Matrix-Based Multigrid

Theory and Applications

By: Yair Shapira

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Hardcover | 2 July 2008 | Edition Number 2

At a Glance

Hardcover
344 Pages

Edition Type
Revised

Dimensions(cm)
23.5 x 16.51 x 2.54

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This book introduces and analyzes the multigrid approach for the numerical solution of large sparse linear systems arising from the discretization of elliptic partial differential equations. Special attention is given to the powerful matrix-based-multigrid approach, which is particularly useful for problems with variable coefficients and nonsymmetric and indefinite problems. This approach applies not only to model problems on rectangular grids but also to more realistic applications with complicated grids and domains and discontinuous coefficients.

Matrix-Based Multigrid can be used as a textbook in courses in numerical analysis, numerical linear algebra, and numerical PDEs at the advanced undergraduate and graduate levels in computer science, math, and applied math departments. The theory is written in simple algebraic terms and therefore requires preliminary knowledge in basic linear algebra and calculus only. Because it is self contained and includes useful exercises, the book is also suitable for self study by research students, researchers, engineers, and others interested in the numerical solution of partial differential equations.

Industry Reviews

From the reviews of the second edition:

"Shapira delivers a systematic and unified presentation of the multigrid method that is used for the efficient solution of partial differential equations. ... The notations are consistent and the presentation is self-contained. The book is recommended to readers involved in the field of computational science and engineering, from the postgraduate to the expert level. Additionally, the book is suitable for courses in numerical analysis, numerical linear algebra, scientific computing, and numerical solution of partial differential equations." (George A. Gravvanis, ACM Computing Reviews, May, 2009)

"This book provides an introduction into this area. Basically, it presupposes only a sound knowledge of analysis and linear algebra and introduces all other necessary concepts on its own. ... Many exercises are included. The presentation is well suited for seminars in this area." (H. Muthsam, Monatshefte f¼r Mathematik, Vol. 156 (3), March, 2009)

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You Can Find This Book In

Non-Fiction Mathematics Algebra Calculus & Mathematical Analysis Differential Calculus & Equations Numerical Analysis Science Physics

