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.

Table of Contents:
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