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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

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Outline Background Steepest Descent Conjugate Gradient

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

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

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

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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.

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Directional Derivatives: first, the one dimension derivative:

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Directional Derivatives : Along the Axes…

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Directional Derivatives : In general direction…

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Directional Derivatives

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In the plane The Gradient: Definition in

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The Gradient: Definition

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

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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，

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Steepest Descent

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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)

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

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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:

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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 x 0 :

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Conjugate Gradient

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Gradient

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4 共轭斜量法与最速下降法的比较：

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