ACCELERATED LANDWEBER ITERATIONS FOR THE SOLUTION OF ILL-POSED EQUATIONS
ACCELERATED LANDWEBER ITERATIONS FOR THE SOLUTION OF ILL-POSED EQUATIONS
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DOI：
10.1007/bf01385727
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发表时间：
1991-01-01
影响因子：
2.1
通讯作者：
HANKE, M
中科院分区：
文献类型：
--
作者：
HANKE, M
本文考虑了线性半迭代法在线性不适定问题和病态矩阵方程近似解中的潜力。提出了几种有效的两步法，其中大多数已在文献中介绍过。规定了与解的平滑性有关的某些条件，就可以形成最优收敛速度的概念。为了说明这个概念的性质，推导出了各种正、反结果。如果问题的右侧数据被噪声污染，可以使用半迭代方法作为正则化方法。假设无扰动问题的迭代收敛速度为最优，则正则化近似具有阶最优精度。为了得到这些结果，我们将多项式的特定性质与求解不适定问题的基本理论联系起来。应用速降多项式的最新结果来回答制动的一个开放问题。给出了数值算例，并与共轭梯度法进行了比较。
In this paper, the potentials of so-called linear semiiterative methods are considered for the approximate solution of linear ill-posed problems and ill conditioned matrix equations. Several efficient two-step methods are presented, most of which have been introduced earlier in the literature. Stipulating certain conditions concerning the smoothness of the solution, a notion of optimal speed of convergence may be formulated. Various direct and converse results are derived to illustrate the properties of this concept.If the problem's right hand side data are contaminated by noise, semiiterative methods may be used as regularization methods. Assuming optimal rate of convergence of the iteration for the unperturbed problem, the regularized approximations will be of order optimal accuracy.To derive these results, specific properties of polynomials are used in connection with the basic theory of solving ill-posed problems. Rather recent results on fast decreasing polynomials are applied to answer an open question of Brakhage.Numerical examples are given including a comparison to the method of conjugate gradients.