Wavelets Theory and Its Applications (Forum for Interdisciplinary Mathematics)

✍ Scribed by Mehra

• Publisher: Springer
• Year: 2018
• Tongue: English
• Leaves: 185
• Edition: 1st ed. 2018
• Category: Library

✦ Synopsis

This book provides comprehensive information on the conceptual basis of wavelet theory and it applications. Maintaining an essential balance between mathematical rigour and the practical applications of wavelet theory, the book is closely linked to the wavelet MATLAB toolbox, which is accompanied, wherever applicable, by relevant MATLAB codes. The book is divided into four parts, the first of which is devoted to the mathematical foundations. The second part offers a basic introduction to wavelets. The third part discusses wavelet-based numerical methods for differential equations, while the last part highlights applications of wavelets in other fields. The book is ideally suited as a text for undergraduate and graduate students of mathematics and engineering.

✦ Table of Contents

Preface
Why Another Book on Wavelets
Structure of the Book
Acknowledgements
Contents
About the Author
Part I Mathematical Foundation
1 Preliminaries
1.1 Vector Space
1.2 Normed Space
1.3 Inner Product Space
1.4 Hilbert Space
1.5 Projection
1.6 Function Series
2 Fourier Analysis
2.1 Fourier Series
2.2 Fourier Transform
2.3 Time–Frequency Analysis
References
Part II Introduction to Wavelets
3 Wavelets on Flat Geometries
3.1 Multiresolution Analysis
3.1.1 Wavelets and Fourier Transform
3.1.2 Different Ways to Start Multiresolution
3.1.3 Evaluation of Scaling Functions and Wavelets
3.2 Daubechies Wavelet
3.2.1 Construction of Daubechies Wavelet on the Real Line (mathbbR)
3.2.2 Construction of Periodized Daubechies Wavelet
3.2.3 Construction of Daubechies Wavelet on the Interval
3.3 Coiflet Wavelet
3.4 Shannon Wavelet
3.5 Meyer Wavelet
3.6 Hermitian Wavelets
3.7 Morlet Wavelet
3.8 Biorthogonal Wavelet
3.8.1 Splines
3.8.2 Spline Wavelets
3.8.3 Compactly Supported Biorthogonal Spline Wavelet
3.9 Battle–Lemarie Wavelet
3.10 Interpolating Wavelet
3.11 Comparison of Different Wavelets
3.12 Wavelets in Higher Dimensions
3.13 Non-MRA Wavelet
References
4 Wavelets on Arbitrary Manifolds
4.1 Second-Generation Wavelets
4.1.1 Lifting Scheme
4.1.2 Spherical Wavelet
4.2 Diffusion Wavelet
4.3 Spectral Graph Wavelet
References
5 Wavelet Transform
5.1 Continuous Wavelet Transform
5.2 Discrete Wavelet Transform
5.2.1 Daubechies Wavelet Transform
5.2.2 Diffusion Wavelet and Inverse Diffusion Wavelet Transform
5.2.3 Spectral Graph Wavelet Transform (SGWT)
5.3 Discretized CWT Versus DWT
References
Part III Wavelets Based Numerical Methods for Differential Equations
6 Introduction to Numerical Methods
6.1 Finite Difference Methods (FDM)
6.2 Compact Finite Difference Methods
6.3 Spectral Methods
6.4 Finite Element Methods (FEM)
6.5 Finite Volume Methods (FVM)
References
7 Wavelet-Galerkin Methods
7.1 Different Ways of Handling Nonlinearities
7.1.1 Connection Coefficients Approach
7.1.2 Pseudo Approach
7.2 Different Ways of Handling Boundary Conditions
7.3 Differentiation Projection Matrix
7.4 The Basic Paradigm of Wavelet-Galerkin Method
References
8 Wavelet Collocation Methods
8.1 Differentiation Projection Matrix
8.2 The Basic Paradigm of Wavelet Collocation Method
References
9 Other Wavelet-Based Numerical Methods
9.1 Wavelet Optimized Numerical Methods
9.1.1 Wavelet Optimized Finite Difference Methods
9.1.2 Wavelet Optimized Finite Element Methods
9.1.3 Wavelet Optimized Finite Volume Methods
9.2 Space–Time Adaptive Methods
9.3 A Lagrange Particle Wavelet Methods
9.4 Wavelet–Taylor Galerkin Methods
References
Part IV Applications of Wavelets in Other Fields
10 Applications of Wavelet in Inverse Problems
10.1 Different Approaches to Solve Inverse Problems
10.1.1 Regularization
10.1.2 Regularization by Filtering
10.1.3 Variational Regularization Methods
10.1.4 Iterative Regularization Methods
10.2 Wavelet-Based Approaches to Solve Inverse Problems
10.2.1 Wavelet–Vaguelette Decomposition
10.2.2 The Vaguelette–Wavelet Decomposition Method
10.2.3 Generalized Wavelet-Galerkin Method
10.2.4 Wavelet Domain Linear Inversion
References
11 Other Useful Applications of Wavelet
11.1 Wavelet and Preconditioners
11.2 Wavelet and Turbulence
11.3 Wavelet and Multigrid Methods
11.3.1 Multigrid Framework
11.3.2 Similarities Between Multigrid and Multiresolution
11.4 Wavelet and Integral Equations
References