The Averaged Kaczmarz Iteration for Solving Inverse Problems
The Averaged Kaczmarz Iteration for Solving Inverse Problems
复制标题
求解反问题的平均 Kaczmarz 迭代
DOI：
10.1137/17m1146178
复制
发表时间：
2018-01-01
影响因子：
2.1
通讯作者：
Haltmeier, Markus
中科院分区：
文献类型：
--
作者：
Li, Housen;Haltmeier, Markus
我们介绍了一种新的迭代正则化方法来解决反问题，可以写为系统的线性或非线性方程在希尔伯特空间。建议的平均Kaczmarz（AVEK）方法可以被看作是Landweber和Kaczmarz方法之间的混合方法。与Kaczmarz方法一样，该方法每次迭代更新只需要评估一个直接子问题和一个伴随子问题。另一方面，与Landweber迭代类似，它使用了先前辅助迭代的平均值，这增加了稳定性。我们提出了一个收敛性分析的AVEK迭代。此外，详细的数值研究的层析图像重建问题，即有限数据问题的光声层析。因此，AVEK与其他迭代正则化方法，包括标准的Landweber和Kaczmarz迭代，以及最近提出的加速版本的基础上误差最小化松弛策略进行了比较。
We introduce a new iterative regularization method for solving inverse problems that can be written as systems of linear or non-linear equations in Hilbert spaces. The proposed averaged Kaczmarz (AVEK) method can be seen as a hybrid method between the Landweber and the Kaczmarz method. As the Kaczmarz method, the proposed method only requires evaluation of one direct and one adjoint sub-problem per iterative update. On the other, similar to the Landweber iteration, it uses an average over previous auxiliary iterates which increases stability. We present a convergence analysis of the AVEK iteration. Further, detailed numerical studies are presented for a tomographic image reconstruction problem, namely the limited data problem in photoacoustic tomography. Thereby, the AVEK is compared with other iterative regularization methods including standard Landweber and Kaczmarz iterations, as well as recently proposed accelerated versions based on error minimizing relaxation strategies.