论文标题
Kaczmarz-Tanabe类型方法的标准形式和收敛理论解决线性系统
The standard forms and convergence theory of the Kaczmarz-Tanabe type methods for solving linear systems
论文作者
论文摘要
在本文中，我们考虑了两种Kaczmarz-Tanabe类型方法的标准形式，一种是源自Kaczmarz方法，另一种是从对称的Kaczmarz方法中得出的。作为计算机断层扫描中著名的图像重建方法，Kaczmarz方法简单易于实现，但其收敛速度却很慢，因此对称的Kaczmarz方法也是如此。当获得Kaczmarz-Tanabe型方法的标准形式时，可以在随后的迭代中连续使用其迭代矩阵。此外，迭代矩阵可以存储在图像重建设备中，这使Kaczmarz方法和对称Kaczmarz方法像同时迭代的重建技术（SIRT）一样使用。同时，理论分析表明，对称Kaczmarz-Tanabe方法的收敛速率比Kaczmarz-Tanabe方法更好，但比两步kaczmarz-Tanabe方法的收敛速度稍差，该方法已通过数值进行了验证。数值实验还表明，Kaczmarz-Tanabe方法的收敛速率和对称Kaczmarz-Tanabe方法比SIRT方法的收敛速率更好。
In this paper, we consider the standard forms of two kinds of Kaczmarz-Tanabe type methods, one is derived from the Kaczmarz method and the other is derived from the symmetric Kaczmarz method. As a famous image reconstruction method in computerized tomography, the Kaczmarz method is simple and easy to implement, but its convergence speed is slow, so is the symmetric Kaczmarz method. When the standard forms of the Kaczmarz-Tanabe type methods are obtained, their iteration matrices can be used continuously in the subsequent iterations. Moreover, the iteration matrices can be stored in the image reconstruction devices, which enables the Kaczmarz method and the symmetric Kaczmarz method to be used like the simultaneous iterative reconstructive techniques (SIRT). Meanwhile, theoretical analysis shows that the convergence rate of the symmetric Kaczmarz-Tanabe method is better than that of the Kaczmarz-Tanabe method but is slightly worse than that of two-step Kaczmarz-Tanabe method, which is verified numerically. Numerical experiments also show that the convergence rates of the Kaczmarz-Tanabe method and the symmetric Kaczmarz-Tanabe method are better than those of the SIRT methods.
