1. Introduction to Scientific Computing II
Conjugate Gradients
Dr. Miriam Mehl

2. Steepest Descent – Basic Idea
solution of SLE
minimization
iterative
one-dimensional minima
direction of steepest descent?

3. Steepest Descent – Algorithm

4. Steepest Descent – Algorithm II

5. Steepest Descent – Example
initial error
after 1 iteration
after 10 iterations

6. Steepest Descent – Example
1/128
1/64
1/32
1/16 h
48,629
11,576
2,744
646
iterations

7. Steepest Descent – Convergence
Poisson with 5-point-stencil
like Jacobi

8. Steepest Descent – Convergence

9. Conjugate Gradients – Basic Idea
solution of SLE
minimization
iterative
one-dimensional minima
no repeating search directions

10. Steepest Descent – Principle

11. Conjugate Gradients – Principle

12. CG – Algorithm

13. Steepest Descent – Example
initial error
after 1 iteration
after 10 iterations

14. Conjugate Gradients – Example
initial error
after 1 iteration
after 10 iterations

15. Conjugate Gradients – Example
322
157 76 35
iterations cg
1/128
1/64
1/32
1/16 h
48,629
11,576
2,744
646
iterations sd
16,129
3,969
961
225
#unknowns

16. CG – Convergence Poisson with 5-point-stencil like SOR
no parameter adjustment

17. PCG – Idea convergence rate cg: Solve system M-1Ax=M-1b
better condition number k
M-1 easy to apply

18. PCG – Algorithm

19. PCG – Algorithm