Strong underrelaxation in Kaczmarz's method for inconsistent systems

Censor, Yair; Eggermont, P. P. B.; Gordon, Dan

doi:10.1007/bf01396307

Cited by 179 publications

(118 citation statements)

References 6 publications

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“…This error is sharp in general [13]. Modified Kaczmarz algorithms can also be used to solve the least squares version of this problem, see for example [4,5,8,2] and the references therein.…”

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confidence: 99%

Two-Subspace Projection Method for Coherent Overdetermined Systems

Needell

Ward

2012

J Fourier Anal Appl

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Abstract. We present a Projection onto Convex Sets (POCS) type algorithm for solving systems of linear equations. POCS methods have found many applications ranging from computer tomography to digital signal and image processing. The Kaczmarz method is one of the most popular solvers for overdetermined systems of linear equations due to its speed and simplicity. Here we introduce and analyze an extension of the Kaczmarz method which iteratively projects the estimate onto a solution space given from two randomly selected rows. We show that this projection algorithm provides exponential convergence to the solution in expectation. The convergence rate significantly improves upon that of the standard randomized Kaczmarz method when the system has coherent rows. We also show that the method is robust to noise, and converges exponentially in expectation to the noise floor. Experimental results are provided which confirm that in the coherent case our method significantly outperforms the randomized Kaczmarz method.

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confidence: 99%

Two-Subspace Projection Method for Coherent Overdetermined Systems

Needell

Ward

2012

J Fourier Anal Appl

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“…Whitney and Meany prove that if the relaxation parameters tend to zero that the iterates converge to the least squares solution [WM67]. Further results using relaxation have also been obtained, see for example [CEG83,Tan71,HN90,ZF12]. An alternative to relaxation parameters was recently proposed by Zouzias and Freris [ZF12] as the REK method described by (3).…”

Section: Related Work and Discussionmentioning

confidence: 99%

Randomized block Kaczmarz method with projection for solving least squares

Needell

Zhao

Zouzias

2015

Linear Algebra and its Applications

139

107

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The Kaczmarz method is an iterative method for solving overcomplete linear systems of equations Ax = b. The randomized version of the Kaczmarz method put forth by Strohmer and Vershynin iteratively projects onto a randomly chosen solution space given by a single row of the matrix A and converges linearly 1 in expectation to the solution of a consistent system. In this paper we analyze two block versions of the method each with a randomized projection, designed to converge in expectation to the least squares solution, often faster than the standard variants. Our approach utilizes both a row and column-paving of the matrix A to guarantee linear convergence when the matrix has consistent row norms (called nearly standardized), and a single column-paving when the row norms are unbounded. The proposed methods are an extension of the block Kaczmarz method analyzed by Needell and Tropp and the Randomized Extended Kaczmarz method of Zouzias and Freris. The contribution is thus two-fold; unlike the standard Kaczmarz method, our results demonstrate convergence to the least squares solution of inconsistent systems (both methods in the nearly standardized case and the second method in other cases). By using appropriate blocks of the matrix this convergence can be significantly accelerated, as is demonstrated by numerical experiments.

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“…al [4] proved that for sufficiently small values of λ the algorithm converges to a least squares solution of a weighted version of Eqn. (1).…”

Section: D-reconstruction Algorithmmentioning

confidence: 99%

Motion and Positional Error Correction for Cone Beam 3D-Reconstruction with Mobile C-Arms

Bodensteiner

Darolti

Schumacher

et al. 2007

Medical Image Computing and Computer-Assisted Intervention – MICCAI 2007

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Abstract. CT-images acquired by mobile C-arm devices can contain artefacts caused by positioning errors. We propose a data driven method based on iterative 3D-reconstruction and 2D/3D-registration to correct projection data inconsistencies. With a 2D/3D-registration algorithm, transformations are computed to align the acquired projection images to a previously reconstructed volume. In an iterative procedure, the reconstruction algorithm uses the results of the registration step. This algorithm also reduces small motion artefacts within 3D-reconstructions. Experiments with simulated projections from real patient data show the feasibility of the proposed method. In addition, experiments with real projection data acquired with an experimental robotised C-arm device have been performed with promising results.

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Strong underrelaxation in Kaczmarz's method for inconsistent systems

Cited by 179 publications

References 6 publications

Two-Subspace Projection Method for Coherent Overdetermined Systems

Two-Subspace Projection Method for Coherent Overdetermined Systems

Randomized block Kaczmarz method with projection for solving least squares

Motion and Positional Error Correction for Cone Beam 3D-Reconstruction with Mobile C-Arms

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