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

Censor, Yair; Gordon, Dan; Gordon, Rachel

doi:10.1016/s0167-8191(00)00100-9

Cited by 191 publications

(179 citation statements)

References 22 publications

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“…The Cimmino feasibility algorithm [22][23][24][25][26] is used in this paper as the method of choice to determine directly a "best" solution for the simple inverse problem.…”

Section: Searching With the Cimmino Feasibility Algorithmmentioning

confidence: 99%

“…These weights are applied within the algorithm. 23,26 Importance weighting of the volumes in the domain of calculation allows relatively more of the error of the compromise solution to fall on the less important volumes. It also allows additional constraint points to augment those of the discretization grid, e.g., extra constraint points to better define the peripheral dose of the prostate.…”

Section: Searching With the Cimmino Feasibility Algorithmmentioning

confidence: 99%

“…The Cimmino feasibility algorithm is an iterative linear algorithm which was first applied to radiotherapy inverse problems by Censor et al [23][24][25] The algorithm is safer than most common optimization algorithms outlined earlier since it always converges and, if the prescribed dose constraints are not all satisfied, it reverts to the least-squares solution. 26 …”

Section: Introductionmentioning

confidence: 99%

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Optimized interstitial PDT prostate treatment planning with the Cimmino feasibility algorithm

Altschuler

Zhu

et al. 2005

Medical Physics

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The primary aim of this study was to determine whether optimized photodynamic therapy (PDT) treatment planning (seeking optimized positions, lengths, and strengths of the light sources to satisfy a given dose prescription) can improve dose coverage to the prostate and the sparing of critical organs relative to what can be achieved by the standard PDT plan. The Cimmino algorithm and search procedures based on that algorithm were tested for this purpose. A phase I motexafin lutetium (MLu)-mediated photodynamic therapy protocol is ongoing at the University of Pennsylvania. PDT for the prostate is performed with cylindrical diffusing fibers of various lengths inserted perpendicular to a base plate to obtain longitudinal coverage by a matrix of parallel catheters. The standard plan for the protocol uses sources of equal strength with equal spaced (1-cm) loading. Uniform optical properties were assumed. Our algorithms produce plans that cover the prostate and spare the urethra and rectum with less discrepancy from the dose prescription than the standard plan. The Cimmino feasibility algorithm is fast enough that changes to the treatment plan may be made in the operating room before and during PDT to optimize light delivery.

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“…The Cimmino feasibility algorithm [22][23][24][25][26] is used in this paper as the method of choice to determine directly a "best" solution for the simple inverse problem.…”

Section: Searching With the Cimmino Feasibility Algorithmmentioning

confidence: 99%

Section: Searching With the Cimmino Feasibility Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimized interstitial PDT prostate treatment planning with the Cimmino feasibility algorithm

Altschuler

Zhu

et al. 2005

Medical Physics

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“…As follows from estimates (22), (23), (29), (33), (36), (38), (41) The paper [1] reviews the literature on direct and iterative methods of computing weighted pseudoinverses and weighted normal pseudosolutions. Noteworthy are the papers [31][32][33][34], which contain some algorithmic results that use weighted pseudoinversion with singular weights and consider problems of parallel computations for the algorithms proposed.…”

Section: Remark 12mentioning

confidence: 99%

Representations and expansions of weighted pseudoinverse matrices, iterative methods, and problem regularization. II. Singular weights

2008

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The paper reviews studies on the representations and expansions of weighted pseudoinverse matrices with positive semidefinite weights and on the construction of iterative methods and regularized problems for the calculation of weighted pseudoinverses and weighted normal pseudosolutions based on these representations and expansions. The use of these methods to solve constrained least squares problems is examined.

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“…in the denominator of Equation (1.2) suggested to Censor et al [8] that, when A is sparse, this denominator might be replaced with…”

Section: Introduction and Notationmentioning

confidence: 99%

Bounds on the largest singular value of a matrix and the convergence of simultaneous and block‐iterative algorithms for sparse linear systems

Byrne¹

2009

Int Trans Operational Res

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We obtain the following upper bounds for the eigenvalues of the matrix A † A. For any a in the interval [0, 2] letand c a and r a the maxima of the c aj and r ai , respectively. Then no eigenvalue of the matrix A † A exceeds the maximum ofover all j. Therefore, no eigenvalue of A † A exceeds c a r a .Using these bounds, it follows that, for the matrix G with entriesno eigenvalue of G † G exceeds one, provided that, for some a in the interval [0, 2], we have α i ≤ r −1 ai , and β j ≤ c −1 aj . Using this result, we obtain convergence theorems for several iterative algorithms for solving the problem Ax = b, including the CAV, BICAV, CARP1, SART, SIRT, and the block-iterative DROP and SART methods.

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

Cited by 191 publications

References 22 publications

Optimized interstitial PDT prostate treatment planning with the Cimmino feasibility algorithm

Optimized interstitial PDT prostate treatment planning with the Cimmino feasibility algorithm

Representations and expansions of weighted pseudoinverse matrices, iterative methods, and problem regularization. II. Singular weights

Bounds on the largest singular value of a matrix and the convergence of simultaneous and block‐iterative algorithms for sparse linear systems

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