Convergence rates of the Kaczmarz–Tanabe method for linear systems

Kang, Chuan-gang

doi:10.1016/j.cam.2021.113577

Cited by 6 publications

(2 citation statements)

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“…In order to keep the advantages of Kaczmarz method and overcome its disadvantages, many scholars consider the subsequence {y k } of sequence {x k }, where y k = x k•m . Kang [13] give the following iterative scheme of Kaczmarz's subsequence {y k }, i.e.,…”

Section: Introductionmentioning

confidence: 99%

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The standard forms of the Kaczmarz-Tanabe type methods and their convergence theory

Kang¹

2022

Preprint

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In this paper, we consider the standard form of two kinds of Kaczmarz-Tanabe type methods, one derived from the Kaczmarz method and the other derived from the symmetric Kaczmarz method. As a famous image reconstruction method in computed tomography, the Kaczmarz method has both advantage and disadvantage. The advantage are simple and easy to implement, while the disadvantages are slow convergence speed, and the symmetric Kaczmarz method is the same. For the standard form of this method, once the iterative matrix is generated, it can be used continuously in the subsequent iterations. Moreover, the iterative matrix can be stored in the image reconstructive devices, which makes the Kaczmarz method and the symmetric Kaczmarz method can be used like the simultaneous iterative reconstructive techniques (SIRT). Meanwhile, theoretical analysis shows that the convergence rate of symmetric Kaczmarz method is better than the Kaczmarz method but is slightly worse than that of two iterations Kaczmarz method, which is verified numerically. Numerical experiments also show that the convergence rates of the Kaczmarz method and the symmetric Kaczmarz method are better than the SIRT methods and slightly worse than CGMN method in some cases. However, the Kaczmarz Tanabe type methods have better problem adaptability.

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Section: Introductionmentioning

confidence: 99%

“…Theorem 1.1. [13] For any matrix A with non-zero rows, let {y k , k ≥ 0} be the sequence of vectors generated by (1.5), then there holds…”

Section: Introductionmentioning

confidence: 99%