The regularizing effect of the Golub-Kahan iterative bidiagonalization and revealing the noise level in the data
The regularizing effect of the Golub-Kahan iterative bidiagonalization and revealing the noise level in the data
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DOI：
10.1007/s10543-009-0239-7
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发表时间：
2009-09
影响因子：
1.5
通讯作者：
I. Hnětynková;M. Plešinger;Z. Strakoš
中科院分区：
文献类型：
--
作者：
I. Hnětynková;M. Plešinger;Z. Strakoš
基于Golub-Kahan迭代双对角化的正则化技术属于解决大型不适定问题的流行方法之一。首先，使用双对角化算法将原问题投影到一个低维子空间上，该算法本身代表了投影正则化的一种形式。然而，投影问题继承了原始问题的一部分不适定性，因此必须应用某种形式的内部正则化。整个过程的停止准则是基于投影（小）problem.In本文中，我们考虑一个不适定的问题与噪声的右手边（观察向量），其中的噪声水平是未知的正则化。我们展示了如何从Golub-Kahan迭代双对角化的信息可以用于估计噪声水平。这样的信息可以是有用的构造有效的停止准则，在解决不适定问题。
Regularization techniques based on the Golub-Kahan iterative bidiagonalization belong among popular approaches for solving large ill-posed problems. First, the original problem is projected onto a lower dimensional subspace using the bidiagonalization algorithm, which by itself represents a form of regularization by projection. The projected problem, however, inherits a part of the ill-posedness of the original problem, and therefore some form of inner regularization must be applied. Stopping criteria for the whole process are then based on the regularization of the projected (small) problem.In this paper we consider an ill-posed problem with a noisy right-hand side (observation vector), where the noise level is unknown. We show how the information from the Golub-Kahan iterative bidiagonalization can be used for estimating the noise level. Such information can be useful for constructing efficient stopping criteria in solving ill-posed problems.