Elsevier S & T

Numerical Analysis of Wavelet Methods

Cohen, A.

In stock
No reviews
50.750 KD inc. VAT
License
Table of contents
  • Cover
  • Contentsix
  • Introductionxi
  • Notationsxvii
  • Chapter 1. Basic examples1
  • 1.1 Introduction1
  • 1.2 The Haar system2
  • 1.3 The Schauder hierarchical basis10
  • 1.4 Multivariate constructions16
  • 1.5 Adaptive approximation22
  • 1.6 Multilevel preconditioning33
  • 1.7 Conclusions40
  • 1.8 Historical notes41
  • Chapter 2. Multiresolution approximation43
  • 2.1 Introduction43
  • 2.2 Multiresolution analysis45
  • 2.3 Refinable functions57
  • 2.4 Subdivision schemes63
  • 2.5 Computing with refinable functions70
  • 2.6 Wavelets and multiscale algorithms74
  • 2.7 Smoothness analysis83
  • 2.8 Polynomial exactness89
  • 2.9 Duality, orthonormality and interpolation94
  • 2.10 Interpolatory and orthonormal wavelets99
  • 2.11 Wavelets and splines107
  • 2.12 Bounded domains and boundary conditions120
  • 2.13 Point values, cell averages, finite elements131
  • 2.14 Conclusions150
  • 2.15 Historical notes151
  • Chapter 3. Approximation and smoothness155
  • 3.1 Introduction155
  • 3.2 Function spaces159
  • 3.3 Direct estimates165
  • 3.4 Inverse estimates171
  • 3.5 Interpolation and approximation spaces174
  • 3.6 Characterization of smoothness classes182
  • 3.7 LP-unstable approximation and 0 < p < 1187
  • 3.8 Negative smoothness and LP-spaces199
  • 3.9 Bounded domains208
  • 3.10 Boundary conditions216
  • 3.11 Multilevel preconditioning226
  • 3.12 Conclusions239
  • 3.13 Historical notes240
  • Chapter 4. Adaptivity243
  • 4.1 Introduction243
  • 4.2 Nonlinear approximation in Besov spaces248
  • 4.3 Nonlinear wavelet approximation in Lp254
  • 4.4 Adaptive finite element approximation262
  • 4.5 Other types of nonlinear approximations267
  • 4.6 Adaptive approximation of operators276
  • 4.7 Nonlinear approximation and PDE's289
  • 4.8 Adaptive multiscale processing296
  • 4.9 Adaptive space refinement306
  • 4.10 Conclusions317
  • 4.11 Historical notes318
  • References321
  • Index335
Book details
  • Vendor Elsevier S & T
  • SKU 9780444511249
  • ISBN-13 9780080537856
  • Author Cohen, A.
  • Category Mathematics
  • Subject Infinity

Do you have questions about this book?

Ask an expert!

Description eBook delivery & access Reviews
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.