Semi-iterative Regularization in Hilbert Scales
Semi-iterative Regularization in Hilbert Scales
复制标题
希尔伯特量表的半迭代正则化
DOI：
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发表时间：
2006
影响因子：
2.9
通讯作者：
H. Egger
中科院分区：
文献类型：
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作者：
H. Egger
在本文中，我们研究了希尔伯特尺度中半迭代正则化方法针对线性不适定问题和扰动数据的正则化性质。
众所周知，标准 Landweber 迭代可以通过多项式加速方法显着加速，从而获得最佳收敛速度，这可以通过几种有效的两步方法获得，例如 $
Brakhage 的 u$-方法。之前观察到，如果在希尔伯特尺度中执行 Landweber 迭代，可以获得相似的收敛速度，即产生最佳收敛速率的相似迭代次数。
我们表明，两种想法的结合可以进一步加速，与 $
u$-方法或希尔伯特尺度中的 Landweber 迭代。通过几个例子和数值测试说明了理论结果，包括与共轭梯度法的比较。
In this paper we investigate the regularizing properties of semi-iterative regularization methods in Hilbert scales for linear ill-posed problems and perturbed data.
It is well known that standard Landweber iteration can be remarkably accelerated by polynomial acceleration methods leading to optimal speed of convergence, which can be obtained by several efficient two-step methods, e.g., the $
u$-methods by Brakhage. It was observed earlier that a similar speed of convergence, i.e., similar iteration numbers yielding optimal convergence rates, can be obtained if Landweber iteration is performed in Hilbert scales.
We show that a combination of both ideas allows for a further acceleration, yielding optimal convergence rates with only the square root of iterations as compared to the $
u$-methods or Landweber iteration in Hilbert scales. The theoretical results are illustrated by several examples and numerical tests, including a comparison to the method of conjugate gradients.