1. Steepest Decent and Conjugate Gradients (CG)

2. Solving of the linear equation system

3. Steepest Decent and Conjugate Gradients (CG) Solving of the linear equation system Problem: dimension n too big, or not enough time for gauss elimination Iterative methods are used to get an approximate solution.

4. Steepest Decent and Conjugate Gradients (CG) Solving of the linear equation system Problem: dimension n too big, or not enough time for gauss elimination Iterative methods are used to get an approximate solution. Definition Iterative method: given starting point, do steps hopefully converge to the right solution

5. starting issues

6. Solving is equivalent to minimizing

7. starting issues Solving is equivalent to minimizing A has to be symmetric positive definite:

8. starting issues

9. starting issues If A is also positive definite the solution of is the minimum

10. starting issues If A is also positive definite the solution of is the minimum

11. starting issues error: The norm of the error shows how far we are away from the exact solution, but can’t be computed without knowing of the exact solution.

12. starting issues error: The norm of the error shows how far we are away from the exact solution, but can’t be computed without knowing of the exact solution. residual: can be calculated

13. Steepest Decent

14. We are at the point. How do we reach ?

15. Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( )

16. Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( ) how far should we go?

17. Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( ) how far should we go? Choose so that is minimized:

18. Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( ) how far should we go? Choose so that is minimized:

19. Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( ) how far should we go? Choose so that is minimized:

20. Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( ) how far should we go? Choose so that is minimized:

21. Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( ) how far should we go? Choose so that is minimized:

22. Steepest Decent We are at the point. How do we reach ? Idea: go into the direction in which decreases most quickly ( ) how far should we go? Choose so that is minimized:

23. Steepest Decent one step of steepest decent can be calculated as follows:

24. Steepest Decent one step of steepest decent can be calculated as follows: stopping criterion: or with an given small It would be better to use the error instead of the residual, but you can’t calculate the error.

25. Steepest Decent Method of steepest decent:

26. Steepest Decent As you can see the starting point is important!

27. Steepest Decent As you can see the starting point is important! When you know anything about the solution use it to guess a good starting point. Otherwise you can choose a starting point you want e.g..

28. Steepest Decent - Convergence

29. Definition energy norm:

30. Steepest Decent - Convergence Definition energy norm: Definition condition: ( is the largest and the smallest eigenvalue of A)

31. Steepest Decent - Convergence Definition energy norm: Definition condition: ( is the largest and the smallest eigenvalue of A) convergence gets worse when the condition gets larger

32. Conjugate Gradients

33. is there a better direction?

34. Conjugate Gradients is there a better direction? Idea: orthogonal search directions

35. Conjugate Gradients is there a better direction? Idea: orthogonal search directions

36. Conjugate Gradients is there a better direction? Idea: orthogonal search directions only walk once in each direction and minimize

37. Conjugate Gradients is there a better direction? Idea: orthogonal search directions only walk once in each direction and minimize maximal n steps are needed to reach the exact solution

38. Conjugate Gradients is there a better direction? Idea: orthogonal search directions only walk once in each direction and minimize maximal n steps are needed to reach the exact solution has to be orthogonal to

39. Conjugate Gradients example with the coordinate axes as orthogonal search directions:

40. Conjugate Gradients example with the coordinate axes as orthogonal search directions: Problem: can’t be computed because (you don’t know !)

41. Conjugate Gradients new idea: A-orthogonal

42. Conjugate Gradients new idea: A-orthogonal Definition A-orthogonal: A-orthogonal (reminder: orthogonal: )

43. Conjugate Gradients new idea: A-orthogonal Definition A-orthogonal: A-orthogonal (reminder: orthogonal: ) now has to be A-orthogonal to

44. Conjugate Gradients new idea: A-orthogonal Definition A-orthogonal: A-orthogonal (reminder: orthogonal: ) now has to be A-orthogonal to

45. Conjugate Gradients new idea: A-orthogonal Definition A-orthogonal: A-orthogonal (reminder: orthogonal: ) now has to be A-orthogonal to can be computed!

46. Conjugate Gradients A set of A-orthogonal directions can be found with n linearly independent vectors and conjugate Gram- Schmidt (same idea as Gram-Schmidt).

47. Conjugate Gradients Gram-Schmidt: linearly independent vectors

48. Conjugate Gradients Gram-Schmidt: linearly independent vectors

49. Conjugate Gradients Gram-Schmidt: linearly independent vectors conjugate Gram-Schmidt:

50. Conjugate Gradients A set of A-orthogonal directions can be found with n linearly independent vectors and conjugate Gram- Schmidt (same idea as Gram-Schmidt). CG works by setting (makes conjugate Gram- Schmidt easy)

51. Conjugate Gradients A set of A-orthogonal directions can be found with n linearly independent vectors and conjugate Gram- Schmidt (same idea as Gram-Schmidt). CG works by setting (makes conjugate Gram- Schmidt easy) with

52. Conjugate Gradients

53. Conjugate Gradients

54. Conjugate Gradients

55. Conjugate Gradients

56. Conjugate Gradients

57. Conjugate Gradients

58. Conjugate Gradients

59. Conjugate Gradients

60. Conjugate Gradients

61. Conjugate Gradients

62. Conjugate Gradients

65. Method of Conjugate Gradients:

66. Conjugate Gradients - Convergence

68. Conjugate Gradients - Convergence for steepest decent for CG Convergence of CG is much better!