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NUMERICAL ANALYSIS OF WAVELET METHODS SMIA32H
A. Cohen, Web page:, http://www.ann.jussieu.fr/~cohen
Since their introduction in the 1980's, wavelets have become a powerful tool in mathematical analysis, with applications such as image compression, statistical estimation and numerical simulation of partial differential equations. One of their main attractive features is the ability to accurately represent fairly general functions with a small number of adaptively chosen wavelet coefficients, as well as to characterize the smoothness of such functions from the numerical behaviour of these coefficients. The theoretical pillar that underlies such properties involves approximation theory and function spaces, and plays a pivotal role in the analysis of wavelet-based numerical methods.
This book offers a self-contained treatment of wavelets, which includes this theoretical pillar and it applications to the numerical treatment of partial differential equations. Its key features are:
1. Self-contained introduction to wavelet bases and related numerical algorithms, from the simplest examples to the most numerically useful general constructions.
2. Full treatment of the theoretical foundations that are crucial for the analysis
of wavelets and other related multiscale methods : function spaces, linear and nonlinear approximation, interpolation theory.
3. Applications of these concepts to the numerical treatment of partial differential equations : multilevel preconditioning, sparse approximations of differential and integral operators, adaptive discretization strategies.
Introduction. Notations. 1. Basic examples. 1.1 Introduction. 1.2 The Haar system. 1.3 The Schauder hierarchical basis. 1.4 Multivariate constructions. 1.5 Adaptive approximation. 1.6 Multilevel preconditioning. 1.7 Conclusions. 1.8 Historical notes. 2. Multiresolution approximation. 2.1 Introduction. 2.2 Multiresolution analysis. 2.3 Refinable functions. 2.4 Subdivision schemes. 2.5 Computing with refinable functions. 2.6 Wavelets and multiscale algorithms. 2.7 Smoothness analysis. 2.8 Polynomial exactness. 2.9 Duality, orthonormality and interpolation. 2.10 Interpolatory and orthonormal wavelets. 2.11 Wavelets and splines. 2.12 Bounded domains and boundary conditions. 2.13 Point values, cell averages, finite elements. 2.14 Conclusions. 2.15 Historical notes. 3. Approximation and smoothness. 3.1 Introduction. 3.2 Function spaces. 3.3 Direct estimates. 3.4 Inverse estimates. 3.5 Interpolation and approximation spaces. 3.6 Characterization of smoothness classes. 3.7 L p -unstable approximation and 0< p <1. 3.8 Negative smoothness and L p -spaces. 3.9 Bounded domains. 3.10 Boundary conditions. 3.11 Multilevel preconditioning. 3.12 Conclusions. 3.13 Historical notes. 4. Adaptivity. 4.1 Introduction. 4.2 Nonlinear approximation in Besov spaces. 4.3 Nonlinear wavelet approximation in L p . 4.4 Adaptive finite element approximation. 4.5 Other types of nonlinear approximations. 4.6 Adaptive approximation of operators. 4.7 Nonlinear approximation and PDE's. 4.8 Adaptive multiscale processing. 4.9 Adaptive space refinement. 4.10 Conclusions. 4.11 Historical notes. References. Index.