arXiv subjects

Chuan-gang Kang

Publications and source records attributed to Chuan-gang Kang.

4 recordsLinked to original sources

arXiv2023

The standard forms and convergence theory of the Kaczmarz-Tanabe type methods for solving linear systems

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.

math.NA↗

arXiv2023

The standard form and convergence theory of the relaxation Kaczmarz-Tanabe method for solving linear systems

The Kaczmarz method is a popular iterative method for solving consistent, overdetermined linear system such as medical imaging in computerized tomography. The Kaczmarz's iteration repeatedly scans all equations in order, which leads to lower computational efficiency especially in solving a large scale problem. The standard form of Kaczmarz-Tanabe's iteration proposed recently effectively overcomes the computational redundancy problem of the Kaczmarz method. In this paper, we introduce relaxation parameters ${\bf u}=(μ_1,\ldots,μ_m)$ into the Kaczmarz-Tanabe method based on the relaxation Kaczmarz method, and consider the standard form and convergence of this combination. Moreover, we analyze and prove the sufficient conditions for convergence of the relaxation Kaczmarz-Tanabe method, i.e., $μ_i\in (0,2)$. Numerical experiments show the convergence characteristics of the relaxation Kaczmarz-Tanabe method corresponding to these parameters.

math.NA↗

arXiv2022

Convergence rates of the Kaczmarz-Tanabe method for linear systems

In this paper, we investigate the Kaczmarz-Tanabe method for exact and inexact linear systems. The Kaczmarz-Tanabe method is derived from the Kaczmarz method, but is more stable than that. We analyze the convergence and the convergence rate of the Kaczmarz-Tanabe method based on the singular value decomposition theory, and discover two important factors, i.e., the second maximum singular value of $Q$ and the minimum non-zero singular value of $A$, that influence the convergence speed and the amplitude of fluctuation of the Kaczmarz-Tanabe method (even for the Kaczmarz method). Numerical tests verify the theoretical results of the Kaczmarz-Tanabe method.

math.NA↗

arXiv2022

Error estimates of Kaczmarz and randomized Kaczmarz methods

The Kaczmarz method is an iterative projection scheme for solving con-sistent system $Ax = b$. It is later extended to the inconsistent and ill-posed linear problems. But the classical Kaczmarz method is sensitive to the correlation of the adjacent equations. In order to reduce the impact of correlation on the convergence rate, the randomized Kaczmarz method and randomized block Kaczmarz method are proposed, respectively. In the current literature, the error estimate results of these methods are established based on the error $\|x_k-x_*\|_2$, where $x_*$ is the solution of linear system $Ax=b$. In this paper, we extend the present error estimates of the Kaczmarz and randomized Kaczmarz methods on the basis of the convergence theorem of Kunio Tanabe, and obtain some general results about the error $\|x_k-P_{N(A)}x_0-x^\dagger\|_2$.

math.NA↗