Projection method for solving a singular system of linear equations and its applications

Tanabe, Kunio

doi:10.1007/bf01436376

Cited by 397 publications

(162 citation statements)

References 5 publications

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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

153

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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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

153

107

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“…Since the matrix equation obtained here is not the diagonal dominance, the Gauss-Seidel method, which is one of the popular iterative methods, is not suitable. Kaczmarz method has been proved to converge for any system of linear equations (10) (22) and is used as relaxation method in this paper.…”

Section: Multigrid Methodsmentioning

confidence: 99%

“…Especially, it is referred to the case γ = γ s = 1, γ j = 0 as V-cycles and γ s = 2, γ j = 0 as W-cycles. Kaczmarz iteration scheme is as follows (22) :…”

Section: Multigrid Methodsmentioning

confidence: 99%

Application of Multigrid Moment Method to Scattering of a Gaussian Beam by a Dielectric Cylinder

Yokota

2004

IEEJ Trans. FM

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MemberScattering of a Gaussian beam by a dielectric cylinder is analyzed by using the moment methods combined with the multigrid method. The scattered fields can be obtained by the volume equivalent theorem. The integral equation is converted to the matrix form by the moment methods. As numerical examples, the scattering of a Gaussian beam by a circular dielectric cylinder is considered. The CPU time and residual norm are examined for the conventional multigrid cycle. Also, the modified multigrid cycle proposed here is compared with the conventional one. The scattered near fields are calculated and are compared with those obtained by the method of eigenfunction expansion.

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“…The properties (8)- (12) were proved in the paper [31]. But, independently on the general approach presented in this section, a theorem of the form Theorem 1 was proved in [31], the extension of the form (35)-(37) was first proposed in [25], a constrained version of Kaczmarz method in [20] and a constrained version of Kaczmarz Extended method in [28].…”

Section: Remarkmentioning

confidence: 97%

“…Examples in this sense are given by the projection algorithms appearing in image reconstruction from projections: Kaczmarz, Cimmino, Landweber, DW, SART, etc (see e.g. [6,7,8,16,13,18,20,21,31] and references therein).…”

Section: Remarkmentioning

confidence: 99%

A general extending and constraining procedure for linear iterative methods

Nicola

Petra

Popa

et al. 2012

International Journal of Computer Mathematics

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Abstract. Algebraic Reconstruction Techniques (ART), on their both successive or simultaneous formulation, have been developed since early 70's as efficient "row action methods" for solving the image reconstruction problem in Computerized Tomography. In this respect, two important development directions were concerned with, firstly their extension to the inconsistent case of the reconstruction problem, and secondly with their combination with constraining strategies, imposed by the particularities of the reconstructed image. In the first part of our paper we introduce extending and constraining procedures for a general iterative method of ART type and we propose a set of sufficient assumptions that ensure the convergence of the corresponding algorithms. As an application of this approach, we prove that Cimmino's simultaneous reflections method satisfies this set of assumptions, and we derive extended and constrained versions for it. Numerical experiments with all these versions are presented on a head phantom widely used in the image reconstruction literature. We also considered hard thresholding constraining used in sparse approximation problems and applied it successfully to a 3D particle image reconstruction problem.

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Projection method for solving a singular system of linear equations and its applications

Cited by 397 publications

References 5 publications

Randomized block Kaczmarz method with projection for solving least squares

Randomized block Kaczmarz method with projection for solving least squares

Application of Multigrid Moment Method to Scattering of a Gaussian Beam by a Dielectric Cylinder

A general extending and constraining procedure for linear iterative methods

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