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

Steepest Descent – Algorithm II

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

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

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

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

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 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

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

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

PCG – Algorithm

PCG – Algorithm