Iterative Methods for Sparse Linear Systems Second Edition

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viii CONTENTS 6.11.4 Convergence of GMRES . . . . . . . . . . . . . . 215 6.12 Block Krylov Methods . . . . . . . . . . . . . . . . . . . . . 218 7 Krylov Subspace Methods Part II 229 7.1 Lanczos Biorthogonalization . . . . . . . . . . . . . . . . . . 229 7.1.1 The Algorithm . . . . . . . . . . . . . . . . . . . 229 7.1.2 Practical Implementations . . . . . . . . . . . . . 232 7.2 The Lanczos Algorithm for Linear Systems . . . . . . . . . . . 234 7.3 The BCG and QMR Algorithms . . . . . . . . . . . . . . . . . 234 7.3.1 The Biconjugate Gradient Algorithm . . . . . . . 234 7.3.2 Quasi-Minimal Residual Algorithm . . . . . . . . 236 7.4 Transpose-Free Variants . . . . . . . . . . . . . . . . . . . . . 241 7.4.1 Conjugate Gradient Squared . . . . . . . . . . . . 241 7.4.2 BICGSTAB . . . . . . . . . . . . . . . . . . . . 244 7.4.3 Transpose-Free QMR (TFQMR) . . . . . . . . . 247 8 Methods Related to the Normal Equations 259 8.1 The Normal Equations . . . . . . . . . . . . . . . . . . . . . . 259 8.2 Row Projection Methods . . . . . . . . . . . . . . . . . . . . 261 8.2.1 Gauss-Seidel on the Normal Equations . . . . . . 261 8.2.2 Cimmino’s Method . . . . . . . . . . . . . . . . 263 8.3 Conjugate Gradient and Normal Equations . . . . . . . . . . . 266 8.3.1 CGNR . . . . . . . . . . . . . . . . . . . . . . . 266 8.3.2 CGNE . . . . . . . . . . . . . . . . . . . . . . . 267 8.4 Saddle-Point Problems . . . . . . . . . . . . . . . . . . . . . . 268 9 Preconditioned Iterations 275 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 9.2 Preconditioned Conjugate Gradient . . . . . . . . . . . . . . . 276 9.2.1 Preserving Symmetry . . . . . . . . . . . . . . . 276 9.2.2 Efficient Implementations . . . . . . . . . . . . . 279 9.3 Preconditioned GMRES . . . . . . . . . . . . . . . . . . . . . 282 9.3.1 Left-Preconditioned GMRES . . . . . . . . . . . 282 9.3.2 Right-Preconditioned GMRES . . . . . . . . . . 284 9.3.3 Split Preconditioning . . . . . . . . . . . . . . . 285 9.3.4 Comparison of Right and Left Preconditioning . . 285 9.4 Flexible Variants . . . . . . . . . . . . . . . . . . . . . . . . . 287 9.4.1 Flexible GMRES . . . . . . . . . . . . . . . . . . 287 9.4.2 DQGMRES . . . . . . . . . . . . . . . . . . . . 290 9.5 Preconditioned CG for the Normal Equations . . . . . . . . . . 291 9.6 The Concus, Golub, and Widlund Algorithm . . . . . . . . . . 292 10 Preconditioning Techniques 297 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 10.2 Jacobi, SOR, and SSOR Preconditioners . . . . . . . . . . . . 298
CONTENTS ix 10.3 ILU Factorization Preconditioners . . . . . . . . . . . . . . . 301 10.3.1 Incomplete LU Factorizations . . . . . . . . . . . 301 10.3.2 Zero Fill-in ILU (ILU(0)) . . . . . . . . . . . . . 307 10.3.3 Level of Fill and ILU(p) . . . . . . . . . . . . . . 311 10.3.4 Matrices with Regular Structure . . . . . . . . . . 315 10.3.5 Modified ILU (MILU) . . . . . . . . . . . . . . . 319 10.4 Threshold Strategies and ILUT . . . . . . . . . . . . . . . . . 321 10.4.1 The ILUT Approach . . . . . . . . . . . . . . . . 321 10.4.2 Analysis . . . . . . . . . . . . . . . . . . . . . . 323 10.4.3 Implementation Details . . . . . . . . . . . . . . 325 10.4.4 The ILUTP Approach . . . . . . . . . . . . . . . 327 10.4.5 The ILUS Approach . . . . . . . . . . . . . . . . 330 10.4.6 The Crout ILU Approach . . . . . . . . . . . . . 332 10.5 Approximate Inverse Preconditioners . . . . . . . . . . . . . . 336 10.5.1 Approximating the Inverse of a Sparse Matrix . . 337 10.5.2 Global Iteration . . . . . . . . . . . . . . . . . . 337 10.5.3 Column-Oriented Algorithms . . . . . . . . . . . 339 10.5.4 Theoretical Considerations . . . . . . . . . . . . 341 10.5.5 Convergence of Self Preconditioned MR . . . . . 343 10.5.6 Approximate Inverses via bordering . . . . . . . . 346 10.5.7 Factored inverses via orthogonalization: AINV . . 348 10.5.8 Improving a Preconditioner . . . . . . . . . . . . 349 10.6 Reordering for ILU . . . . . . . . . . . . . . . . . . . . . . . 350 10.6.1 Symmetric permutations . . . . . . . . . . . . . . 350 10.6.2 Nonsymmetric reorderings . . . . . . . . . . . . . 352 10.7 Block Preconditioners . . . . . . . . . . . . . . . . . . . . . . 354 10.7.1 Block-Tridiagonal Matrices . . . . . . . . . . . . 