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
2.3 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
Outline ØBackground ØSteepest Descent ØConjugate Gradient
1 Background The min(max) problem min f(X X But we learned in calculus how to solve that kind of question
1 Background • The min(max) problem: • But we learned in calculus how to solve that kind of question! min f (x) x
real world” problem Connectivity shapes (isenburg,gumhold, gotsman) mesh=c=(v, e), geometry, What do we get only from C without geometry
“real world” problem • Connectivity shapes (isenburg,gumhold,gotsman) • What do we get only from C without geometry? mesh {C (V, E), geometry}
Motivation-"real world problem first we introduce error functionals and then try to minimize them E(x∈R3)=2(x-x)|-) (i,j)∈E E,(x∈R3)=∑L(x)2 (x)=∑x i(i,j)∈E
Motivation- “real world” problem • First we introduce error functionals and then try to minimize them: 2 3 ( , ) ( ) 1 n s i j i j E E x x x ( , ) 1 ( ) i j i i i j E L x x x d 3 2 1 ( ) ( ) n n r i i E x L x
Motivation-"real world problem Then we minimize E(C, h)=argmin(1-h)E, (x)+E, (x) x∈R >High dimension non-linear problem Conjugate gradient method is maybe the most popular optimization technique based on what we'll see here
Motivation- “real world” problem ØThen we minimize: ØHigh dimension non-linear problem. ØConjugate gradient method is maybe the most popular optimization technique based on what we’ll see here. 3 ( , ) arg min 1 ( ) ( ) n s r x E C E x E x
Directional derivatives first. the one dimension derivative .1 -0.05 d -.1 40.15 dx
Directional Derivatives: first, the one dimension derivative:
Directional derivatives Along the axes of(, y) y of(x, y) oX
x f x y ( , ) y f x y ( , ) Directional Derivatives : Along the Axes…
Directional derivatives In general direction v∈R21 of(x, y OV
v f x y ( , ) 2 v R v 1 Directional Derivatives : In general direction…
Directional Derivatives 42024 of(, y) of(x, y) ay ax
Directional Derivatives x f x y ( , ) y f x y ( , )