Parallel Scientific Computing

ISBN-13: 9781848215818 / Angielski / Twarda / 2016 / 372 str.

Frederic Magoules; François–Xavier Roux

Opis

Scientific computing has become an indispensable tool in numerous fields, such as physics, mechanics, biology,
finance and industry. For example, it enables us, thanks to efficient algorithms adapted to current computers, to
simulate, without the help of models or experimentations, the deflection of beams in bending, the sound level in a theater room or a fluid flowing around an aircraft wing.
This book presents the scientific computing techniques applied to parallel computing for the numerical simulation of large-scale problems; these problems result from systems modeled by partial differential equations. Computing concepts will be tackled via examples.
Implementation and programming techniques resulting from the finite element method will be presented for direct solvers, iterative solvers and domain decomposition methods, along with an introduction to MPI and OpenMP.

Spis treści

Preface xi

Introduction  xv

Chapter 1. Computer Architectures  1

1.1. Different types of parallelism 1

1.1.1. Overlap, concurrency and parallelism  1

1.1.2. Temporal and spatial parallelism for arithmetic logic units 4

1.1.3. Parallelism and memory 6

1.2. Memory architecture 7

1.2.1. Interleaved multi–bank memory 7

1.2.2. Memory hierarchy 8

1.2.3. Distributed memory  13

1.3. Hybrid architecture  14

1.3.1. Graphics–type accelerators  14

1.3.2. Hybrid computers 16

Chapter 2. Parallelization and Programming Models 17

2.1. Parallelization 17

2.2. Performance criteria  19

2.2.1. Degree of parallelism 19

2.2.2. Load balancing 21

2.2.3. Granularity 21

2.2.4. Scalability 22

2.3. Data parallelism  25

2.3.1. Loop tasks 25

2.3.2. Dependencies 26

2.3.3. Examples of dependence 27

2.3.4. Reduction operations 30

2.3.5. Nested loops  31

2.3.6. OpenMP  34

2.4. Vectorization: a case study  37

2.4.1. Vector computers and vectorization 37

2.4.2. Dependence  38

2.4.3. Reduction operations 39

2.4.4. Pipeline operations  41

2.5. Message–passing  43

2.5.1. Message–passing programming 43

2.5.2. Parallel environment management  44

2.5.3. Point–to–point communications  45

2.5.4. Collective communications  46

2.6. Performance analysis 49

Chapter 3. Parallel Algorithm Concepts  53

3.1. Parallel algorithms for recurrences 54

3.1.1. The principles of reduction methods 54

3.1.2. Overhead and stability of reduction methods  55

3.1.3. Cyclic reduction  57

3.2. Data locality and distribution: product of matrices 58

3.2.1. Row and column algorithms 58

3.2.2. Block algorithms  60

3.2.3. Distributed algorithms 64

3.2.4. Implementation  66

Chapter 4. Basics of Numerical Matrix Analysis  71

4.1. Review of basic notions of linear algebra  71

4.1.1. Vector spaces, scalar products and orthogonal projection 71

4.1.2. Linear applications and matrices 74

4.2. Properties of matrices 79

4.2.1. Matrices, eigenvalues and eigenvectors 79

4.2.2. Norms of a matrix 80

4.2.3. Basis change  83

4.2.4. Conditioning of a matrix 85

Chapter 5. Sparse Matrices  93

5.1. Origins of sparse matrices  93

5.2. Parallel formation of sparse matrices: shared memory 98

5.3. Parallel formation by block of sparse matrices: distributed memory 99

5.3.1. Parallelization by sets of vertices  99

5.3.2. Parallelization by sets of elements  101

5.3.3. Comparison: sets of vertices and elements 101

Chapter 6. Solving Linear Systems  105

6.1. Direct methods 105

6.2. Iterative methods  106

Chapter 7. LU Methods for Solving Linear Systems  109

7.1. Principle of LU decomposition  109

7.2. Gauss factorization  113

7.3. Gauss Jordan factorization  115

7.3.1. Row pivoting  118

7.4. Crout and Cholesky factorizations for symmetric matrices  121

Chapter 8. Parallelization of LU Methods for Dense Matrices 125

8.1. Block factorization  125

8.2. Implementation of block factorization in a message–passing environment 130

8.3. Parallelization of forward and backward substitutions 135

Chapter 9. LU Methods for Sparse Matrices 139

9.1. Structure of factorized matrices 139

9.2. Symbolic factorization and renumbering  142

9.3. Elimination trees  147

9.4. Elimination trees and dependencies 152

9.5. Nested dissections 153

9.6. Forward and backward substitutions 159

Chapter 10. Basics of Krylov Subspaces 161

10.1. Krylov subspaces 161

10.2. Construction of the Arnoldi basis  164

Chapter 11. Methods with Complete Orthogonalization for Symmetric Positive Definite Matrices 167

11.1. Construction of the Lanczos basis for symmetric matrices 167

11.2. The Lanczos method 168

11.3. The conjugate gradient method 173

11.4. Comparison with the gradient method 177

11.5. Principle of preconditioning for symmetric positive definite matrices 180

Chapter 12. Exact Orthogonalization Methods for Arbitrary Matrices  185

12.1. The GMRES method 185

12.2. The case of symmetric matrices: the MINRES method 193

12.3. The ORTHODIR method  196

12.4. Principle of preconditioning for non–symmetric matrices 198

Chapter 13. Biorthogonalization Methods for Non–symmetric Matrices 201

13.1. Lanczos biorthogonal basis for non–symmetric matrices  201

13.2. The non–symmetric Lanczos method  206

13.3. The biconjugate gradient method: BiCG  207

13.4. The quasi–minimal residual method: QMR  211

13.5. The BiCGSTAB 217

Chapter 14. Parallelization of Krylov Methods 225

14.1. Parallelization of dense matrix–vector product  225

14.2. Parallelization of sparse matrix–vector product based on node sets 227

14.3. Parallelization of sparse matrix–vector product based on element sets  229

14.3.1. Review of the principles of domain decomposition 229

14.3.2. Matrix–vector product  231

14.3.3. Interface exchanges 233

14.3.4. Asynchronous matrix–vector product with non–blocking communications  236

14.3.5. Comparison: parallelization based on node and element sets  236

14.4. Parallelization of the scalar product  238

14.4.1. By weight 239

14.4.2. By distributivity 239

14.4.3. By ownership 240

14.5. Summary of the parallelization of Krylov methods  241

Chapter 15. Parallel Preconditioning Methods 243

15.1. Diagonal  243

15.2. Incomplete factorization methods  245

15.2.1. Principle  245

15.2.2. Parallelization  248

15.3. Schur complement method 250

15.3.1. Optimal local preconditioning 250

15.3.2. Principle of the Schur complement method  251

15.3.3. Properties of the Schur complement method 254

15.4. Algebraic multigrid 257

15.4.1. Preconditioning using projection  257

15.4.2. Algebraic construction of a coarse grid  258

15.4.3. Algebraic multigrid methods  261

15.5. The Schwarz additive method of preconditioning  263

15.5.1. Principle of the overlap 263

15.5.2. Multiplicative versus additive Schwarz methods 265

15.5.3. Additive Schwarz preconditioning 268

15.5.4. Restricted additive Schwarz: parallel implementation 269

15.6. Preconditioners based on the physics  275

15.6.1. Gauss Seidel method  275

15.6.2. Linelet method  276

Appendices  279

Appendix 1  281

Appendix 2  301

Appendix 3  323

Bibliography 339

Index  343