Wavelet Methods - Elliptic Boundary Value Problems and Control Problems

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.

Wavelets for Control Problems

Vorwort
Wavelets for Control Problems

Autorentext
Prof. Dr. Angela Kunoth, Universität Bonn

Klappentext
This research monograph deals with applying recently developed wavelet methods to stationary operator equations involving elliptic differential equations. Particular emphasis is placed on the treatment of the boundary and the boundary conditions.
While wavelets have since their discovery mainly been applied to problems in signal analysis and image compression, their analytic power has also been recognized for problems in Numerical Analysis. Together with the functional analytic framework for
differential and integral quations, one has been able to conceptually discuss questions which are relevant for the fast numerical solution of such problems: preconditioning,
stable discretizations, compression of full matrices, evaluation of difficult norms, and adaptive refinements. The present text focusses on wavelet methods for elliptic
boundary value problems and control problems to show the conceptual strengths of wavelet techniques.

Inhalt
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 FiniteDimensional 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.