Index

ADVC, 138 BEM-FEM coupling, 5 ADVC code, 164, 165 Benson, M.W., 205 advection-diffusion-reaction, 131 Benzi, M., 205 ADVENTURE, 165, 182, 184 BiCGStab, see stabilized bi-conjugate ADVENTURE project, 138 gradient affine transformation, 231, 235, 237 Blue Gene/L, 20, 167 AIAC, see asynchronous iteration asyn- Bonnet, M., 230 chronous communication boundary conditions, 5, 230 AISM, see approximate inverse with the essential, 248 Sherman-Morrison periodic, 251, 254 Akiba, H., 9, 137 slip, 249 application layer, 125 Burger’s equation, 23 approximate inverse, 204 Burrage, K., 20 approximate inverse with the Sherman- Morrison, 203, 205, 211, 215 CG, see conjugate gradient asynchronous iteration asynchronous CGCG, see coarse grid conjugate gradient communication, 106, 107, 112, Cholesky factorization, 204 113, 122, 125, 126, 130, 134 Cnaga, M.E., 172 asynchronous iteration model, 105 coarse grid, 3, 150, 158, 162, 180 asynchronous methods, 71–73, 76–79 coarse grid conjugate gradient, 10, 137– asynchronous model, 9 170 coarse grid correction, 171 Bahi, J.M., 8, 105 coarse grid problem, 137 balancing domain decomposition, 9–11, coarse grid projection, 4 137, 138, 157, 158, 160–169, coarse propagator, 34 171, 173–175, 177, 179–183, coarse simulation, 27 187, 189, 194, 198 coarse solver, 25 balancing domain decomposition by con- coarse spacial grid, 30 straints, 3 coarsening, 29 bandwidth, 72 communication layer, 125 Bavestrello, H., 174 complexity, 24, 39 BDD, see domain decomposition balanc- complexity analysis, 26 ing compressed row storage, 256, 258 BDDC, see balancing domain decomposi- congruent subdomains, 229, 236, 237, tion by constraints 255

267 268 Substructuring Techniques and Domain Decompostion Methods conjugate gradient, 10, 127, 128, 130 Fagakis, Y., 175 conjugate gradient method, 148 Farhat, C., 9, 174, 180 conjugate projected gradient method, 148, fault tolerant, 71 149, 161, 171 FETI, see finite element tearing and inter- contact, 5, 190, 197 connecting convergence, 101 finite element, 2, 138, 171, 229, 231 analysis, 27, 79 vector-valued, 234 condition, 83 finite element tearing and interconnecting, detection, 96, 112 3, 4, 9, 10, 138, 139, 168, 173– global, 97 175 local, 96 finite element tearing and interconnecting properties, 27 - dual primal, 174 Courant-Friedrichs-Lewy, 31 Frobenius norm, 205 Couturier, R., 8, 105 Fujitsu GP7000F, 264 CPG, see conjugate projected gradient method Gander, M., 21 CSR, see compressed row storage Gander, M.J., 48 Gauss Seidel method, 48, 73 daemon, 123 generalised minimal residual method, De Roeck, Y.-H., 180 203–205, 214, 217, 218, 221, 223 deadlock, 125 generalised minimal residuals, 12 decomposer, 3 GMRES, see generalised minimal resid- see decomposition congruent, 255 uals, generalised minimal residual method decoupled parametric systems, 51 greedy algorithm, 12, 204–207 Dirichlet boundary conditions, 179, 230 grid computing, 7, 105 discrete harmonic expansion, 144 grid environment, 134 distributed algorithms, 45 Grid’5000, 107, 126, 128, 133, 134 distributed computing, 7, 71 Grid’5000 architecture, 9 divide and conquer, 2 G´eradin, M., 180 domain decomposition, 2, 137, 173–202, 229 Hackbush, W., 21 balancing, 3 harmonic analysis, 4 overlapping, 33 heterogeneous problems, 4 Dryja, M., 180 dual-primal, 5 I/O functions, 184 dual-primal FETI, 10 IBM Blue Gene/L, 137, 166 dynamic substructuring, 2, 3 interface conditions, 3 interface nodes, 179 Earth Simulator, 175 interface problem, 2, 177 efficiency, 231 inverse preconditioners, 11, 204 elliptic problems, 6 iterative algorithms, 107 Euler’s method, 7 iterative coefficient, 60 European Option example, 57 iterative methods, 71 eXtended finite element method, 5 iterative solvers, 2, 32, 113, 229 Index 269 iterative substructuring method, 137, 138, meta-cluster, 106, 109–111, 130, 134 149 Metis, 183, 197 minimum residual scheme, 203–205, 210, JACE, 8, 9, see Java asynchronous com- 211, 215, 217, 218, 221, 223, 225 puting environment Miranker, W.L., 47 JACK library, 8, 71, 93 Miyamura, T., 5, 10, 171 Jacobi iterative, 48 model reduction, 29 Jacobi method, 8, 73, 75 Montero, G., 205 Java, 8, 105, 128 Moriya, K., 11, 203, 205 Java asynchronous computing environ- mortar interfaces, 5 ment, 105, 107, 122–125, 128 MPC, see multi-point constraints Klawonn, A., 175 MPI Library, 8, 93, 94 Krylov solvers, 204 MTBF, see mean time between failure Krylov subspace, 203, 231 multi-models, 4 Krylov subspace methods, 12 multi-physics, 4 Krylov subspace solvers, 204 multi-point constraints, 5, 10 multi-splitting algorithm, 8, 9 Lagrange multiplier, 11 multi-step methods, 7 Lagrangian method, 173 multigrid, 4, 21 Lai, C.-H., 7, 45 multipoint constraint, 171–174, 176, 177, Laiymani, D., 8, 105 179, 181, 183–185, 187, 189, Lapalace equation, 144 194, 197, 198 Laplace transform, 46, 50, 53, 61, 63, 64 multipoint constraints, 180 large scale analysis, 137 multishooting technique, 21 latency, 72, 95 multisplitting algorithms, 105, 113 Le Tallec, P., 180 multisplitting Newton algorithm, 121, 131 Lelarasmee, E., 21 Linger, W., 47 Nakajima, K., 173 LINPACK benchmark, 166 NAS benchmark, 9 Lions, J.-L., 6, 19, 47 NAS parallel benchmark, 126, 127 local cluster, 108 Navier equations, 13, 235 local solver, 3 Neuman condition, 145 local space, 153 Neumann method, 155 LU decomposition, 148 Neumann preconditioner, 137, 161 LU preconditioner, 204 Neumann-Neumann preconditioner, 180 lump-scaling, 4 Newton iteration, 12 Maday, Y., 6, 19, 47 Newton method, 215 Magoul`es, F., 1, 8, 71 Newton scheme, 205, 215, 217 Mandel, J., 9 Newton’s linearisation, 61 master DOFs, 173 Newton’s method, 63 master-slave method, 173 Nikishn, A.A., 205 matrix-vector multiplication, 254 NIO protocol, 125 mean time between failure, 72 Nodera, T., 11, 203 message passing, 106 nonlinear parabolic problem, 54 270 Substructuring Techniques and Domain Decompostion Methods nonlinear problems, 7 coarse, 34 fine, 34 O’Leary, D.P., 114, 115 propagators, 28 Ogino, M., 180 OpenMP library, 13, 230, 231, 262, 264 reduced system, 204, 214 optimal control, 7 remote method invocation, 125 order reduction, 2 rigid body motion, 10 ordinary differential equations, 19 Rixen, D., 1, 174 orthogonal decomposition, 145, 147, 162 Ruehli, A.E., 21 orthogonal projection, 229, 248, 251, 254 Runge Kutta methods, 7 orthogonal transformation, 229, 231, 240 Runge-Kutta methods, 20