354 10.7.2 General Matrices . . . . . . . . . . . . . . . . . . 356 10.8 Preconditioners for the Normal Equations . . . . . . . . . . . 356 10.8.1 Jacobi, SOR, and Variants . . . . . . . . . . . . . 357 10.8.2 IC(0) for the Normal Equations . . . . . . . . . . 357 10.8.3 Incomplete Gram-Schmidt and ILQ . . . . . . . . 360 11 Parallel Implementations 369 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 11.2 Forms of Parallelism . . . . . . . . . . . . . . . . . . . . . . . 370 11.2.1 Multiple Functional Units . . . . . . . . . . . . . 370 11.2.2 Pipelining . . . . . . . . . . . . . . . . . . . . . 370 11.2.3 Vector Processors . . . . . . . . . . . . . . . . . 371 11.2.4 Multiprocessing and Distributed Computing . . . 371 11.3 Types of Parallel Architectures . . . . . . . . . . . . . . . . . 371 11.3.1 Shared Memory Computers . . . . . . . . . . . . 372 11.3.2 Distributed Memory Architectures . . . . . . . . 374 11.4 Types of Operations . . . . . . . . . . . . . . . . . . . . . . . 376
Page 1: Iterative Methods for Sparse Linear
Page 4 and 5: vi CONTENTS 2.1.2 The Convection Di
Page 8 and 9: x CONTENTS 11.5 Matrix-by-Vector Pr
Page 10 and 11: xii CONTENTS
Page 12 and 13: xiv PREFACE underlying architecture
Page 14 and 15: xvi PREFACE The rest could be skipp
Page 16 and 17: xviii PREFACE
Page 18 and 19: xx PREFACE Apart from two recent vo
Page 21 and 22: Chapter 1 BACKGROUND IN LINEAR ALGE
Page 23 and 24: 1.2. SQUARE MATRICES AND EIGENVALUE
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1.12. PROJECTION OPERATORS 37 In ma
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1.13. BASIC CONCEPTS IN LINEAR SYST
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Chapter 2 DISCRETIZATION OF PDES Pa
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2.1. PARTIAL DIFFERENTIAL EQUATIONS
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2.2. FINITE DIFFERENCE METHODS 51 f
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2.2. FINITE DIFFERENCE METHODS 53 u
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2.2. FINITE DIFFERENCE METHODS 55 B
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2.2. FINITE DIFFERENCE METHODS 57
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2.2. FINITE DIFFERENCE METHODS 59 i
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2.2. FINITE DIFFERENCE METHODS 61 T
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2.3. THE FINITE ELEMENT METHOD 63 o
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2.3. THE FINITE ELEMENT METHOD 65 F
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2.3. THE FINITE ELEMENT METHOD 67 i
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2.4. MESH GENERATION AND REFINEMENT
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2.5. FINITE VOLUME METHOD 71 3 (a)
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2.5. FINITE VOLUME METHOD 73 This g
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Chapter 3 SPARSE MATRICES As descri
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3.2. GRAPH REPRESENTATIONS 77 Figur
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3.3. PERMUTATIONS AND REORDERINGS 7
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3.4. STORAGE SCHEMES 93 containing
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3.5. BASIC SPARSE MATRIX OPERATIONS
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3.6. SPARSE DIRECT SOLUTION METHODS
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3.7. TEST PROBLEMS 99 3 4 1 4 ✻ a
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3.7. TEST PROBLEMS 101 Figure 3.13:
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3.7. TEST PROBLEMS 103 P-3.8 Consid
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Chapter 4 BASIC ITERATIVE METHODS T
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4.1. JACOBI, GAUSS-SEIDEL, AND SOR
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4.2. CONVERGENCE 115 4.2.1 General
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4.2. CONVERGENCE 117 Then, the eige
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4.2. CONVERGENCE 119 From this and
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4.2. CONVERGENCE 121 cannot be an i
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4.2. CONVERGENCE 123 4.2.5 Property
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4.2. CONVERGENCE 125 Consider now a
