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By C. Bardos, J. M. Lasry, M. Schatzman

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Example text

One wants to know then whether the rate of convergence of the preconditioned conjugate gradient method is appreciably decreased. 13 to be the ratio between bounds on the number of iterations required by a> and co*. $(a) is a very slowly increasing function of a for a > 1: Thus the rate of convergence of the conjugate gradient method with SSOR preconditioning is remarkably insensitive to the estimates of \i and <5. 61). 61) is equivalent to Solve for/c = 0, 1,2,.... 74) is the more advantageous formulation for computation.

The difference between measuring errors in the Euclidean and energy norms becomes more pronounced as the ellipsoids become more eccentric. This is illustrated in Fig. 4, wher It is appropriate to mention here that the technique of preconditioning, which will be introduced later in this section, has the effect of replacing H by a matrix whose level surfaces are significantly less eccentric than those of H. This reduces the disparity between the Euclidean and energy norms. We shall now derive the rate of convergence of the method of steepest descent in the energy norm.

1. Level surfaces (N = 2) in the neighborhood of a local minimizer. 1 QUADRATIC FUNCTIONALS 7 gradient vector g(x) is perpendicular to Lk at x and points in the direction in which the functional increases most rapidly. " Generally speaking, numerical methods for finding a strong local minimizer of a functional perform best when the level surfaces in the neighborhood of the minimizer are spheres and poorly when they show a pronounced distortion from spheres. The quantity where Sk is the set of all points interior to Lk, is a measure of the distortion of Lk from spherical form.

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