Two New Variants of the Simpler Block GMRES Method with Vector Deflation and Eigenvalue Deflation for Multiple Linear Systems

Tajaddini, Azita; Wu, Gang; Saberi-Movahed, Farid; Azizizadeh, Najmeh

doi:10.1007/s10915-020-01376-w

Cited by 7 publications

(3 citation statements)

References 37 publications

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“…It is known that block Krylov algorithms can handle all columns at once, and thus has a few practical advantages in terms of data locality and commnication efficiency. Block GMRES was described by Vital [273] in 1990 and then studied by Simoncini and Gallopoulos [239,240]; see also [131,220,194,213,133,43,42,2,3,108,158,257]. Block GMRES has also been extended to handle the single right-hand side system (1); see, e.g., [55,15,130].…”

Section: Extensions To Other Problemsmentioning

confidence: 99%

GMRES algorithms over 35 years

Zou

2021

Preprint

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Section: Extensions To Other Problemsmentioning

confidence: 99%

GMRES algorithms over 35 years

Zou

2021

Preprint

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“…PCA method uses the linear combination of principal component vectors to minimize the mean square error. Mathematically, it can be verified that the PCA method can use eigenvalues to realize projection vector [13].…”

Section: Eoretical Basis Of Pcamentioning

confidence: 99%

Recognition Algorithm of Popular Elements of Ethnic Minority Traditional Clothing Based on PCA

2021

Scientific Programming

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Image recognition of ethnic minority costumes is helpful for people to understand, carry forward, and inherit national culture. Taking the minority clothing image as the research object, the image enhancement and threshold segmentation are completed; the principal component features of the minority clothing image are extracted by PCA method; and the image matching degree is obtained according to the principle of minimizing the Euclidean distance. Finally, the calculation process of the PCA method is optimized by a wavelet transform algorithm to realize the recognition of popular elements of minority traditional clothing. The comparative experimental results show that the PCA + BP neural network algorithm is better than the other two recognition algorithms in recognition rate and recognition time.

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“…When solving sequences of linear systems as Equation (1.1), attractive approaches are those that can exploit information generated during the solution of a given system to accelerate the convergence for the next ones. Deflated restarting implements a similar idea between the cycles in the generalized minimum residual norm method (GMRES) [19,21,27]; it is realized by using a deflation subspace containing a few approximate eigenvectors deemed to hamper the convergence of the Krylov subspace methods [11][12][13].…”

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confidence: 99%

A Block Minimum Residual Norm Subspace Solver with Partial Convergence Management for Sequences of Linear Systems

Giraud¹,

Jing²,

Xiang³

2022

SIAM J. Matrix Anal. Appl.

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We are concerned with the iterative solution of linear systems with multiple right-hand sides available one group after another with possibly slowly-varying left-hand sides. For such sequences of linear systems, we first develop a new block minimum norm residual approach that combines two main ingredients. The first component exploits ideas from GCRO-DR [SIAM J. Sci. Comput., 28(5) (2006), pp. 1651-1674, enabling to recycle information from one solve to the next. The second component is the numerical mechanism to manage the partial convergence of the right-hand sides, referred to as inexact breakdown detection in IB-BGMRES [Linear Algebra Appl., 419 (2006), pp. 265-285], that enables the monitoring of the rank deficiency in the residual space basis expanded block-wise.Secondly, for the class of block minimum norm residual approaches, that relies on a block Arnoldi-like equality between the search space and the residual space (e.g., any block GMRES or block GCRO variants), we introduce new search space expansion policies defined on novel criteria to detect the partial convergence. These novel detection criteria are tuned to the selected stopping criterion and targeted convergence threshold to best cope with the selected normwise backward error stopping criterion, enabling to monitor the computational effort while ensuring the final accuracy of each individual solution. Numerical experiments are reported to illustrate the numerical and computational features of both the new block Krylov solvers and the new search space block expansion polices.

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Two New Variants of the Simpler Block GMRES Method with Vector Deflation and Eigenvalue Deflation for Multiple Linear Systems

Cited by 7 publications

References 37 publications

GMRES algorithms over 35 years

GMRES algorithms over 35 years

Recognition Algorithm of Popular Elements of Ethnic Minority Traditional Clothing Based on PCA

A Block Minimum Residual Norm Subspace Solver with Partial Convergence Management for Sequences of Linear Systems

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