Introduction to Scientific Computing II

1. Overview Michael Bader Introduction to Scientific Computing II

2. Recall: Scientific Computing “Pipeline”

3. Topic #1 – SLE(numerical treatment, implementation) ???

4. Topic #2 – Molecular Dynamics(entire pipeline for one application)

5. Prerequisites • discretisation of PDEs • linear algebra • Gaussian elimination • basics on iterative solvers • Jacobi, Gauss-Seidel, SOR, MG • matlab

6. Organization • lecture (90 min/week) • theory • methods • simple examples • tutorials (45 min/week) • more examples • make your own experiences

7. What Determines the Grading? • written exam at the end of the semester • no weighting of tutorials however: solving tutorials is essential • for understanding and remembering subjects • for your success in the exam

8. Course Material • slides (short, only headwords) • exercise sheets • make your own lecture notes! • find your own solutions! • solutions presented in the tutorials

9. Contact • for questions contact us after the lectures • or fix a date per emailMichael Bader: bader@in.tum.deWolfgang Eckhardt:eckhardw@in.tum.de

10. Introduction to Scientific Computing II From Gaussian Elimination to Multigrid – A Recapitulation

11. What’s the Problem to be Solved? Application Scenario Partial Differential Equations Modelling Scientific Computing I Finite Elements Finite Differences (Finite Volumes) Scientific Computing I Numerical Programming II Systems of linear equations LU, Richardson, Jacobi, Gauss-Seidel, SOR, MG Scientific Computing I, Scientific Computing Lab, Numerical Programming I More on this!!!

12. Example Equation v v v v v v v v v v v v v v v two-dimensional Poisson equation • heat equation • diffusion • membranes • … grid + finite differences

13. Typical SLE sparse band structure

14. Example

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 (LU)

22. Gaussian Elimination (LU)

23. Gaussian Elimination (LU)

24. Gaussian Elimination – Costs • Storage: (for an n-by-n grid) • matrix has N = n2 rows • in L and U: n new non-zeros per row • therefore: O(Nn) = O(n3) bytes • In 3D: • N = n3rows, n2 new non-zeros • therefore: O(Nn2) = O(n5) bytes

25. Gaussian Elimination – Costs • Operations: • matrix has N = n2 rows • for each row, eliminate n non-zeros in column below • addition of rows requ. O(n) operations • therefore: O(Nn2) = O(n4) operations • In 3D: • N = n3rows, n2 new non-zeros • therefore: O(Nn4) = O(n7) operations

26. Gaussian Elimination – Costs • Storage: (for an n-by-n grid) • 2D: O(Nn) = O(n3) bytes • 3D: O(Nn2) = O(n5) bytes • Computation: • 2D: O(Nn2) = O(n4) operations • 3D: O(Nn4) = O(n7) operations • Even for problems of modest size (n = 100-1000)  Gaussian Elimination is unfeasible

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

28. Relaxation Methods problem: order an amount of peas on a straight line (corresponds to solving uxx=0)

29. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

30. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

31. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

32. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

33. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

34. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

35. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

36. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

37. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

38. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

39. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

40. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

41. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

42. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

43. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

44. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

45. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

46. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

47. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

48. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

49. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours

50. Relaxation Methods – Gauss-Seidel sequentially place peas on the line between two neighbours • we get a smooth curve instead of a straight line • global error is locally (almost) invisible