Generalized locally Toeplitz sequences. Vol.2

✍ Scribed by Garoni C., Serra-Capizzano S

• Publisher: Springer
• Year: 2018
• Tongue: English
• Leaves: 199
• Category: Library

✦ Table of Contents

Preface......Page 5
Contents......Page 7
About the Authors......Page 10
1 Notes to the Reader......Page 11
2.1.1 General Notation and Terminology......Page 12
2.1.2 Multi-index Notation......Page 16
2.2 Multivariate Trigonometric Polynomials......Page 21
2.3 Multivariate Riemann-Integrable Functions......Page 25
2.4 Matrix Norms......Page 26
2.5 Tensor Products and Direct Sums......Page 27
2.6.1 The Notion of Singular Value and Eigenvalue Distribution......Page 33
2.6.2 Clustering and Attraction......Page 35
2.6.3 Zero-Distributed Sequences......Page 36
2.6.4 Sparsely Unbounded and Sparsely Vanishing Sequences of Matrices......Page 37
2.6.5 Spectral Distribution of Sequences of Perturbed Hermitian Matrices......Page 39
2.7.1 Definition of a.c.s. and a.c.s. Topology......Page 40
2.7.2 The a.c.s. Tools for Computing Singular Value and Spectral Distributions......Page 42
2.7.3 The a.c.s. Algebra......Page 43
2.7.4 Some Criteria to Identify a.c.s.......Page 44
2.7.5 An Extension of the Concept of a.c.s.......Page 45
3.1 Multilevel Toeplitz Matrices and Multilevel Toeplitz Sequences......Page 48
3.2 Basic Properties of Multilevel Toeplitz Matrices......Page 51
3.3 Schatten p-Norms of Multilevel Toeplitz Matrices......Page 55
3.4 Multilevel Circulant Matrices......Page 60
3.5 Singular Value and Spectral Distribution of Multilevel …......Page 64
3.6 Extreme Eigenvalues of Hermitian Multilevel Toeplitz Matrices......Page 67
4.1 Multilevel LT Operator......Page 69
4.1.1 Definition of Multilevel LT Operator......Page 70
4.1.2 Properties of the Multilevel LT Operator......Page 75
4.3 Fundamental Examples of Multilevel LT Sequences......Page 80
4.3.2 Sequences of Multilevel Diagonal Sampling Matrices......Page 81
4.3.3 Multilevel Toeplitz Sequences......Page 87
4.4 Singular Value and Spectral Distribution of a Finite Sum of Multilevel LT Sequences......Page 90
4.5 Algebraic Properties of Multilevel LT Sequences......Page 93
4.6 Characterizations of Multilevel LT Sequences......Page 94
5.1 Equivalent Definitions of Multilevel GLT Sequences......Page 99
5.2 Singular Value and Spectral Distribution of Multilevel GLT Sequences......Page 101
5.3 Approximation Results for Multilevel GLT Sequences......Page 103
5.3.1 Characterizations of Multilevel GLT Sequences......Page 107
5.3.2 Sequences of Multilevel Diagonal Sampling Matrices......Page 108
5.4 The Multilevel GLT Algebra......Page 111
5.5 Algebraic-Topological Definitions of Multilevel GLT Sequences......Page 117
6 Summary of the Theory......Page 119
7.1.1 Multilevel GLT Preconditioning......Page 129
7.1.2 Multilevel Arrow-Shaped Sampling Matrices......Page 130
7.2 Applications to PDE Discretizations: An Introduction......Page 132
7.3 FD Discretization of Convection-Diffusion-Reaction PDEs......Page 135
7.4 FE Discretization of Convection-Diffusion-Reaction PDEs......Page 144
7.5 B-Spline IgA Collocation Discretization of Convection-Diffusion-Reaction PDEs......Page 154
7.6 Galerkin B-Spline IgA Discretization of Convection-Diffusion-Reaction PDEs......Page 168
7.7 Galerkin B-Spline IgA Discretization of Second-Order Eigenvalue Problems......Page 182
8 Future Developments......Page 186
References......Page 189
Index......Page 192