Wavelet Methods - Elliptic Boundary Value Problems and Control Problems

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.

Autorentext

Prof. Dr. Angela Kunoth, Universität Bonn

Klappentext

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.

Inhalt

A General Concept - Multiscale Decomposition of Function Spaces - Wavelets on Bounded Domains and Manifolds - Elliptic Boundary - Value Problems - The Fictitious Domain-Lagrange Multiplier Approach - Stability for the Saddle Point Problem: The LBB Condition - Least Squares Formulation of General Saddle Point Problems - Wavelet Representation of Least Squares Systems - Preconditioning and Computational Work - Control Problems - Weakly Coupled Saddle Point Problems - Semi-Iterative Methods - A Fully Iterative Method for Coupled Saddle Point Problems - Complexity Analysis - Numerical Examples