P-FETI method, see primal substructuring Saad, Y., 204, 205 Papadrakakis, M., 175 Saint-Georges, S., 172 parallel algorithms, 45 Sangiovanni-Vincentelli, A.L., 21 parallel architectures, 108 scalability, 21 parallel computing, 71, 171 scaling techniques, 4 parallel machines, 108 Schnabel, R.B., 204 parallel processing, 137 Schultz, M.H., 204 parallel structural analysis, 138 Schur parallelism efficiency, 24, 39 complement, 2, 12, 148, 180, 203– parareal, 19–44, 47 205, 207, 214, 217, 223 Parmetis, 197 dual, 4 partial differential equations, 19 primal, 4 PC cluster, 11, 12, 204 Schwarz iteration, 2 PCPG, see preconditioned conjugate pro- Schwarz method, 2, 137–139, 143, 144, jected gradient method 148, 149, 152, 155, 156, 161, penalty method, 173 164, 168 periodicity, 230 Sherman-Morrison formula, 205 Perron-Frobenius theorem, 80 Shioya, R., 175 Picard-Lindel¨of iteration, 48 SIAC, see synchronous iterations- Poisson equation, 144, 230 asynchronous communications Poisson problem, 8 similarity transformation methods, 51 poro-elastic models, 5 similarity transforms, 49 preconditioned conjugate projected gradi- Simonicini, V., 204 ent method, 178 SISC, see synchronous iterations- preconditioner, 3–5, 171, 204, 205 synchronous communications preconditioning, 173, 204 slave DOFs, 173 predictor, 6 sparseness, 148 predictor corrector approach, 19 sparsity, 204 primal substructuring, 173, 175 spawner, 124 programming environment, 92 SPOOLES, 182 project gradient methods, 172 stability, 40 projections, 3 stabilized bi-conjugate gradient, 12 propagator, 27 Stokes problem, 4 Index 271 structural analysis, 138 subdomains non-overlapping, 3, 237 overlapping, 3 transformed, 230 substructuring, 3 asynchronous, 84 iterative, 139–143 methods, 71–104 synchronous, 84 Suzuki, A., 12, 229 synchronous communications, 106 synchronous iteration, 105 synchronous iterations-synchronous com- munications, 111 synchronous methods, 73, 75–77 synchronous model, 9 Szyld, D.B., 204

Tabata, M., 12, 229 temporal domain, 7 temporal integration, 45, 47 time-marching scheme, 46 TOP500, 166 transformation methods, 7, 45, 46, 48 transient problems, 7 Turinici, G., 6, 19, 47 user application, 125 Uzawa iteration, 4

Vanderwalle, S., 21 Vandewall, S., 48 Venet,C.,8,71 virtual control, 35 viscosity, 23 waveform relaxation method, 134 weak formulations, 246 White, R.E., 114, 115 Widlund, O.B., 175, 180 worker, 124

Zhang, L., 11, 203 Zienkiewicz, O.C., 230 mmx