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Unformatted text preview: E (y) = 1 yT Ay ; bT y: 2 (9.3.1) The necessary condition for a minimum, E 0 (y) = Ay ; b = 0 implies that y = x, the solution of (9.1.1). (9.3.2a) 9.3. Conjugate Gradient Methods 31 If we de ne the residual r(y) = b ; Ay (9.3.2b) E 0(y) = ;r(y): (9.3.2c) then (9.3.2a) may be written as The level surfaces E (y) = C (a constant) of (9.3.1) are ellipsoids in <N with a common center at x. Since the gradient of a function is in the direction of steepest increase, to minimize a E (y) at a point x(0) , we could move in a direction opposite to the gradient of the level surface through x(0) . From (9.3.2c), the gradient at x(0) is E 0(x(0) ) = ;r(x(0) ) = ;r(0) : Let our subsequent guess x(1) for the minimum x be x(1) = x(0) + 0r(0) and let us calculate the distance 0 moved in the negative gradient direction r(0) so as to minimize E (x(1) ). Using (9.3.1), we have 1 E (x(1) ) = E (x(0) + 0r(0) ) = 2 (x(0) + 0r(0) )T A(x(0) + 0r(0) ) ; bT (x(0) + 0r(0) ): Di erentiating with respect to 0 d E (x(0) + r(0) ) = (x(0) )T A ; bT + (r(0) )T A]r(0) = 0: 0 0 d0 Using (9.3.2b) (0) )T (0) = (r T r (0) : 0 (r(0) ) Ar With subsequent iterates computed in the same manner, the process is called the method of steepest descent. A pseudocode algorithm of the method appears in Figure 9.3.1. Some comments on the procedure and method follow. 1. The calculation of r( +1) shown in the algorithm follows de nition (9.3.2b) thus, r( +1) = b ; Ax( +1) = b ; A(x( ) + r( ) ) = r( ) ; Ar( ) : (9.3.3) Formula (9.3.3) is less susceptible to the accumulation of round-o error than direct computation using (9.3.2b). 32 Solution Techniques for Elliptic Problems procedure steepest descent r(0) = b ; Ax(0) =0 while not converged do = (r( ))T r( ) =(r( ) )T Ar( ) ( +1) = x( ) + r( ) x r( +1) = r( ) ; Ar( ) = +1 end while Figure 9.3.1: A steepest-descent algorithm. 2. The algorithm only has one matrix multiplication and two vector multiplications per step. When solving partial di erential equations, it is not necessary to store the matrix A. The product Ar( ) can be obtained directly from th...
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This document was uploaded on 03/16/2014 for the course CSCI 6840 at Rensselaer Polytechnic Institute.

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