General Concept.- 3 Wavelets.- 3.1 Preliminaries.- 3.2 Multiscale Decomposition of Function Spaces - Uniform Refinements.- 3.3 Wavelets on an Interval.- 3.4 Wavelets on Manifolds.- 3.5 Multiscale Decomposition of Function Spaces - Non-Uniform Refinements.- 4 Elliptic Boundary Value Problems.- 4.1 General Saddle Point Problems.- 4.2 Elliptic Boundary Value Problems as Saddle Point Problems.- 4.3 Numerical Studies.- 5 Least Squares Problems.- 5.1 Introduction.- 5.2 General Setting.- 5.3 Least Squares Formulation of General Saddle Point Problems.- 5.4 Wavelet Representation of Least Squares Systems.- 5.5 Truncation.- 5.6 Preconditioning and Computational Work.- 5.7 Numerical Experiments.- 6 Control Problems.- 6.1 Introduction.- 6.2 The Continuous Case: Two Coupled Saddle Point Problems.- 6.3 Discretization and Preconditioning.- 6.4 The Discrete Finite-Dimensional Problem.- 6.5 Iterative Methods for WSPP($$ hat Lambda ,Lambda $$).- 6.6 Alternative Iterative Methods for the Coupled System.- 6.7 Outlook into Nonlinear Problems and Adaptive Strategies.- References.
About the author
Prof. Dr. Angela Kunoth, Universität Bonn
Summary
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.
Foreword
Wavelets for Control Problems
