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2.3 共轭斜量法 （ Conjugate Gradient Methods)

2.3 共轭斜量法 （ Conjugate Gradient Methods). 属于一种迭代法，但如果不考虑计算过程的舍入误差， CG 算法只用有限步就收敛于方程组的精确解. Outline. Background Steepest Descent Conjugate Gradient. 1 Background. The min(max) problem: But we learned in calculus how to solve that kind of question!. “real world” problem.

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2.3 共轭斜量法 （ Conjugate Gradient Methods)

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  1. 2.3 共轭斜量法（Conjugate Gradient Methods) 属于一种迭代法，但如果不考虑计算过程的舍入误差，CG算法只用有限步就收敛于方程组的精确解

  2. Outline • Background • Steepest Descent • Conjugate Gradient

  3. 1 Background • The min(max) problem: • But we learned in calculus how to solve that kind of question!

  4. “real world” problem • Connectivity shapes (isenburg,gumhold,gotsman) • What do we get only from C without geometry?

  5. Motivation- “real world” problem • First we introduce error functionals and then try to minimize them:

  6. Motivation- “real world” problem • Then we minimize: • High dimension non-linear problem. • Conjugate gradient method is maybe the most popular optimization technique based on what we’ll see here.

  7. Directional Derivatives: first, the one dimension derivative:

  8. Directional Derivatives : Along the Axes…

  9. Directional Derivatives : In general direction…

  10. Directional Derivatives

  11. The Gradient: Definition in In the plane

  12. The Gradient: Definition

  13. 基本思想 • Modern optimization methods • A method to solve quadratic function minimization: (A is symmetric and positive definite)

  14. 2 最速下降法 （Steepest Descent） （1）概念：将 点的修正方向取为该点的负梯度方向 ，即为最速下降方向，该方法进而称之为最速下降法. （2）计算公式：任意取定初始向量，

  15. Steepest Descent

  16. 3 共轭斜量法（Conjugate Gradient） • Modern optimization methods : “conjugate direction” methods. • A method to solve quadratic function minimization: (A is symmetric and positive definite)

  17. Conjugate Gradient • Originally aimed to solve linear problems: • Later extended to general functions under rational of quadratic approximation to a function is quite accurate.

  18. Conjugate Gradient • The basic idea: decompose the n-dimensional quadratic problem into n problems of 1-dimension • This is done by exploring the function in “conjugate directions”. • Definition: A-conjugate vectors:

  19. Conjugate Gradient • If there is an A-conjugate basis then: • N problems in 1-dimension (simple smiling quadratic) • The global minimizer is calculated sequentially starting from x0:

  20. Conjugate Gradient

  21. Gradient Conjugate Gradient

  22. 4 共轭斜量法与最速下降法的比较：

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