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4.3. ALTERNATING DIRECTION METHODS
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Chapter 5 PROJECTION METHODS Most o
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5.1. BASIC DEFINITIONS AND ALGORITH
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5.2. GENERAL THEORY 137 5.2 General
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5.2. GENERAL THEORY 139 ✻✯ ❥
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5.2. GENERAL THEORY 141 Thus, an n-
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5.3. ONE-DIMENSIONAL PROJECTION PRO
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5.3. ONE-DIMENSIONAL PROJECTION PRO
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5.4. ADDITIVE AND MULTIPLICATIVE PR
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Chapter 6 KRYLOV SUBSPACE METHODS P
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6.2. KRYLOV SUBSPACES 159 Proof. Th
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6.3. ARNOLDI’S METHOD 161 which s
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6.3. ARNOLDI’S METHOD 163 from a
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6.4. ARNOLDI’S METHOD FOR LINEAR
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6.5. GMRES 171 6. Fortunately, an i
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6.5. GMRES 173 6.5.2 The Householde
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6.5. GMRES 175 Define the rotation
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6.5. GMRES 177 3. The residual vect
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6.5. GMRES 179 6.5.4 Breakdown of G
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6.5. GMRES 181 note that if ¯ Hm i
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6.5. GMRES 183 then b − Axm2 = |
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6.5. GMRES 185 Matrix Iters Kflops
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6.5. GMRES 187 Proposition 6.12 Ass
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6.5. GMRES 189 Proof. The following
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6.5. GMRES 191 In the situation whe
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6.5. GMRES 193 QMRES/DQGMRES. The k
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6.6. THE SYMMETRIC LANCZOS ALGORITH
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6.7. THE CONJUGATE GRADIENT ALGORIT
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6.8. THE CONJUGATE RESIDUAL METHOD
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6.9. GCR, ORTHOMIN, AND ORTHODIR 20
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6.10. THE FABER-MANTEUFFEL THEOREM
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6.11. CONVERGENCE ANALYSIS 209 One
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6.11. CONVERGENCE ANALYSIS 211 Ther
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6.11. CONVERGENCE ANALYSIS 213 over
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6.11. CONVERGENCE ANALYSIS 215 Ther
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ℑm(z) 6.11. CONVERGENCE ANALYSIS
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6.12. BLOCK KRYLOV METHODS 219 Kryl
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6.12. BLOCK KRYLOV METHODS 221 Now
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6.12. BLOCK KRYLOV METHODS 223 wher
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6.12. BLOCK KRYLOV METHODS 225 P-6.
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6.12. BLOCK KRYLOV METHODS 227 is p
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Chapter 7 KRYLOV SUBSPACE METHODS P
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7.1. LANCZOS BIORTHOGONALIZATION 23
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7.1. LANCZOS BIORTHOGONALIZATION 23
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7.3. THE BCG AND QMR ALGORITHMS 235
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7.4. TRANSPOSE-FREE VARIANTS 241 AL
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7.4. TRANSPOSE-FREE VARIANTS 243 th
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7.4. TRANSPOSE-FREE VARIANTS 245 Ac
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7.4. TRANSPOSE-FREE VARIANTS 247 4.
