New convergence results on the global GMRES method for diagonalizable matrices

Bellalij, M.; Jbilou, Khalide; Sadok, H.

doi:10.1016/j.cam.2007.09.016

Cited by 21 publications

(4 citation statements)

References 13 publications

(25 reference statements)

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“…We use the global version of GMRES for the solution of nonsymmetric saddle point problems with multiple right-hand sides (1.1). This version of GMRES has been introduced in [13] and studied in [4]. The method is based on the global version of the standard Arnoldi process; see for example [1].…”

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confidence: 99%

Preconditioned global Krylov subspace methods for solving saddle point problems with multiple right-hand sides

Badahmane¹,

Bentbib²,

Sadok³

2019

etna

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In the present paper, we propose a preconditioned global approach as a new strategy to solve linear systems with several right-hand sides coming from saddle point problems. The preconditioner is obtained by replacing a (2,2)-block in the original saddle-point matrix A by another well-chosen block. We apply the global GMRES method to solve this new problem with several right-hand sides and give some convergence results. Moreover, we analyze the eigenvalue distribution and the eigenvectors of the proposed preconditioner when the first block is positive definite. We also compare different preconditioned global Krylov subspace algorithms (CG, MINRES, FGMRES, GMRES) with preconditioned block (CG, GMRES) algorithms. Numerical results show that our preconditioned global GMRES method is competitive with other preconditioned global Krylov subspace and preconditioned block Krylov subspace methods for solving saddle point problems with several right-hand sides.

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confidence: 99%

Preconditioned global Krylov subspace methods for solving saddle point problems with multiple right-hand sides

Badahmane¹,

Bentbib²,

Sadok³

2019

etna

Self Cite

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“…In 1972, Frieden first made the maximum entropy method apply to image restoration, and proposed the maximum entropy algorithm that combined the object and noise restoration. Subsequently, Frieden and other scholars also proposed to use maximum entropy restoration problem transformed into the improvement of the deconvolution restoration [7], and get a maximum entropy fast algorithm that based on closed form solution. But Matthew and other scholars used first-order Taylor expansion instead of the original entropy expression, and also presents a maximum entropy fast algorithm.…”

Section: The Research Of Non-blind Image Restoration Methodsmentioning

confidence: 99%

Research Motion Blurred Image Restoration Algorithms

Li¹,

Rao²,

Yang³

2015

IJSIP

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Subject in the process of image acquisition and the relative games between the camera or camera image fuzzy, it seriously affected the further image feature extraction, target recognition and image analysis. In order to meet the needs of reality, get high quality image, we must be used image restoration techniques for fuzzy image restoration. Based on the detailed study on the basis of the theory of wavelet transform, we confirmed by experiments in motion blurred image of wavelet and decomposition level .The method to identify the fuzzy parameters, under the condition of without noise and with noise on the four classic image restoration algorithm, the simulation experiment using the peak signal-to-noise ratio, mean square error, average gray level gradient method and Laplace operator method of four kinds of image quality evaluation index to evaluate the healing effect and recovery result diagram and its quality evaluation. Through the scene shooting motion blurred image point spread function of parameter identification and image restoration, it demonstrates that the proposed point spread function's parameters identification method is sound and effective.

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“…GMRES 算法还可用于求解诸如最优控制、滤波估计、去耦、降阶等控制理论中的微分 Riccati 方程 [27] 。在大特征值问题和边值问题中会出现多元线性系统，线性控制、滤波理论、图像修复等方面包含了著名的 Lyapunov 矩阵方程、Sylvester 矩阵方程和 Stein 矩阵方程，这些方程同样是典型的多元线性系统问题，全局 GMRES 算法正好为这些问题的解决提供了一个很好的工具，不同的数值实验更显示出该方法收敛行为方面的优势 [28,29] 。 GMRES 算法还用于求解 Toeplitz 方程、Helmholtz 方程和 Navier-Stokes 方程等，预处理 GMRES 并行算法也得到了很好的应用 [30][31][32][33] 。在太阳物理的研究中，我国科学家颜毅华于 1995 年首次推导出太阳常 alpha 无力场的边界积分表示，并用边界元方法实现了数值求解 [34] ； Li 等人 2007 年对颜毅华的算法进行了改进，他们引入 GMRES 算法来解决边界元方程组；由此，对 10,000 阶以上的矩阵，用 GMRES 算法使得计算效率提高了 1000~9000 倍 [35] 。 [20] 实型 Laplace 变换的线性方程组光谱延迟修正技术 [21] 微分代数方程的初始值问题控制、光辐射和流体力学 [23][24][25][26] 近海水域控制方程、光学辐射传输方程、计算流体力学 Euler 方程控制理论 [27] 微分 Riccati 方程大型奇异值问题 [28] 广义希尔维斯特矩阵方程太阳物理研究 …”

Section: Gmres 算法的应用简况unclassified

An Overview of Recent Developments and Applications of the GMRES Method

马¹

2013

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Abstract:The generalized minimum residual method (GMRES) is widely applied in the scientific and engineering computations due to its general merit of fast convergence. This paper presents a summary introduction of the GMRES method for its historical development and practical applications, with an emphasis on its recent status. We start with a summary on the origin of the method, followed by some notable variants, together with some recent developments. Then, we introduce some recent applications of the GMRES method in various research fields, pointing out its connection to and impact on these fields. Finally, we provide an outlook on the further development and applications of the GMRES method.Keywords: GMRES; Historical Development; Practical Application

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New convergence results on the global GMRES method for diagonalizable matrices

Cited by 21 publications

References 13 publications

Preconditioned global Krylov subspace methods for solving saddle point problems with multiple right-hand sides

Preconditioned global Krylov subspace methods for solving saddle point problems with multiple right-hand sides

Research Motion Blurred Image Restoration Algorithms

An Overview of Recent Developments and Applications of the GMRES Method

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