一类PCG迭代的强初值效应

预处理共轭梯度法的大量计算结果都表明,当迭代终止标准要求将余量的模减小某个倍数时,初值的选取对迭代次数仅有微弱影响。然而,本文给出的一类算例却表明,采用零初值或不同的随机初值,迭代次数之间会出现数倍的差别。同一种随机初值对不同参数模型问题的迭代次数也有很大差别。这种强初值效应对于方法的研究和比较是不利的。本文讨论和分析了这种现象。

ON THE SERIOUS VARIATION OF NUMBERS OF PCG ITERATIONS CAUSED BY INITIAL GUESSES

It was shown in literatures of the preconditioned conjugate gradient (PCG) that the initial guess gives little influence upon the number of PCG iterations when the stopping criterion requests to reduce the residual norm by a factor. However, the examples reported here show that using a zero initial gress or different random initial guess causes the number of PCG iterations varies seriously. Moreover, the number of iterations still varies seriously for the different parameter of the modeb when the same random initial guess is used. This variation should be avoided since it may cause confusions when different methods are compared. This paper shows that if the preconditioner is given, the series {r_k} determined uniquely by the linear systems provided a zero initial guess is used. To avoid the serious variation in the number of PCG iterations, using the zero initial guess is a good choice.