Intro -- Contents -- List of Algorithms -- Preface -- Acknowledgements -- List of Figures -- List of Tables -- 1 Introduction -- 1.1 Some aspects of the history of wavelets -- 1.2 The scope of this book -- 1.3 Outline -- 2 Multiscale approximation and multiresolution -- 2.1 The Haar system -- 2.1.1 Projection by interpolation -- 2.1.2 Orthogonal projection -- 2.2 Piecewise linear systems -- 2.3 Similar properties -- 2.3.1 Stability -- 2.3.2 Refinement relation -- 2.3.3 Multiresolution -- 2.3.4 Locality -- 2.4 Multiresolution analysis on the real line -- 2.4.1 The scaling function -- 2.4.2 When does a mask define a refinable function? -- 2.4.3 Consequences of the refinability -- 2.5 Daubechies orthonormal scaling functions -- 2.6 B-splines -- 2.6.1 Centralized B-splines -- 2.7 Dual scaling functions associated to B-splines -- 2.8 Multilevel projectors -- 2.9 Approximation properties -- 2.9.1 A general framework -- 2.9.2 Stability properties -- 2.9.3 Error estimates -- 2.10 Plotting scaling functions -- 2.10.1 Subdivision -- 2.10.2 Cascade algorithm -- 2.11 Periodization -- 2.12 Exercises and programs -- 3 Elliptic boundary value problems -- 3.1 A model problem -- 3.1.1 Variational formulation -- 3.1.2 Existence and uniqueness -- 3.2 Variational formulation -- 3.2.1 Operators associated by the bilinear form -- 3.2.2 Reduction to homogeneous boundary conditions -- 3.2.3 Stability -- 3.3 Regularity theory -- 3.4 Galerkin methods -- 3.4.1 Discretization -- 3.4.2 Stability -- 3.4.3 Error estimates -- 3.4.4 L[sub(2)]-estimates -- 3.4.5 Numerical solution -- 3.5 Exercises and programs -- 4 Multiresolution Galerkin methods -- 4.1 Multiscale discretization -- 4.2 Multiresolution multiscale discretization -- 4.2.1 Piecewise linear multiresolution -- 4.2.2 Periodic boundary value problems -- 4.2.3 Common properties -- 4.3 Error estimates.

4.4 Some numerical examples -- 4.5 Setup of the algebraic system -- 4.5.1 Refinable integrals -- 4.5.2 The right-hand side -- 4.5.3 Quadrature -- 4.6 The BPX preconditioner -- 4.7 MultiGrid -- 4.8 Numerical examples for the model problem -- 4.9 Exercises and programs -- 5 Wavelets -- 5.1 Detail spaces -- 5.1.1 Updating -- 5.1.2 The Haar system again -- 5.2 Orthogonal wavelets -- 5.2.1 Multilevel decomposition -- 5.2.2 The construction of wavelets -- 5.2.3 Wavelet projectors -- 5.3 Biorthogonal wavelets -- 5.3.1 Biorthogonal complement spaces -- 5.3.2 Biorthogonal projectors -- 5.3.3 Biorthogonal B-spline wavelets -- 5.4 Fast Wavelet Transform (FWT) -- 5.4.1 Decomposition -- 5.4.2 Reconstruction -- 5.4.3 Efficiency -- 5.4.4 A general framework -- 5.5 Vanishing moments and compression -- 5.6 Norm equivalences -- 5.6.1 Jackson inequality -- 5.6.2 Bernstein inequality -- 5.6.3 A characterization theorem -- 5.7 Other kinds of wavelets -- 5.7.1 Interpolatory wavelets -- 5.7.2 Semiorthogonal wavelets -- 5.7.3 Noncompactly supported wavelets -- 5.7.4 Multiwavelets -- 5.7.5 Frames -- 5.7.6 Curvelets -- 5.8 Exercises and programs -- 6 Wavelet-Galerkin methods -- 6.1 Wavelet preconditioning -- 6.2 The role of the FWT -- 6.3 Numerical examples for the model problem -- 6.3.1 Rate of convergence -- 6.3.2 Compression -- 6.4 Exercises and programs -- 7 Adaptive wavelet methods -- 7.1 Adaptive approximation of functions -- 7.1.1 Best N-term approximation -- 7.1.2 The size and decay of the wavelet coefficients -- 7.2 A posteriori error estimates and adaptivity -- 7.2.1 A posteriori error estimates -- 7.2.2 Ad hoc refinement strategies -- 7.3 Infinite-dimensional iterations -- 7.4 An equivalent l[sub(2)] problem: Using wavelets -- 7.5 Compressible matrices -- 7.5.1 Numerical realization of APPLY -- 7.5.2 Numerical experiments for APPLY -- 7.6 Approximate iterations.

