1 / 13

Preconditioners for the Space-Time Solution of Large-Scale PDE Applications

This presentation discusses the use of preconditioners for space-time solutions of large-scale PDE applications. Topics covered include motivation, space-time formulations, parallelism in time, and numerical experiments. The results show the effectiveness of different preconditioners in improving performance.

seawell
Download Presentation

Preconditioners for the Space-Time Solution of Large-Scale PDE Applications

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Preconditioners for the Space-Time Solution of Large-Scale PDE Applications Danny Dunlavy, Andy Salinger Sandia National Laboratories Albuquerque, New Mexico, USA SIAM Parallel Processing February 23, 2006 SAND2006-1075C Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company,for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

  2. Motivation • Large-scale Transient Applications • Space-Time Formulations • Transient calculations: • Initial conditions and parameter • Space-time formulations: • Parallelism in time (and space) • Intermediate/final values • Integrated values • Periodic orbits • Applications • Current: Fluid flow (MPSalsa) • Planned: Semiconductor devices (Charon) Fluid/structure problems (Aria/Sierra) SIAM Parallel Processing 2006

  3. Space-Time Formulation Instead, solve for all solutions at once: Transient Simulation of: First solve: where Then solve: Then solve: … and with Newton solve: Solve system with GMRES (right preconditioning) SIAM Parallel Processing 2006

  4. Space-Time Preconditioners = Solve/Precondition • Global • Sequential • Parallel • Block Diag • “Parareal” (Multilevel) = Multiply, Add SIAM Parallel Processing 2006

  5. Each processor owns 1 time step for the entire spatial domain proc 0: proc 0: proc 0: proc 1: proc 0: proc 1: proc 2: proc 0: proc 0: Each processor owns 4 time steps for ¼ of the spatial domain proc 3: proc 0: proc 1: Each processor owns 2 time steps for ½ of the spatial domain Space and Time Partitioned Independently Ex: 4 Time Steps on 4 Procs Spatial Domains Space-Time Domains Proc 0: Proc 1: Proc 2: Proc 3: SIAM Parallel Processing 2006

  6. Preliminary Analysis – Computational Time Time Integration Sequential (preconditioning only, 1 time domain) Sequential (preconditioning only, Nproc time domains) Parallel (Nproc time domains) Parareal (Nproc time domains) Global (Nproc time domains) SIAM Parallel Processing 2006

  7. Demonstration Problem • Frank-Kamenetskii explosion model • Extended to include reactant consumption term • 5 scalar PDEs • 5 unknowns: insulated axis of symmetry SIAM Parallel Processing 2006

  8. Numerical Experiments • Methods • MPSalsa: FEM: 64 x 48 elements, time steps: 32, unknowns: 509,600 • Trilinos: Newton (NOX) : 4–7 iterations GMRES (Aztec) : 400 max. outer, 200 max. inner iterations ILUk (Ifpack) : k=1 (fill) Continuation in (LOCA): 1 step • Fixed Number of Spatial Domains (4) • Processors: 4 8 16 32 64 128 • Time Domains: 1 2 4 8 16 32 • How much can parallelism in time speed up the solve? • Fixed Number of Processors (32) • Spatial domains: 1 2 4 8 16 32 • Time domains: 32 16 8 4 2 1 • How can space-time parallelism be used most effectively? SIAM Parallel Processing 2006

  9. Results – Fixed Number of Spatial Domains (4) Processors 4 8 16 32 64 128 Time Domains 1 2 4 8 16 32 Sequential (1e-6, P) 236 164 131 115 108 104 Sequential (1e-2, P) 217 139 94 74 67 65 Sequential (P, 1e-3) 931 636 477 380 352 357 Parallel (1e-6, 1e-3) 331 210 148 116 98 93 Parallel (P, 1e-3) 943 477 246 108 61 53 Block Diag (P, 1e-3) 1027 523 263 110 64 53 Global (1e-3) 958 491 244 105 57 46 Parareal (1e-6, P) 237 112 145 119 Parareal (P, 1e-3) 950 277 181 106 Preconditioner (block solve tolerance, GMRES tolerance); P = preconditioning only SIAM Parallel Processing 2006

  10. Results – Fixed Number of Spatial Domains (4) Best Results Sequential (1e-2, P) Parallel (P, 1e-3) Global (1e-3) SIAM Parallel Processing 2006

  11. Results – Fixed Number of Processors (32) Spatial Domains 32 16 8 4 2 1 Time Domains 1 2 4 8 16 32 Sequential (1e-6, P) 72 71 87 100 168 122 Sequential (1e-2, P) 55 52 59 66 103 84 Sequential (P, 1e-3) 551 310 339 359 548 625 Parallel (1e-6, 1e-3) 117 95 99 107 154 170 Parallel (P, 1e-3) 548 217 162 135 84 70 Block Diag (P, 1e-3) 550 204 161 137 88 69 Global (1e-3) 365 172 143 125 81 57 Parareal (1e-6, P) 70 75 110 226 Parareal (P, 1e-3) 551 188 184 399 Preconditioner (block solve tolerance, GMRES tolerance); P = preconditioning only SIAM Parallel Processing 2006

  12. Summary • Conclusions • Several preconditioners improve performance of space-time solves • Achieve time parallelism for serial codes (fixed spatial domains) • Future Work • More time steps (study limits of time parallelism) • Comparison of analysis to experimental timing results • Periodic orbit tracking • Initial guesses for Newton (mesh refinement/preconditioning) • Other time discretizations (p-refinement) • Adaptive time steps (r-adaptivity) and time domain partitioning SIAM Parallel Processing 2006

  13. Thank You MS44 – Parallel Space-Time Algorithms Friday, 9:45 – 11:45 AM (Carmel Room) Space-Time Solution of Large-Scale PDE Applications Andy Salinger, 11:15 – 11:40 AM Danny Dunlavy dmdunla@sandia.gov Andy Salinger agsalin@sandia.gov SIAM Parallel Processing 2006

More Related