1. Introduction to Scientific Computing II
Steepest Descent & 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 – Principle

4. Steepest Descent – Principle

5. Steepest Descent – Algorithm

6. Steepest Descent – Algorithm II

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

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

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

10. Steepest Descent – Convergence

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

12. Steepest Descent – Principle

13. Conjugate Gradients – Principle

14. CG – Algorithm

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

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

17. 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

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