Unformatted text preview: e di erence
scheme. For example, when solving a problem for Poisson's equation using centered
di erences (8.1.2), we could compute Ar( ) at grid point (j k) as
(
)
Ar(jk) = rjk) ; x(rj(+1 k + rj(;)1 k ) ; y (rj( k)+1 + rj( k);1): The matrix A is not stored and the vector r( ) is stored in the mesh coordinates.
3. Using (9.3.3) and the de nition of given in the algorithm of Figure 9.3.1, we
have
(r( +1) )T r( ) = (r( ) )T r( ) ; (r( ))T Ar( ) = 0:
Thus, the search directions r( ) , k = 0 1 : : : , are \orthogonal" in the sense that
(r( +1) )T r( ) = 0:
4. The steepest descent method converges from any initial guess x(0) when A and
AT A;1 are positive de nite thus, symmetry of A is not necessary ( 5], Section
14.1).
5. Let us introduce the strain energy norm kxk2 := xT Ax:
A (9.3.4a) 9.3. Conjugate Gradient Methods 33 Then ( 4], Section 5.3) kx( +1) ; xk2 (1 ; (1A) )kx( ) ; xk2
A
A
2
where 2 is the condition number of A in the L2 norm as given by (9.3.4b) 2 (A) = max = min (9.3.4c) and max and min are the maximum and minimum eigenvalues of A. Convergence
is slow when (A) is large and A is said to be ill conditioned in this case. When this
occurs, the level surfaces of E (y) are often elongated ellipses with the minimum
x lying at the bottom of a narrow valley with steep walls. Successive iterates
tend to wander back and forth across the valley making little progress towards the
minimum.
Example 9.3.1. Let's solve a simple problem with 1
A = 2 10
1
to illustrate some geometrical aspects of the method of steepest descent. With b = 0,
the minimum of (9.3.1) is at the origin. We select the initial guess x(0) = 0:1972 0:4443]T
to be such that E (x(0) ) = 1 (Figure 9.3.2). Since b = 0, we obtain r(0) = ;0:8387 ;4:6406] 0 = 0:0990 x(1) = 0:1141 ;0:0153]T : 1 = 0:5255 x(2) = 0:0022 0:0050]T : The second iteration produces r(1) = ;0:2130 0:0385]T The iteration seems to be converging to the minimum at the origin. These two iterates
and the level surfaces E (y) = 1, 0.75, 0.5, 0.25 are shown in Figure 9.3.2. The initial
iteration proceeds \downhill" in a direction opposite to the gradient at x(0) until a local
minimum is reached. The second iterate proceeds from there. 34 Solution Techn...
View
Full Document
 Spring '14
 JosephE.Flaherty
 Articles with example pseudocode, Gauss–Seidel method, Jacobi method, Iterative method, elliptic problems

Click to edit the document details