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Unformatted text preview: iques for Elliptic Problems
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure 9.3.2: Convergence of the method of steepest descent for the problem of Example
9.3.1. Ellipses have level values of 1, 0.75, 0.5, and 0.25 (outer to inner). The rst two
iterations are shown to be converging to the solution at (0 0).
9.3.1 The Conjugate Gradient Method The remedy for the slow convergence encountered with the method of steepest descent
is to choose other search directions. With the conjugate gradient method, we choose x( +1) = x( ) + p( ) (9.3.5a) p( ) = r( ) + (x( ) ; x( ;1) ): (9.3.5b) with
Thus, the new search direction is a linear combination of the steepest descent search
direction and the previous \correction" x( ) ; x( ;1) . The parameters and are to
Using (9.3.5a), we rewrite (9.3.5b) in the form p( ) = r( ) + ;1 p ( ;1) = r( ) + ;1p( ;1) (9.3.5c) 9.3. Conjugate Gradient Methods 35 with the goal of specifying and ;1 so that convergence is as fast as possible. As
with steepest descent, we choose to minimize E (x( +1) ). In this case,
E (x( +1) ) = E (x( ) + p( ) ) = 1 (x( ) + p( ) )T A(x( ) + p( ) ) ; bT (x( ) + p( ) ):
Di erentiating with respect to E 0 (x( ) + p( ) ) = (x( ) + p( ) )T Ap( ) ; bT p( ) = 0
or, using (9.3.2b) (;r( ) + p( ) A)T p( ) = 0: Thus,
= (r ) ) p ( )
= 0 1 ::: :
(p( )T Ap
Let's develop a few properties and relationships that will be interesting in their own
right and useful for calculating . First, we'll use (9.3.2b) and (9.3.5a) to write r( +1) = b ; Ax( +1) = b ; A(x( ) + p( ) ) = r( ) ; Ap( ) : (9.3.8) Taking an inner product with p( ) to obtain
(r( +1) )T p( ) = (r( ))T p( ) ; (p( ) )T Ap( ) :
Using (9.3.7) to eliminate reveals the orthogonality condition
(r( +1) )T p( ) = 0: (9.3.9a) Next, take the inner product of (9.3.5c) with r( +1) to obtain
(r( +1) )T p( +1) = (r( +1) )T r( +1) + (r( +1) )T p( )
or, using (9.3.9a), (r( +1) )T p( +1) = (r( +1) )T r( +1) : If we select p(0) = r(0)...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.
- Spring '14