Journal Article10.1016/S0167-8191(00)00100-9
Component averaging: An efficient iterative parallel algorithm for large and sparse unstructured problems
Yair Censor,Dan Gordon,Rachel Gordon +2 more
- 01 May 2001
- Vol. 27, Iss: 6, pp 777-808
247
TL;DR: In this article, Component averaging (CAV) is introduced as a new iterative parallel technique suitable for large and sparse unstructured systems of linear equations, which simultaneously projects the current iterate onto all the system's hyperplanes, and is thus inherently parallel.
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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.
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Citations
Globally convergent image reconstruction for emission tomography using relaxed ordered subsets algorithms
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Convergence studies on iterative algorithms for image reconstruction
Ming Jiang,Ge Wang +1 more
TL;DR: It is found that in all cases the limit is the sum of the minimum norm solution of a weighted least-squares problem and an oblique projection of the initial image onto the null space of the system matrix.
261
AIR Tools II: Algebraic Iterative Reconstruction Methods, Improved Implementation
TL;DR: A MATLAB software package with efficient, robust, and flexible implementations of algebraic iterative reconstruction (AIR) methods for computing regularized solutions to discretizations of inverse problems, based on a new modular design.
Block-Iterative Algorithms with Diagonally Scaled Oblique Projections for the Linear Feasibility Problem
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TL;DR: A block-iterative algorithmic scheme for the solution of systems of linear inequalities and/or equations and its convergence is formed and a mathematical study of the convergence of the algorithms is studied.
References
A parallel projection method for overdetermined nonlinear systems of equations
TL;DR: A generalization of the classical method of Cimmino for linear systems for overdetermined nonlinear systems and a practical strategy for improving the global convergence properties of the method are introduced.
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Iterative algorithms for large partitioned linear systems, with applications to image reconstruction
TL;DR: It is shown that some well-known iterative methods of image reconstruction fall into the class of algorithms under consideration, and are thus covered by the convergence theory, and a novel application to truly three-dimensional image reconstruction is described.
Algebraic reconstruction techniques can be made computationally efficient (positron emission tomography application)
Gabor T. Herman,L.B. Meyer +1 more
TL;DR: Algebraic reconstruction techniques (ART) are iterative procedures for recovering objects from their projections as discussed by the authors, which is claimed to produce high-quality reconstructions with excellent computational efficiency This is demonstrated by an example based on a particular (but realistic) medical imaging task, showing that ART can match the performance of the standard expectation-maximization approach for maximizing likelihood, but at an order of magnitude less computational cost.
A Projection-Based Algorithm for Consistent and Inconsistent Constraints
TL;DR: It is shown that the iterates generated by the algorithm converge weakly to a global minimizer of $\hat J$ provided the set of fixed points of the algorithm is nonempty.
The Landweber iteration and projection onto convex sets
TL;DR: For circulant systems, the Landweber iteration with a special step size is shown to be equivalent to a variation of the projection method, and it is conjectured that the projection technique has a better convergence rate.