1. Introduction to Scientific Computing II From Gaussian Elimination to Multigrid – A Recapitulation Dr. Miriam Mehl

2. Tasks – SLE ???

3. Tasks – Molecular Dynamics

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

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

6. What Determines the Grading? written exam at the end of the semester no weighting of tutorials !!!! solving tutorials is essential !!!! -for understanding and remembering subjects -for your success in the exam

7. Materials slides (short, only headwords) exercise sheets  make your own lecture notes!  find your own solutions!  solutions presented in the tutorials

8. Rules for questions ask or fix a date per email Dr. Miriam Mehl: mehl@in.tum.de Martin Buchholz: buchholm@in.tum.de

9. Introduction to Scientific Computing II From Gaussian Elimination to Multigrid – A Recapitulation Dr. Miriam Mehl

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

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

12. Typical SLE sparse band structure

13. Example

14. Gaussian Elimination (LU)

23. Gaussian Elimination – Costs 2D: O(N 4 ) 3D: O(N 7 )

24. Gaussian Elimination – Costs 2D hallo hruntime (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

25. Gaussian Elimination – Costs 3D hallo hruntime (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

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

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

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

29. Relaxation Methods – Gauss-Seidel

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

49. 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 Relaxation Methods – Gauss-Seidel

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

51. Relaxation Methods – Jacobi place peas on the line between two neighbours in parallel

52. Relaxation Methods – Jacobi place peas on the line between two neighbours in parallel

53. Relaxation Methods – Jacobi place peas on the line between two neighbours in parallel

54. Relaxation Methods – Jacobi place peas on the line between two neighbours in parallel we get a high plus a low frequency oscillation  these fequencies are locally (almost) invisible

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

56. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

57. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

58. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

59. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

60. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

61. Relaxation Methods – SOR

62. sequentially correct location of peas a little more than to the line between two neighbours

63. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

64. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

65. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

66. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

67. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

68. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

69. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

70. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

71. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

72. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

73. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

74. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

75. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

76. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

77. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours

78. Relaxation Methods – SOR sequentially correct location of peas a little more than to the line between two neighbours better than GS and J, but still not optimal

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

80. Relaxation Methods – Hierarchical place peas on the line between two neighbours in parallel, but in a hierarchical way from coarse to smooth

81. Relaxation Methods – Hierarchical place peas on the line between two neighbours in parallel, but in a hierarchical way from coarse to smooth

82. Relaxation Methods – Hierarchical place peas on the line between two neighbours in parallel, but in a hierarchical way from coarse to smooth

83. Relaxation Methods – Hierarchical place peas on the line between two neighbours in parallel, but in a hierarchical way from coarse to smooth exact solution in one step  unfortunately only in 1D, 2D and 3D: multigrid