Matrix based multigrid: theory and applications
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Kluwer Academic Publishers
2003
|
| Series: | Numerical methods and algorithms
2 |
| Subjects: | |
| Online Access: | Table of contents Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. [215]-221) |
| Physical Description: | XVI, 221 S. Ill. |
| ISBN: | 1402074859 |
Staff View
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| 245 | 1 | 0 | |a Matrix based multigrid |b theory and applications |c Yair Shapira |
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Record in the Search Index
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| adam_text | Contents
List of Figures ix
List of Tables xiii
Preface xv
1. THE MULTILEVEL MULTISCALE APPROACH 1
1 The Multilevel Multiscale Concept 1
2 The Integer Number 2
3 The Division Algorithm 4
4 The Greatest Common Divider Algorithm 5
5 Multilevel Refinement 5
6 Examples from Computer Science 6
7 Self Similarity 7
8 The Wavelet Transform 7
9 Mathematical Induction and Recursion 7
10 The Product Algorithm 9
11 Preliminary Notations and Definitions 10
12 Application to Pivoting 15
13 The Fourier Transform 16
Part I The Problem and Solution Methods
2. PDES AND DISCRETIZATION METHODS 25
1 Standard Lemmas about Symmetric Matrices 25
2 Elliptic Partial Differential Equations 30
3 The Diffusion Equation 31
4 The Finite Difference Discretization Method 31
5 Finite Differences for the Poisson Equation 34
6 The Finite Volume Discretization Method 36
vi MATRIX BASED MULTIGR1D
1 The Finite Element Discretization Method 38
8 Structured and Unstructured Grids 41
3. ITERATIVE LINEAR SYSTEM SOLVERS 43
1 Iterative Sparse Linear System Solvers 43
2 The Jacobi, Gauss Seidel, and Kacmarz Relaxation Methods 44
3 Reordering by Colors 45
4 Cache Oriented Reordering 47
5 Symmetric Gauss Seidel Relaxation 50
6 The Preconditioned Conjugate Gradient (PCG) Method 51
7 Incomplete LU Factorization (ILU) 53
8 Parallelizable ILU Relaxation 53
9 Parallelizable Gauss Seidel Relaxation 58
4. MULTIGRID ALGORITHMS 61
1 The Two Grid Method 61
2 The Multigrid Method 63
3 Geometric Multigrid 64
4 Matrix Based Multigrid 66
5 Algebraic Multigrid 66
Part II Multigrid for Structured Grids
5. THE AUTOMUG METHOD 73
1 Properties of the AutoMUG Method 73
2 Cyclic Reduction 73
3 The 2 D Case 75
4 The AutoMUG Method 76
5 The AutoMUG(g) Method 77
6. APPLICATIONS IN IMAGE PROCESSING 79
1 The Denoising Problem 80
2 The Denoising Algorithm for Grayscale Images 80
3 The Denoising Algorithm for RGB Color Images 82
4 Examples 85
7. THE BLACK BOX MULTIGRID METHOD 91
1 Definition of BBMG 91
2 Application to Problems with Discontinuous Coefficients 93
Contents vii
8. THE INDEFINITE HELMHOLTZ EQUATION 99
1 The Helmholtz Equation 99
2 Adequate Discretization of the Indefinite Helmholtz Equation 101
3 Multigrid for the Indefinite Helmholtz Equation 103
4 Definition of BBMG2 103
5 Computational Two Level Analysis 105
6 Multiple Coarse Grid Corrections 108
7 The Size of the Coarsest Grid 110
8 Numerical Examples 111
9. MATRIX BASED SEMICOARSENING 115
1 Flow of Information in Elliptic Problems 116
2 Sequential Block ILU Factorization 119
3 The Domain Decomposition Solver 121
4 Reordered Block ILU Factorization 123
5 Matrix Based Semicoarsening Multigrid Method 125
6 A Deblurring Problem 126
Part III Multigrid for Semi Structured Grids
10. MULTIGRID FOR LOCALLY REFINED MESHES 133
1 Locally Refined Meshes 133
2 Multigrid and Hierarchical Basis Linear System Solvers 136
3 The Two Level Method 137
4 Matrix Induced Inner Products and Norms 143
5 Properties of the Two Level Method 145
6 Isotropic Diffusion Problems 148
7 The Multi Level Method 150
8 Upper Bound for the Condition Number 152
9 The V(1,1),AFAC, and AFACx Cycles 155
10 Scaling the Coefficient Matrix 161
11 Black Box Multigrid Version for Semi Structured Grids 163
12 Conclusions 165
viii MATRIX BASED MULTIGRID
Part IV Multigrid for Unstructured Grids
11. DOMAIN DECOMPOSITION 171
1 Advantages of the Domain Decomposition Approach 171
2 The Domain Decomposition Multigrid Method 172
