1. Introduction to Scientific Computing II
Multigrid
Miriam Mehl, Michael Bader

2. Multigrid – Algorithm iterate (GS) on the fine grid
restrict residual to the coarse grid
solve coarse grid equation for the error
interpolate error to the fine grid
correct fine grid solution

3. Multigrid – Things to Choose
smoother
relation step sizes coarse – fine
grid transfer operators
restriction
interpolation
processing order of grid levels

4. Multigrid – Convergence
two grid analysis
h-independent convergence
for ‚good‘ components

5. Two Grid – Multigrid Example: 2D Poisson 5-point-stencil h
two-grid analysis
V-cycle
1/32
0.042
1/64
0.044
1/128
1/256
0.043
1/512
1/1024
1/2048

6. Multigrid – Some Rules smoother optimal smoothing
not(!) optimal convergence
small number of smoothing iterations!

7. Multigrid – Some Rules grid coarsening standard: doubling of h
exceptions:
anisotropic operators
adaptively refined grids
unstructured grids/general SLEs

8. Multigrid – Some Rules restriction/interpolation
order of restriction + order of interpolation > order of discretisation

9. Multigrid – Some Rules
V-cycle
faster
W-cycle
more robust

10. Multigrid – Parallelisation
parallel smoothing
parallel restriction and interpolation
parallel stopping criteria

11. Multigrid – Parallel Smoothing
Gauss Seidel
 Jacobi-like operation!!!
processor 1
processor 2

12. Multigrid – Parallel Smoothing
Gauss Seidel
processor 1
processor 2

13. Multigrid – Parallel Smoothing
Gauss Seidel
processor 1
processor 2

14. Multigrid – Parallel Smoothing
Gauss Seidel
 Jacobi-like operation!!!
processor 1
processor 2

15. Multigrid – Parallel Smoothing
Gauss Seidel  different result than sequential GS!!!
processor 1
processor 2

16. Multigrid – Parallel Smoothing
Alternatives: Red-Black Gauss-Seidel
processor 1
processor 2

17. Multigrid – Parallel Smoothing
Alternatives: Red-Black Gauss-Seidel
processor 1
processor 2

18. Multigrid – Parallel Smoothing
Alternatives: Red-Black Gauss-Seidel
processor 1
processor 2

19. Multigrid – Parallel Smoothing
Alternatives: Red-Black Gauss-Seidel
processor 1
processor 2

20. Multigrid – Parallel Smoothing
Alternatives: Red-Black Gauss-Seidel
processor 1
processor 2

21. Multigrid – Parallel Smoothing
Alternatives: Red-Black Gauss-Seidel

22. Multigrid – Parallel Smoothing
Alternatives: damped Jacobi
processor 1
processor 2

23. Multigrid – Parallel Smoothing
Alternatives: damped Jacobi
processor 1
processor 2

24. Multigrid – Parallel Smoothing
Alternatives: damped Jacobi
processor 1
processor 2

25. Multigrid – Parallel Smoothing
red-black GS: robust and fast smoothing
further reading: Irad Yavneh, Multigrid smoothing factors for red-black Gauss-Seidel relaxation applied to a class of elliptic operators, SIAM Journal on Numerical Analysis, 32 (4), 1995
Jun Zhang, Acceleration of five-point red-black Gauss-Seidel in multigrid for Poisson equation, Applied Mathematics and Computation, 80(1), 1996
damped Jacobi: good smoothing

26. Ferienakademie, Sarntal, Sep 23 – Oct 5, 2012
Universität
Erlangen-Nürnberg
Technische Universität
München
Universität Stuttgart
Ferienakademie, Sarntal, Sep 23 – Oct 5, 2012
Course 4: Scales and Scalability as Challenges for CSE
(Bader, Schweitzer, Wellein)
Deadline: Yesterday