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Paperback

Published: 29th November 2010

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Preface	p. xxi
Concepts and Preliminaries
The Multilevel-Multiscale Approach	p. 5
The Multilevel-Multiscale Concept	p. 5
The Integer Number	p. 6
Division of Integers	p. 8
The Greatest Common Divisor	p. 9
Multilevel Refinement	p. 10
Example in Computer Science	p. 10
Multilevel in Mathematical Logic	p. 11
Multilevel in Language	p. 11
Multilevel Programming	p. 12
Object-Oriented Programming	p. 12
Example in Data Structures	p. 13
The Sorting Problem	p. 13
Parallelism	p. 13
Self-Similarity	p. 14
The Wavelet Transform	p. 14
Mathematical Induction and Recursion	p. 14
The Tower Problem	p. 15
The Parallel Product Algorithm	p. 17
Multilevel in Statistics	p. 18
Multilevel in Music	p. 19
Exercises	p. 19
Preliminaries	p. 23
Preliminary Notation and Definitions	p. 23
Application in Pivoting	p. 31
Standard Lemmas about Symmetric Matrices	p. 31
The Fourier Transform	p. 38
Exercises	p. 41
Partial Differential Equations and Their Discretization
Finite Differences and Volumes	p. 49
Elliptic PDEs	p. 49
The Diffusion Equation	p. 50
The Finite-Difference Discretization Method	p. 51
Error Estimate	p. 53
Finite Differences for the Poisson Equation	p. 54
Error Estimate for Diffusion Problems	p. 55
The Indefinite Helmholtz Equation	p. 55
Adequate Discretization of the Helmholtz Equation	p. 57
Adequate Discretization of Highly Anisotropic Equations	p. 59
Oblique Anisotropy	p. 60
Finite Differences for the Convection-Diffusion Equation	p. 61
The Finite-Volume Discretization Method	p. 63
Exercises	p. 65
Finite Elements	p. 67
The Finite-Element Discretization Method	p. 67
The Weak Formulation	p. 67
The Discrete Weak Formulation	p. 69
Bilinear Finite Elements	p. 71
Triangulation	p. 72
Diagonal Dominance in the Isotropic Case	p. 74
Diagonal Dominance in the Anisotropic Case	p. 76
Locally Refined Meshes	p. 77
The Refinement Step	p. 78
Adaptive Mesh Refinement	p. 80
Exercises	p. 82
The Numerical Solution of Large Sparse Linear Systems
Iterative Linear System Solvers	p. 89
Iterative Sparse Linear System Solvers	p. 89
Relaxation Methods	p. 90
The Jacobi Relaxation Method	p. 90
The Damped Jacobi Relaxation Method	p. 91
The Block Jacobi Relaxation Method	p. 91
The Gauss-Seidel Relaxation Method	p. 91
The Block-Gauss-Seidel Relaxation Method	p. 92
Reordering by Colors	p. 92
Four-Color Reordering	p. 95
Cache-Oriented Reordering	p. 96
Symmetric Gauss-Seidel Relaxation	p. 99
The Preconditioned Conjugate Gradient Method	p. 100
Incomplete LU Factorization (ILU)	p. 102
Parallelizable ILU Version	p. 102
Nonsymmetric and Indefinite Problems	p. 104
Numerical Comparison	p. 105
The Normal Equations	p. 106
Exercises	p. 106
The Multigrid Iteration	p. 109
The Two-Grid Method	p. 109
Transfer and Coarse-Grid Operators	p. 112
The Multigrid Method	p. 112
Geometric Multigrid	p. 114
Variational Multigrid	p. 115
Domain Decomposition and Variational Multigrid	p. 116
Domain Decomposition and Black-Box Multigrid	p. 118
Domain Decomposition and Algebraic Multigrid	p. 120
The Algebraic Multilevel Method	p. 122
Algebraic Multigrid	p. 124
Semicoarsening	p. 126
Exercises	p. 128
Multigrid for Structured Grids
Automatic Multigrid	p. 135
Properties of the AutoMUG Method	p. 135
Cyclic Reduction	p. 136
The Two-Dimensional Case	p. 137
Definition of the AutoMUG Method	p. 138
The AutoMUG(q) Version	p. 141
Exercises	p. 143
Applications in Image Processing	p. 145
The Denoising Problem	p. 145
The Denoising Algorithm for Grayscale Images	p. 146
The Denoising Algorithm for Color Images	p. 147
Numerical Examples	p. 149
Exercises	p. 152
Black-Box Multigrid	p. 155
Definition of Black-Box Multigrid	p. 155
Improvements in Diffusion Problems	p. 156
Using the Right Hand Side	p. 159
Improvement for Problems with Discontinuous Coefficients	p. 160
Exercise	p. 164
The Indefinite Helmholtz Equation	p. 165
Multigrid for the Indefinite Helmholtz Equation	p. 165
Improved Prolongation	p. 166
Improved Black-Box Multigrid	p. 167
Computational Two-Level Analysis	p. 168
Multiple Coarse-Grid Corrections	p. 171
The Size of the Coarsest Grid	p. 174
Numerical Examples	p. 175
Exercises	p. 178
Matrix-Based Semicoarsening	p. 183
The Semicoarsening Approach	p. 183
Flow of Information in Elliptic PDEs	p. 184
Multilevel Line Reordering	p. 185
Block-ILU Factorization	p. 187
The Domain-Decomposition Direct Solver	p. 189
Reordered Block-ILU Factorization	p. 192
Matrix-Based Semicoarsening	p. 193
A Deblurring Problem	p. 195
Exercises	p. 197
Multigrid for Semistructured Grids
Multigrid for Locally Refined Meshes	p. 203
Multigrid and Hierarchical-Basis Linear System Solvers	p. 203
The Two-Level Method	p. 204
Matrix-Induced Inner Products and Norms	p. 210
Properties of the Two-Level Method	p. 211
Isotropic Diffusion Problems	p. 214
Instability and Local Anisotropy	p. 216
The Multilevel Method	p. 216
Upper Bound for the Condition Number	p. 218
Exercises	p. 221
Application to Semistructured Grids	p. 223
Semistructured Grids	p. 223
The V-Cycle	p. 224
The AFAC and AFACx Cycles	p. 224
The Numerical Examples	p. 226
Scaling the Coefficient Matrix	p. 230
A Black-Box Multigrid Version	p. 232
Exercises	p. 233
Multigrid for Unstructured Grids
Domain Decomposition	p. 239
The Domain-Decomposition Approach	p. 239
The Domain-Decomposition Multigrid Method	p. 240
Upper-Bound for the Condition Number	p. 244
High-Order Finite Elements and Spectral Elements	p. 245
Exercises	p. 247
The Algebraic Multilevel Method	p. 249
The Need for Algebraic Multilevel Methods	p. 249
The Algebraic Multilevel Method	p. 250
The Coarsening Procedure	p. 251
The Transfer and Coarse-Level Matrices	p. 252
The Relaxation Method	p. 254
Properties of the Two-Level Method	p. 255
Properties of the Multilevel Method	p. 256
Upper-Bound for the Condition Number	p. 256
The Approximate Schur Complement Method	p. 258
Exercises	p. 258
Applications	p. 261
Highly Anisotropic Equations	p. 261
Two-Step Jacobi Relaxation	p. 262
The Maxwell Equations	p. 263
The Convection-Diffusion Equation	p. 264
ILU Relaxation	p. 267
Towards Algebraic Semicoarsening	p. 268
A Diffusion Problem in a Complicated Domain	p. 268
Exercises	p. 271
Semialgebraic Multilevel for Systems of PDEs	p. 273
Semialgebraic Multilevel Methods	p. 273
Standard Differential Operators	p. 274
The Linear Elasticity Equations	p. 275
The Weak Formulation	p. 275
The Finite-Element Discretization	p. 276
The Semialgebraic Multilevel Preconditioner	p. 277
Preconditioner for the Stokes Equations	p. 278
The Reduced Linear Elasticity Equations	p. 280
Towards Problems with Constraints	p. 281
Towards Semialgebraic Block Lumping	p. 282
A Domain-Decomposition Two-Level Method	p. 283
Exercises	p. 285
Appendices
Time-Dependent Parabolic PDEs	p. 293
Parabolic PDEs	p. 293
The Parabolic Diffusion Equation	p. 293
The Weak Formulation	p. 294
The Semi-Implicit Time Discretization	p. 294
The Finite-Element Discretization	p. 295
Stability Analysis	p. 296
Accuracy of the Numerical Scheme	p. 299
The Algebraic Multilevel Preconditioner	p. 299
Nonlinear Equations	p. 301
Nonlinear PDEs	p. 301
The Residual Equation	p. 301
Defect Correction	p. 302
Geometric Multigrid	p. 303
The Newton Iteration	p. 304
References	p. 305
Index	p. 313
Table of Contents provided by Ingram. All Rights Reserved.

Matrix-Based Multigrid

Theory and Applications

At a Glance

Hardcover

Industry Reviews

Shipping

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You Can Find This Book In

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