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Introduction to Scientific Computing II

Introduction to Scientific Computing II. Preconditioned CG Continued. Dr. Miriam Mehl. Conjugate Gradients – Basic Idea. solution of SLE minimization iterative one-dimensional minima no repeating search directions. Conjugate Gradients – Principle. CG – Convergence.

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Introduction to Scientific Computing II

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  1. Introduction to Scientific Computing II Preconditioned CG Continued Dr. Miriam Mehl

  2. Conjugate Gradients – Basic Idea • solution of SLE • minimization • iterative • one-dimensional minima • no repeating search directions

  3. Conjugate Gradients – Principle

  4. CG – Convergence • Poisson with 5-point-stencil • like SOR • no parameter adjustment

  5. PCG – Idea convergence rate cg: • Solve system M-1Ax=M-1b • better condition number k • M-1 easy to apply

  6. PCG – Algorithm

  7. PCG – Preconditioner • (Jacobi) • SSOR • incomplete Cholesky decomposition • incomplete LU decomposition • (algebraic) multigrid

  8. Dr. Miriam Mehl Introduction to Scientific Computing II Parallel MG – A Scalable Alternative to Parallel CG?

  9. Can we parallelize a solver iteration in a scalable way? Parallel Solvers – Story 1

  10. parallel cg with “optimal” speedup #processors ~ #unknowns Parallel Speedup CG

  11. parallel smoother with optimal speedup #processors ~ #unknowns (on the finest level!) Parallel Speedup MG

  12. Can we parallelize the solver in a scalable way? Parallel Solvers -- Story 2

  13. parallel cg with “optimal” speedup per iteration #processors ~ #unknowns Parallel Speedup CG

  14. parallel smoother with optimal speedup #processors ~ #unknowns (on the finest level!) Parallel Speedup MG

  15. parallel cg with “optimal” speedup #processors ~ #unknowns Parallel Speedup CG

  16. parallel cg with “optimal” speedup #processors ~ #unknowns Parallel Speedup CG

  17. parallel cg with “optimal” speedup #processors ~ #unknowns Parallel Speedup CG

  18. Parallel Speedup CG h h/2 h/4 h/8 O(1/h) total time to solve? O(ln(1/h)/h)

  19. parallel smoother with optimal speedup #processors ~ #unknowns (on the finest level!) #processors = const on all levels time T time T/4 computation > communication Parallel Speedup MG

  20. parallel smoother with optimal speedup #processors ~ #unknowns (on the finest level!) #processors = const on all levels time T time T time T/2 computation < communication Parallel Speedup MG

  21. parallel smoother with optimal speedup #processors ~ #unknowns (on the finest level!) #processors = const on all levels time T #dof < #proc Parallel Speedup MG

  22. parallel smoother with optimal speedup #processors ~ #unknowns (on the finest level!) #processors = const on all levels time T time T time T #dof < #proc Parallel Speedup MG

  23. Parallel Speedup MG level 0 #dof < #proc level L1 Tcomp < Tcomm level L2 Tcomp > Tcomm level Lmax

  24. Parallel Speedup MG level 0 #dof < #proc level L1 Tcomp < Tcomm level L2 d Tcomp > Tcomm level Lmax

  25. Parallel Speedup MG level 0 #dof < #proc level L1 Tcomp < Tcomm level L2 d Tcomp > Tcomm level Lmax

  26. Parallel Speedup MG level 0 #dof < #proc level L1 Tcomp < Tcomm c level L2 d Tcomp > Tcomm level Lmax

  27. Parallel Speedup MG level 0 #dof < #proc level L1 Tcomp < Tcomm c level L2 d Tcomp > Tcomm level Lmax

  28. Parallel Speedup MG level 0 ln(p) #dof < #proc level L1 Tcomp < Tcomm c level L2 d Tcomp > Tcomm level Lmax

  29. Parallel Speedup MG level 0 ln(p)/2 #dof < #proc level L1 Tcomp < Tcomm c level L2 d Tcomp > Tcomm level Lmax

  30. Parallel Speedup MG p 0 total time per iteration? ln(p) ln(p)+c ln(p)+c+d

  31. Parallel Speedup MG h h/2 h/4 h/8 O(1) O(ln(1/h)) total time to solve?

  32. Iterative Solvers – Overview X system matrix s.p.d. or diagonally dominant and othersX system matrix must be s.p.d.!!!

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