1. Introduction to Scientific Computing II
From Gaussian Elimination to Relaxation Methods
Dr. Miriam Mehl

2. Typical SLE
sparse
band structure

3. Example

4. Gaussian Elimination

5. Gaussian Elimination

6. Gaussian Elimination

7. Gaussian Elimination

8. Gaussian Elimination

9. Gaussian Elimination

10. Gaussian Elimination

11. Gaussian Elimination

12. Gaussian Elimination (LU)

13. Gaussian Elimination (LU)

14. Gaussian Elimination (LU)

15. Gaussian Elimination (LU)

16. Gaussian Elimination (LU)

17. Gaussian Elimination (LU)

18. Gaussian Elimination (LU)

19. Gaussian Elimination (LU)

20. Gaussian Elimination (LU)

21. Gaussian Elimination – Costs

22. Gaussian Elimination – Costs
O(1/h)2

23. Gaussian Elimination – Costs
O(1/h)2*(1/h)2

24. Gaussian Elimination – Costs
O(1/h)2*(1/h)2 + O(1/h)

25. Gaussian Elimination – Costs
O(1/h)2*(1/h)2 + O(1/h)*(1/h)2

26. Gaussian Elimination – Costs
O(1/h)2*(1/h)2 + O(1/h)*(1/h)2 + O(1/h)

27. Gaussian Elimination – Costs
O(1/h)2*(1/h)2 + O(1/h)*(1/h)2 + O(1/h)*(1/h)2

28. Gaussian Elimination – Costs
2D: O(1/h)4
3D: O(1/h)7

29. Gaussian Elimination – Costs 2Dh
runtime (HLRB2, 62 TFlop/s)
2-7
0.02 sec
2-8
0.27 sec
2-9
4.4 sec
2-10
1 min 16 sec
2-11
18 min 55 sec
2-12
5 h 02 min 40 sec
2-13
3 d 8 h 37 min 15 sec
hallo

30. Gaussian Elimination – Costs 3D
hallo h
runtime (HLRB2, 62 TFlop/s)
2-6
4 min 44 sec
2-7
10 h 05 min 24 sec
2-8
53 d 19 h 21 min 17 sec
2-9
18 a 313 d 21 h 54 min 22 sec

31. Iterative Solvers – Principle
series of approximations
costs per iteration?
convergence?
stopping criterion?

32. Relaxation Methods – Gauss-Seidel

33. Relaxation Methods – Jacobi