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2010 年教学实践周 7.12-7.16. Cojugate Gradient Method. Zhengru Zhang ( 张争茹 ) [email protected] Office: Math. Building 413(West). Outline. Aim Method of Gauss Elimination Basic Iterative Methods Conjugate Gradient Method Derivation Theory Algorithm References Homework & Project. Aim.

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2010年教学实践周7.12-7.16

Cojugate Gradient Method

Zhengru Zhang (张争茹)

[email protected]

Office: Math. Building 413(West)


Outline

  • Aim

  • Method of Gauss Elimination

  • Basic Iterative Methods

  • Conjugate Gradient Method

    • Derivation

    • Theory

    • Algorithm

  • References

  • Homework & Project


Aim

Solve linear algebraic system like

a11 x1 + a12 x2 + ... + a1n xn = b1

a21 x1 + a22 x2 + ... + a2n xn = b2

...

an1 x1 + an2 x2 + ... + ann xn = bn

Using matrix, the above system can be written as

Ax=b

A is a N x N matrix, b is a N x 1 vector

Consider the case: A is large and sparse



Algorithm of Gaussian Elimination

without Pivoting

U = A, L = I

for k = 1 to N-1

for j = k +1 to N

ljk= ujk/ukk

uj,k:m = uj,k:m – ljkuk,k:m

  • LU Factorization, let A=LU

  • Solve Ly=b

  • Solve Ux=y


Operation Count of Gauss Elimination

  • Gauss Elimination and Back Substitution

  • There are 3 loops

  • There are 2 flops per entry

  • For eachk, the inner loop is repeated for rows k +1, …, N

  • Cost: about About N 3flops


Instability of Gaussian Elimination

without Pivoting

Examples

A2=

A1=

  • Pivoting

  • Partial Pivoting

  • Complete Pivoting

Remedy


Algorithm of Gaussian Elimination

with Partial Pivoting

U = A, L = I

For k = 1 to N-1

for j = k +1 to N

ljk= ujk/ukk

uj,k:m = uj,k:m – ljkuk,k:m


Basic Iterative Methods

  • How to construct iterative sequence?

  • Convergence? Conditions?

  • Convergence rate?


Jacobi iteration

Gauss Seidel iteration

X[k+1] = D-1(L+U) X[k] + D-1 b

B = D-1(L+U)

X[k+1] = (D-L)-1 U X[k] + (D-L)-1 b

B = (D-L)-1 U

  • Iterative method X[k+1] = BX[k]+g converges if and only if

  • (B) < 1

  • Convergence rate

  • ||X[k]-X*||  ||X[1]-X[0]||, whereq =||B||<1


Steepest Decent Method

  • Consider the case: A is symmetric positive definite

  • Quadratic functional

  • (x)= xTAx - 2bTx

  • The solution of Ax=b is equivalent to find the minimizer

  • of the functional(x)

  • Method of optimization: find a direction pk and a step k


Steepest Decent Method

Determine pkand k

  • Suppose that pk is determined. Let’s start from xk

  • Let f() =(xk +  pk)

  • = (xk +  pk)TA(xk + pk)-2bT(xk + pk)

  • = 2pkTApk - 2 rkTpk +(xk)

  • whererk = b - Axk(Residual)

  • By calculas f’() = 2pkTApk- 2rkTpk =0

  • Then let xk+1 = xk + k pk


Algorithm for Steepest Decent Method

  • Verify(xk+1) - (xk) =(xk +k pk) - (xk)

  • = k2pkTApk - 2k rkTpk

  • How to determine the directionpk ?

  • take as the negative gradientpk = rk


Convergence Theorem

Algorithm

Suppose the eigenvalues of A

then there holds

where


Conjugate Gradient Method

Derivation

  • Negative gradient direction rk is the locally steepest

  • decent direction, but it may not be the global one

  • Consider a new direction: combination of rk and pk-1

  • Initially, take p0 = r0 , x1 = x0 + 0p0

  • For step k +1, choose  and  to minimize

  • By calculas


0


Algorithm for CG method

Where  and  are obtained

in a simple form

  • Operations involved:

  • Transpose,

  • Scalar Multiply,

  • Matrix Add,

  • Matrix Multiply


Properties for CG method

Orthogonal

properties

  • Theoretically, CG method is an exact method. Actually,

  • works as an iterative method.

  • Convergence rate:

  • where


References

  • 徐树方,高立,张平文,

  • 数值线性代数,北京大学出版社,北京,2007

  • 袁亚湘,孙文瑜,

  • 最优化理论与方法,科学出版社,北京,2000

  • Yousef Saad,

  • Iterative Methods for Sparse Linear Systems, 2000


Home & Project

Due at the end of this week

Problem: Minimize the functional E(u)=∫(|u|2+u2-2fu )dx

The corresponding Euler-Lagrange equation is

E/u=-2u+2u-2f=0 or -u+u=f

  • Solve the following linear systems using CG method

-u xx + u =f 0<x<1

f=(1+42)sin2x

u(0)=u(1)=0

where

  • Set n = 100, 200, 300, 400, 500

  • Use Matlab to graph the solution (j, uj)


Home & Project

Due at the end of this week

  • Solve the following linear systems using CG method

  • The unknowns can be ordered as below


Where

S Tridiagonal matrix with

diagonal entry:

other entry:

  • Set n=20,40,80,100. Find the solution

  • Use Matlab to graph the solution (i, j, uij)

-u+u=f (x,y)(0,1)(0,1)

u(x,y)=100(x2-x)(y2-y)

f=200(y-y2) + 200(x-x2) + 100(x2-x)(y2-y)



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