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Cojugate Gradient MethodPowerPoint Presentation

Cojugate Gradient Method

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## PowerPoint Slideshow about ' Cojugate Gradient Method' - jameson-carlson

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- Aim
- Method of Gauss Elimination
- Basic Iterative Methods
- Conjugate Gradient Method
- Derivation
- Theory
- Algorithm

- References
- Homework & Project

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

- How to construct iterative sequence?
- Convergence? Conditions?
- Convergence rate?

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

- 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

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’() = 2pkTApk- 2rkTpk =0
- Then let xk+1 = xk + k pk

Algorithm for Steepest Decent Method

- Verify(xk+1) - (xk) =(xk +k pk) - (xk)
- = k2pkTApk - 2k rkTpk
- How to determine the directionpk ?
- take as the negative gradientpk = rk

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

- The corresponding minimizer is
- 0 and 0satisfy
- take
- Let
- In summary,

0

Where and are obtained

in a simple form

- Operations involved:
- Transpose,
- Scalar Multiply,
- Matrix Add,
- Matrix Multiply

Orthogonal

properties

- Theoretically, CG method is an exact method. Actually,
- works as an iterative method.
- Convergence rate:
- where

- 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

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=-2u+2u-2f=0 or -u+u=f

- Solve the following linear systems using CG method

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

f=(1+42)sin2x

u(0)=u(1)=0

where

- Set n = 100, 200, 300, 400, 500
- Use Matlab to graph the solution (j, uj)

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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