Introduction to Scientific Computing II Steepest Descent Conjugate

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Introduction to Scientific Computing II Steepest Descent & Conjugate Gradients Dr. Miriam Mehl

Introduction to Scientific Computing II Steepest Descent & Conjugate Gradients Dr. Miriam Mehl

Steepest Descent – Basic Idea • solution of SLE • minimization • iterative one-dimensional

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

Steepest Descent – Principle

Steepest Descent – Principle

Steepest Descent – Principle

Steepest Descent – Principle

Steepest Descent – Algorithm

Steepest Descent – Algorithm

Steepest Descent – Algorithm II

Steepest Descent – Algorithm II

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

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

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

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

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

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

Steepest Descent – Convergence

Steepest Descent – Convergence

Conjugate Gradients – Basic Idea • solution of SLE • minimization • iterative one-dimensional

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

Steepest Descent – Principle

Steepest Descent – Principle

Conjugate Gradients – Principle

Conjugate Gradients – Principle

CG – Algorithm

CG – Algorithm

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

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 initial error after 1 iteration after 10 iterations

Conjugate Gradients – Example h iterations sd iterations cg #unknowns 1/16 646 35 225

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

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

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

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