Introduction to Scientific Computing II

Conjugate Gradients

Dr. Miriam Mehl

Steepest Descent – Basic Idea

• solution of SLE

• minimization

• iterative

one-dimensional minima

direction of steepest descent?

Steepest Descent – Algorithm

rcuurArrrc

uAbr

T

T

it

1,2,...it for

Steepest Descent – Algorithm II

vcrrrcuu

vrrrc

rAv

uAbr

T

T

1,2,...it for

1

Steepest Descent – Example

initial error

after 1 iteration after 10 iterations

Steepest Descent – Example

1/1281/641/321/16h

48,62911,576

2,744646

iterations

Steepest Descent – Convergence

• Poisson with 5-point-stencil

like Jacobi

minmax

minmax

11

Steepest Descent – Convergence

Conjugate Gradients – Basic Idea

• solution of SLE

• minimization

• iterative

one-dimensional minima

no repeating search directions

Steepest Descent – Principle

Conjugate Gradients – Principle

CG – Algorithm

;

;

;

;

1,2,...it for; ;

pabrp

rrb

pAbarrp

bauu

pApbrra

rpu-Abr

T

TT

Steepest Descent – Example

initial error

after 1 iteration after 10 iterations

Conjugate Gradients – Example

initial error

after 1 iteration after 10 iterations

Conjugate Gradients – Example

3221577635

iterations cg

1/1281/641/321/16h

48,62911,576

2,744646

iterations sd

16,1293,969

961225

#unknowns

CG – Convergence

• Poisson with 5-point-stencil

like SOR

no parameter adjustment

11

PCG – Idea

11

convergence rate cg:

Solve system M-1Ax=M-1b

better condition number

M-1 easy to apply

PCG – Algorithm

pabrMp

rMrb

pAbarr

pbauu

pApbrMra

T

T

T

1

1

1

1,2,...it for

PCG – Algorithm

pabvp

vrbv

pAbarr

pbauu

pApbvra

T

T

T

)r A,,0rations(solver_ite

1,2,...it for