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Introduction to Scientific Computing II

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.

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Introduction to Scientific Computing II

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

  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 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

  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

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