SIAM Journal on Scientific Computing, Ahead of Print.
We propose two new algebraic reconstruction techniques based on Kaczmarz's method that produce a regularized solution to noisy tomography problems. Tomography problems exhibit semiconvergence when iterative methods are employed, and the aim is therefore to stop near the semiconvergence point. Our approach is based on an error gauge that is constructed by pairing standard down-sweep Kaczmarz's method with its up-sweep version; we stop the iterations when this error gauge is minimal. The reconstructions of the new methods differ from standard Kaczmarz iterates in that our final result is the average of the stopped up- and down-sweeps. Even when Kaczmarz's method is supplied with an oracle that provides the exact error---and is therefore able to stop at the best possible iterate---our methods have a lower two-norm error in the vast majority of our test cases. In terms of computational cost, our methods are a little cheaper than standard Kaczmarz equipped with a statistical stopping rule.

中文翻译：

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Kaczmarz迭代的双误差规

《 SIAM科学计算杂志》，预印本。
我们提出了两种基于Kaczmarz方法的新代数重构技术，这些技术可为嘈杂的层析成像问题提供正规化的解决方案。当使用迭代方法时，层析成像问题表现出半收敛性，因此目标是在半收敛点附近停止。我们的方法基于一个误差表，该误差表是通过将标准向下扫描的Kaczmarz方法与其向上扫描的版本配对来构造的；当此误差指标最小时，我们将停止迭代。新方法的重构与标准Kaczmarz迭代的不同之处在于，我们的最终结果是停止上扫和下扫的平均值。即使当Kaczmarz' s方法带有一个提供准确错误的预言器-因此可以在最佳的迭代中停止-我们的方法在我们的绝大多数测试用例中均具有较低的二范数误差。在计算成本方面，我们的方法比配备统计停止规则的标准Kaczmarz便宜一些。