7.6.1 Adaptive Wavelet-Richardson method -- 7.6.2 Adaptive scheme with inner iteration -- 7.6.3 Optimality -- 7.7 Quantitative efficiency -- 7.7.1 Quantitative aspects of the efficiency -- 7.7.2 An efficient modified scheme: Ad hoc strategy revisited -- 7.8 Nonlinear problems -- 7.8.1 Nonlinear variational problems -- 7.8.2 The DSX algorithm -- 7.8.3 Prediction -- 7.8.4 Reconstruction -- 7.8.5 Quasi-interpolation -- 7.8.6 Decomposition -- 7.9 Exercises and programs -- 8 Wavelets on general domains -- 8.1 Multiresolution on the interval -- 8.1.1 Refinement matrices -- 8.1.2 Boundary scaling functions -- 8.1.3 Biorthogonal multiresolution -- 8.1.4 Refinement matrices -- 8.1.5 Boundary conditions -- 8.1.6 Symmetry -- 8.2 Wavelets on the interval -- 8.2.1 Stable completion -- 8.2.2 Spline-wavelets on the interval -- 8.2.3 Further examples -- 8.2.4 Dirichlet boundary conditions -- 8.2.5 Quantitative aspects -- 8.2.6 Other constructions on the interval -- 8.2.7 Software for wavelets on the interval -- 8.2.8 Numerical experiments -- 8.3 Tensor product wavelets -- 8.4 The Wavelet Element Method (WEM) -- 8.4.1 Matching in 1D -- 8.4.2 The setting in arbitrary dimension -- 8.4.3 The WEM in the two-dimensional case -- 8.4.4 Trivariate matched wavelets -- 8.4.5 Software for the WEM -- 8.5 Embedding methods -- 8.6 Exercises and programs -- 9 Some applications -- 9.1 Elliptic problems on bounded domains -- 9.1.1 Numerical realization of the WEM -- 9.1.2 Model problem on the L-shaped domain -- 9.2 More complicated domains -- 9.2.1 Influence of the mapping - A non-rectangular domain -- 9.2.2 Influence of the matching -- 9.2.3 Comparison of the different adaptive methods -- 9.3 Saddle point problems -- 9.3.1 The standard Galerkin discretization: The LBB condition -- 9.3.2 An equivalent l[sub(2)] problem -- 9.3.3 The adaptive wavelet method: Convergence without LBB.

9.4 The Stokes problem -- 9.4.1 Formulation -- 9.4.2 Discretization -- 9.4.3 B-spline wavelets and the exact application of the divergence -- 9.4.4 Bounded domains -- 9.4.5 The divergence operator -- 9.4.6 Compressibility of A and B[sup(T)] -- 9.4.7 Numerical experiments -- 9.4.8 Rate of convergence -- 9.5 Exercises and programs -- A: Sobolev spaces and variational formulations -- A.1 Weak derivatives and sobolev spaces with integer order -- A.2 Sobolev spaces with fractional order -- A.3 Sobolev spaces with negative order -- A.4 Variational formulations -- A.5 Regularity theory -- B: Besov spaces -- B.1 Sobolev and Besov embedding -- B.2 Convergence of approximation schemes -- C: Basic iterations -- References -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- X -- Y -- Z.

A text based on the author's course that introduces graduates to the basics of wavelet methods for partial differential equations and describes the construction and analysis of adaptive wavelet methods.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2018. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.