1. Introduction to Scientific Computing II Overview Michael Bader

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 email Michael Bader: bader@in.tum.de Wolfgang 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? 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!!!

12. 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

13. Typical SLE sparse band structure

14. Example

15. Gaussian Elimination (LU)

24. Gaussian Elimination – Costs Storage: (for an n-by-n grid) matrix has N = n 2 rows in L and U: n new non-zeros per row therefore: O(Nn) = O(n 3 ) bytes In 3D: N = n 3 rows, n 2 new non-zeros therefore: O(Nn 2 ) = O(n 5 ) bytes

25. Gaussian Elimination – Costs Operations: matrix has N = n 2 rows for each row, eliminate n non-zeros in column below addition of rows requ. O(n) operations therefore: O(Nn 2 ) = O(n 4 ) operations In 3D: N = n 3 rows, n 2 new non-zeros therefore: O(Nn 4 ) = O(n 7 ) operations

26. Gaussian Elimination – Costs Storage: (for an n-by-n grid) 2D: O(Nn) = O(n 3 ) bytes 3D: O(Nn 2 ) = O(n 5 ) bytes Computation: 2D: O(Nn 2 ) = O(n 4 ) operations 3D: O(Nn 4 ) = O(n 7 ) 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 u xx =0)

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

30. 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 Relaxation Methods – Gauss-Seidel

50. 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

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

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

55. 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

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

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

62. Relaxation Methods – SOR

63. 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

79. 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

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

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

84. 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