3 Upper Bound for the Condition Number 174
4 High Order Finite Element and Spectral Element Discretizations 177
12. ALGEBRAIC MULTILEVEL METHOD 179
1 The Need for Algebraic Multilevel Methods 179
2 The Algebraic Multilevel Method 182
3 Properties of the Two Level Method 185
4 Properties of the Multilevel Method 186
5 Upper Bound for the Condition Number 187
6 Adequate Discretization of Highly Anisotropic Equations 189
7 Application to the Maxwell Equations 192
8 The Convection Diffusion Equation 193
9 The Approximate Schur Complement Method 199
10 Towards Semi Algebraic Multilevel Methods 199
13. CONCLUSIONS 201
Appendices 205
A C++ Framework for Unstructured Grids 205
References 215
|
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| isbn | 1402074859 |
| language | English |
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| physical | XVI, 221 S. Ill. |
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| spelling | Shapira, Yair Verfasser aut Matrix based multigrid theory and applications Yair Shapira Boston [u.a.] Kluwer Academic Publishers 2003 XVI, 221 S. Ill. txt rdacontent n rdamedia nc rdacarrier Numerical methods and algorithms 2 Includes bibliographical references (p. [215]-221) Analyse numérique matricielle Análise numérica larpcal Equações diferenciais parciais larpcal Méthodes multigrilles (Analyse numérique) Équations aux dérivées partielles - Solutions numériques Differential equations, Partial Numerical solutions Matrices Multigrid methods (Numerical analysis) Matrizenrechnung (DE-588)4126963-9 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Mehrgitterverfahren (DE-588)4038376-3 gnd rswk-swf Mehrgitterverfahren (DE-588)4038376-3 s DE-604 Differentialgleichung (DE-588)4012249-9 s 1\p DE-604 Matrizenrechnung (DE-588)4126963-9 s 2\p DE-604 Numerical methods and algorithms 2 (DE-604)BV016934978 2 http://www.loc.gov/catdir/toc/fy038/2003051640.html Table of contents HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010341206&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Shapira, Yair Matrix based multigrid theory and applications Numerical methods and algorithms Analyse numérique matricielle Análise numérica larpcal Equações diferenciais parciais larpcal Méthodes multigrilles (Analyse numérique) Équations aux dérivées partielles - Solutions numériques Differential equations, Partial Numerical solutions Matrices Multigrid methods (Numerical analysis) Matrizenrechnung (DE-588)4126963-9 gnd Differentialgleichung (DE-588)4012249-9 gnd Mehrgitterverfahren (DE-588)4038376-3 gnd |
| subject_GND | (DE-588)4126963-9 (DE-588)4012249-9 (DE-588)4038376-3 |
| title | Matrix based multigrid theory and applications |
| title_auth | Matrix based multigrid theory and applications |
| title_exact_search | Matrix based multigrid theory and applications |
| title_full | Matrix based multigrid theory and applications Yair Shapira |
| title_fullStr | Matrix based multigrid theory and applications Yair Shapira |
| title_full_unstemmed | Matrix based multigrid theory and applications Yair Shapira |
| title_short | Matrix based multigrid |
| title_sort | matrix based multigrid theory and applications |
| title_sub | theory and applications |
| topic | Analyse numérique matricielle Análise numérica larpcal Equações diferenciais parciais larpcal Méthodes multigrilles (Analyse numérique) Équations aux dérivées partielles - Solutions numériques Differential equations, Partial Numerical solutions Matrices Multigrid methods (Numerical analysis) Matrizenrechnung (DE-588)4126963-9 gnd Differentialgleichung (DE-588)4012249-9 gnd Mehrgitterverfahren (DE-588)4038376-3 gnd |
| topic_facet | Analyse numérique matricielle Análise numérica Equações diferenciais parciais Méthodes multigrilles (Analyse numérique) Équations aux dérivées partielles - Solutions numériques Differential equations, Partial Numerical solutions Matrices Multigrid methods (Numerical analysis) Matrizenrechnung Differentialgleichung Mehrgitterverfahren |
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