A twin error gauge for Kaczmarz's iterations
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We propose two new methods based on Kaczmarz's method that produce a regularized solution to noisy tomography problems. These methods exhibit semi-convergence when applied to inverse problems, and the aim is therefore to stop near the semi-convergence point. Our approach is based on an error gauge that is constructed by pairing Kaczmarz's method with its reverse-ordered method; we stop the iterations when this error gauge is minimum. Our first proposed method stops when the error gauge is at a minimum, the second uses the error gauge to determine step sizes. Our numerical experiments demonstrate that our two methods are superior to the standard Kaczmarz method equipped with state-of-the-art statistical stopping rules. Even compared to Kaczmarz's method equipped with an oracle that provides the exact error -- and is thereby able to stop at the best possible iterate -- our methods perform better in almost 90% of our test cases. In terms of computational cost, our methods are a little cheaper than standard Kaczmarz equipped with a statistical stopping rule.
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