On diagonally relaxed orthogonal projection methods

Yair Censor; Tommy Elfving; Gabor T. Herman; Touraj Nikazad

doi:10.1137/050639399

On diagonally relaxed orthogonal projection methods

Yair Censor, Tommy Elfving, Gabor T. Herman, Touraj Nikazad

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We propose and study a block-iterative projection method for solving linear equations and/or inequalities. The method allows diagonal componentwise relaxation in conjunction with orthogonal projections onto the individual hyperplanes of the system, and is thus called diagonally relaxed orthogonal projections (DROP). Diagonal relaxation has proven useful in accelerating the initial convergence of simultaneous and block-iterative projection algorithms, but until now it was available only in conjunction with generalized oblique projections in which there is a special relation between the weighting and the oblique projections. DROP has been used by practitioners, and in this paper a contribution to its convergence theory is provided. The mathematical analysis is complemented by some experiments in image reconstruction from projections which illustrate the performance of DROP.

Original language	English
Pages (from-to)	473-504
Number of pages	32
Journal	SIAM Journal of Scientific Computing
Volume	30
Issue number	1
DOIs	https://doi.org/10.1137/050639399
State	Published - 2007

Keywords

Block iteration
Convex feasibility
Diagonal relaxation
Projection methods
Simultaneous algorithms

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1137/050639399

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abstract = "We propose and study a block-iterative projection method for solving linear equations and/or inequalities. The method allows diagonal componentwise relaxation in conjunction with orthogonal projections onto the individual hyperplanes of the system, and is thus called diagonally relaxed orthogonal projections (DROP). Diagonal relaxation has proven useful in accelerating the initial convergence of simultaneous and block-iterative projection algorithms, but until now it was available only in conjunction with generalized oblique projections in which there is a special relation between the weighting and the oblique projections. DROP has been used by practitioners, and in this paper a contribution to its convergence theory is provided. The mathematical analysis is complemented by some experiments in image reconstruction from projections which illustrate the performance of DROP.",

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AB - We propose and study a block-iterative projection method for solving linear equations and/or inequalities. The method allows diagonal componentwise relaxation in conjunction with orthogonal projections onto the individual hyperplanes of the system, and is thus called diagonally relaxed orthogonal projections (DROP). Diagonal relaxation has proven useful in accelerating the initial convergence of simultaneous and block-iterative projection algorithms, but until now it was available only in conjunction with generalized oblique projections in which there is a special relation between the weighting and the oblique projections. DROP has been used by practitioners, and in this paper a contribution to its convergence theory is provided. The mathematical analysis is complemented by some experiments in image reconstruction from projections which illustrate the performance of DROP.

KW - Block iteration

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KW - Projection methods

KW - Simultaneous algorithms

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On diagonally relaxed orthogonal projection methods

Abstract

Keywords

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