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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
(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
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
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
Example 9.3.1. Let's solve a simple problem with 1
A = 2 10
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...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
- Spring '14