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7.4. TRANSPOSE-FREE VARIANTS 249 Ne
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7.4. TRANSPOSE-FREE VARIANTS 251 Th
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7.4. TRANSPOSE-FREE VARIANTS 253 15
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7.4. TRANSPOSE-FREE VARIANTS 255 P-
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7.4. TRANSPOSE-FREE VARIANTS 257 A
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Chapter 8 METHODS RELATED TO THE NO
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8.2. ROW PROJECTION METHODS 261 the
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8.2. ROW PROJECTION METHODS 263 ALG
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8.2. ROW PROJECTION METHODS 265 in
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8.3. CONJUGATE GRADIENT AND NORMAL
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8.4. SADDLE-POINT PROBLEMS 269 with
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8.4. SADDLE-POINT PROBLEMS 271 Apar
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8.4. SADDLE-POINT PROBLEMS 273 a. S
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Chapter 9 PRECONDITIONED ITERATIONS
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9.2. PRECONDITIONED CONJUGATE GRADI
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9.3. PRECONDITIONED GMRES 283 Somet
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9.3. PRECONDITIONED GMRES 285 9.3.3
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9.4. FLEXIBLE VARIANTS 287 In most
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9.4. FLEXIBLE VARIANTS 289 Proposit
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9.5. PRECONDITIONED CG FOR THE NORM
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9.6. THE CONCUS, GOLUB, AND WIDLUND
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9.6. THE CONCUS, GOLUB, AND WIDLUND
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Chapter 10 PRECONDITIONING TECHNIQU
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10.2. JACOBI, SOR, AND SSOR PRECOND
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10.3. ILU FACTORIZATION PRECONDITIO
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10.4. THRESHOLD STRATEGIES AND ILUT
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10.5. APPROXIMATE INVERSE PRECONDIT
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10.6. REORDERING FOR ILU 351 become
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10.6. REORDERING FOR ILU 353 Rows 1
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10.7. BLOCK PRECONDITIONERS 355 A b
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10.8. PRECONDITIONERS FOR THE NORMA
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Chapter 11 PARALLEL IMPLEMENTATIONS
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11.3. TYPES OF PARALLEL ARCHITECTUR
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11.4. TYPES OF OPERATIONS 377 among
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11.5. MATRIX-BY-VECTOR PRODUCTS 379
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11.6. STANDARD PRECONDITIONING OPER
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Chapter 12 PARALLEL PRECONDITIONERS
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12.3. POLYNOMIAL PRECONDITIONERS 39
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12.4. MULTICOLORING 407 22 10 23 11
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12.5. MULTI-ELIMINATION ILU 409 D1
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12.5. MULTI-ELIMINATION ILU 411 as
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12.5. MULTI-ELIMINATION ILU 413 1.
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12.6. DISTRIBUTED ILU AND SSOR 415
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12.7. OTHER TECHNIQUES 417 As in th
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12.7. OTHER TECHNIQUES 419 The prec
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12.7. OTHER TECHNIQUES 421 P-12.3 S
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Chapter 13 MULTIGRID METHODS The co
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13.2. MATRICES AND SPECTRA OF MODEL
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13.3. INTER-GRID OPERATIONS 437 and
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13.4. STANDARD MULTIGRID TECHNIQUES
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13.5. ANALYSIS FOR THE TWO-GRID CYC
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13.5. ANALYSIS FOR THE TWO-GRID CYC
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13.6. ALGEBRAIC MULTIGRID 455 to sa
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13.6. ALGEBRAIC MULTIGRID 457 It is
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13.6. ALGEBRAIC MULTIGRID 459 all k
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13.6. ALGEBRAIC MULTIGRID 461 • T
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13.6. ALGEBRAIC MULTIGRID 463 1. f
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13.7. MULTIGRID VS KRYLOV METHODS 4
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13.7. MULTIGRID VS KRYLOV METHODS 4
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Chapter 14 DOMAIN DECOMPOSITION MET
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14.1. INTRODUCTION 471 in Figure 14
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14.1. INTRODUCTION 473 4. Subdomain
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14.2. DIRECT SOLUTION AND THE SCHUR
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14.3. SCHWARZ ALTERNATING PROCEDURE
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14.4. SCHUR COMPLEMENT APPROACHES 4
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14.4. SCHUR COMPLEMENT APPROACHES 4
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14.5. FULL MATRIX METHODS 501 inter
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14.5. FULL MATRIX METHODS 503 Assum
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14.6. GRAPH PARTITIONING 505 14.6.2
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14.6. GRAPH PARTITIONING 507 The ma
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14.6. GRAPH PARTITIONING 509 ALGORI
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14.6. GRAPH PARTITIONING 511 Figure
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14.6. GRAPH PARTITIONING 513 foo wi
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516 BIBLIOGRAPHY [16] O. AXELSSON,
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518 BIBLIOGRAPHY [48] , An iterativ
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520 BIBLIOGRAPHY [83] N. CHRISOCHOI
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522 BIBLIOGRAPHY [117] H. C. ELMAN
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524 BIBLIOGRAPHY [153] G. H. GOLUB
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526 BIBLIOGRAPHY [188] R. KETTLER,
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528 BIBLIOGRAPHY [222] C. C. PAIGE,
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530 BIBLIOGRAPHY [255] Y. SAAD AND
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532 BIBLIOGRAPHY [289] H. A. VAN DE
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534 BIBLIOGRAPHY [324] L. ZHOU AND
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536 INDEX block GMRES, 222-223 mult
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538 INDEX Element-By-Element precon
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540 INDEX ILU, 301-332 Crout varian
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542 INDEX normal, 4, 21 orthogonal,
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544 INDEX by ILQ, 359 EBE, 417 inco
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546 INDEX permutation and reorderin
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