Component averaging: An efficient iterative parallel algorithm for large and sparse unstructured problems

Yair Censor; Dan Gordon; Rachel Gordon

doi:10.1016/S0167-8191(00)00100-9

Component averaging: An efficient iterative parallel algorithm for large and sparse unstructured problems

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Abstract

Component averaging (CAV) is introduced as a new iterative parallel technique suitable for large and sparse unstructured systems of linear equations. It simultaneously projects the current iterate onto all the system's hyperplanes, and is thus inherently parallel. However, instead of orthogonal projections and scalar weights (as used, for example, in Cimmino's method), it uses oblique projections and diagonal weighting matrices, with weights related to the sparsity of the system matrix. These features provide for a practical convergence rate which approaches that of algebraic reconstruction technique (ART) (Kaczmarz's row-action algorithm) - even on a single processor. Furthermore, the new algorithm also converges in the inconsistent case. A proof of convergence is provided for unit relaxation, and the fast convergence is demonstrated on image reconstruction problems of the Herman head phantom obtained within the SNARK93 image reconstruction software package. Both reconstructed images and convergence plots are presented. The practical consequences of the new technique are far reaching for real-world problems in which iterative algorithms are used for solving large, sparse, unstructured and often inconsistent systems of linear equations.

Original language	English
Pages (from-to)	777-808
Number of pages	32
Journal	Parallel Computing
Volume	27
Issue number	6
DOIs	https://doi.org/10.1016/S0167-8191(00)00100-9
State	Published - May 2001

Bibliographical note

Funding Information:
We gratefully acknowledge the fruitful discussions and constructive comments of our colleagues Martin Altschuler, Åke Björck, Lev Bregman, Charles Byrne, Tommy Elfving, Gabor Herman, Krzysztof Kiwiel, Arnold Lent and Robert Lewitt. In particular, our proof of convergence is a modified and revised adaptation of the first author's joint work with C. Byrne [10]. Thanks are also due to the anonymous referees whose detailed comments helped to improve this work. This research was initiated and supported by a research grant from the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities. The work of Y. Censor was also supported by NIH grant HL-28438 at the Medical Image Processing Group (MIPG), Department of Radiology, Hospital of the University of Pennsylvania, Philadelphia, PA, USA. The parallel computations were carried out on the supercomputers of the High Performance Computing Unit (HPCU) of the Israel Inter-University Computation Center (IUCC).

Keywords

Component averaging
Iterative techniques
Oblique projections
Sparse systems
Sparsity-related weights

ASJC Scopus subject areas

Software
Theoretical Computer Science
Hardware and Architecture
Computer Networks and Communications
Computer Graphics and Computer-Aided Design
Artificial Intelligence

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10.1016/S0167-8191(00)00100-9

Cite this

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title = "Component averaging: An efficient iterative parallel algorithm for large and sparse unstructured problems",

abstract = "Component averaging (CAV) is introduced as a new iterative parallel technique suitable for large and sparse unstructured systems of linear equations. It simultaneously projects the current iterate onto all the system's hyperplanes, and is thus inherently parallel. However, instead of orthogonal projections and scalar weights (as used, for example, in Cimmino's method), it uses oblique projections and diagonal weighting matrices, with weights related to the sparsity of the system matrix. These features provide for a practical convergence rate which approaches that of algebraic reconstruction technique (ART) (Kaczmarz's row-action algorithm) - even on a single processor. Furthermore, the new algorithm also converges in the inconsistent case. A proof of convergence is provided for unit relaxation, and the fast convergence is demonstrated on image reconstruction problems of the Herman head phantom obtained within the SNARK93 image reconstruction software package. Both reconstructed images and convergence plots are presented. The practical consequences of the new technique are far reaching for real-world problems in which iterative algorithms are used for solving large, sparse, unstructured and often inconsistent systems of linear equations.",

keywords = "Component averaging, Iterative techniques, Oblique projections, Sparse systems, Sparsity-related weights",

author = "Yair Censor and Dan Gordon and Rachel Gordon",

note = "Funding Information: We gratefully acknowledge the fruitful discussions and constructive comments of our colleagues Martin Altschuler, {\AA}ke Bj{\"o}rck, Lev Bregman, Charles Byrne, Tommy Elfving, Gabor Herman, Krzysztof Kiwiel, Arnold Lent and Robert Lewitt. In particular, our proof of convergence is a modified and revised adaptation of the first author's joint work with C. Byrne [10]. Thanks are also due to the anonymous referees whose detailed comments helped to improve this work. This research was initiated and supported by a research grant from the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities. The work of Y. Censor was also supported by NIH grant HL-28438 at the Medical Image Processing Group (MIPG), Department of Radiology, Hospital of the University of Pennsylvania, Philadelphia, PA, USA. The parallel computations were carried out on the supercomputers of the High Performance Computing Unit (HPCU) of the Israel Inter-University Computation Center (IUCC).",

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N2 - Component averaging (CAV) is introduced as a new iterative parallel technique suitable for large and sparse unstructured systems of linear equations. It simultaneously projects the current iterate onto all the system's hyperplanes, and is thus inherently parallel. However, instead of orthogonal projections and scalar weights (as used, for example, in Cimmino's method), it uses oblique projections and diagonal weighting matrices, with weights related to the sparsity of the system matrix. These features provide for a practical convergence rate which approaches that of algebraic reconstruction technique (ART) (Kaczmarz's row-action algorithm) - even on a single processor. Furthermore, the new algorithm also converges in the inconsistent case. A proof of convergence is provided for unit relaxation, and the fast convergence is demonstrated on image reconstruction problems of the Herman head phantom obtained within the SNARK93 image reconstruction software package. Both reconstructed images and convergence plots are presented. The practical consequences of the new technique are far reaching for real-world problems in which iterative algorithms are used for solving large, sparse, unstructured and often inconsistent systems of linear equations.

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Component averaging: An efficient iterative parallel algorithm for large and sparse unstructured problems

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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