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Relevant bibliographies by topics / Multilevel Krylov Subspace Splitting

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Academic literature on the topic 'Multilevel Krylov Subspace Splitting'

Author: Grafiati

Published: 6 September 2023

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Journal articles on the topic "Multilevel Krylov Subspace Splitting"

1

Bai, Zhong-Zhi. "Regularized HSS iteration methods for stabilized saddle-point problems." IMA Journal of Numerical Analysis 39, no. 4 (July 31, 2018): 1888–923. http://dx.doi.org/10.1093/imanum/dry046.

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Abstract We extend the regularized Hermitian and skew-Hermitian splitting (RHSS) iteration methods for standard saddle-point problems to stabilized saddle-point problems and establish the corresponding unconditional convergence theory for the resulting methods. Besides being used as stationary iterative solvers, this class of RHSS methods can also be used as preconditioners for Krylov subspace methods. It is shown that the eigenvalues of the corresponding preconditioned matrix are clustered at a small number of points in the interval $(0, \, 2)$ when the iteration parameter is close to $0$ and, furthermore, they can be clustered near $0$ and $2$ when the regularization matrix is appropriately chosen. Numerical results on stabilized saddle-point problems arising from finite element discretizations of an optimal boundary control problem and of a Cahn–Hilliard image inpainting problem, as well as from the Gauss–Newton linearization of a nonlinear image restoration problem, show that the RHSS iteration method significantly outperforms the Hermitian and skew-Hermitian splitting iteration method in iteration counts and computing times when they are used either as linear iterative solvers or as matrix splitting preconditioners for Krylov subspace methods, and optimal convergence behavior can be achieved when using inexact variants of the proposed RHSS preconditioners.

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Tian, Zhaolu, Xiaojing Li, and Zhongyun Liu. "A general multi-step matrix splitting iteration method for computing PageRank." Filomat 35, no. 2 (2021): 679–706. http://dx.doi.org/10.2298/fil2102679t.

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Based on the general inner-outer (GIO) iteration method [5,34] and the iteration framework [6], we present a general multi-step matrix splitting (GMMS) iteration method for computing PageRank, and analyze its overall convergence property. Moreover, the same idea can be used as a preconditioning technique for accelerating the Krylov subspace methods, such as GMRES method. Finally, several numerical examples are given to illustrate the effectiveness of the proposed algorithm.

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3

Huang, Yunying, and Guoliang Chen. "A relaxed block splitting preconditioner for complex symmetric indefinite linear systems." Open Mathematics 16, no. 1 (June 7, 2018): 561–73. http://dx.doi.org/10.1515/math-2018-0051.

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AbstractIn this paper, we propose a relaxed block splitting preconditioner for a class of complex symmetric indefinite linear systems to accelerate the convergence rate of the Krylov subspace iteration method and the relaxed preconditioner is much closer to the original block two-by-two coefficient matrix. We study the spectral properties and the eigenvector distributions of the corresponding preconditioned matrix. In addition, the degree of the minimal polynomial of the preconditioned matrix is also derived. Finally, some numerical experiments are presented to illustrate the effectiveness of the relaxed splitting preconditioner.

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4

Xiong, Jin-Song. "Generalized accelerated AOR splitting iterative method for generalized saddle point problems." AIMS Mathematics 7, no. 5 (2022): 7625–41. http://dx.doi.org/10.3934/math.2022428.

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<abstract><p>Generalized accelerated AOR (GAAOR) splitting iterative method for the generalized saddle point problems is proposed in this paper. The iterative scheme and the convergence of the GAAOR splitting method are researched. The eigenvalues distributions of its preconditioned matrix is discussed under {two different choices of the parameter matrix Q}. The resulting GAAOR preconditioner is used to precondition Krylov subspace method such as the restarted generalized minimal residual (GMRES) method for solving the equivalent formulation of the generalized saddle point problems. The theoretical results and effectiveness of the GAAOR splitting iterative method are supported by {some} numerical examples.</p></abstract>

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5

Luo, Jia-Min, Hou-Biao Li, and Wei-Bo Wei. "Block splitting preconditioner for time-space fractional diffusion equations." Electronic Research Archive 30, no. 3 (2022): 780–97. http://dx.doi.org/10.3934/era.2022041.

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<abstract><p>For solving a block lower triangular Toeplitz linear system arising from the time-space fractional diffusion equations more effectively, a single-parameter two-step split iterative method (TSS) is introduced, its convergence theory is established and the corresponding preconditioner is also presented. Theoretical analysis shows that the original coefficient matrix after preconditioned can be expressed as the sum of the identity matrix, a low-rank matrix, and a small norm matrix. Numerical experiments show that the preconditioner improve the calculation efficiency of the Krylov subspace iteration method.</p></abstract>

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Lei, Siu-Long, Xu Chen, and Xinhe Zhang. "Multilevel Circulant Preconditioner for High-Dimensional Fractional Diffusion Equations." East Asian Journal on Applied Mathematics 6, no. 2 (May 2016): 109–30. http://dx.doi.org/10.4208/eajam.060815.180116a.

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AbstractHigh-dimensional two-sided space fractional diffusion equations with variable diffusion coefficients are discussed. The problems can be solved by an implicit finite difference scheme that is proven to be uniquely solvable, unconditionally stable and first-order convergent in the infinity norm. A nonsingular multilevel circulant pre-conditoner is proposed to accelerate the convergence rate of the Krylov subspace linear system solver efficiently. The preconditoned matrix for fast convergence is a sum of the identity matrix, a matrix with small norm, and a matrix with low rank under certain conditions. Moreover, the preconditioner is practical, with an O(NlogN) operation cost and O(N) memory requirement. Illustrative numerical examples are also presented.

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7

Li, Cui-Xia, Yan-Jun Liang, and Shi-Liang Wu. "Generalized Preconditioned MHSS Method for a Class of Complex Symmetric Linear Systems." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/206821.

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Based on the modified Hermitian and skew-Hermitian splitting (MHSS) and preconditioned MHSS (PMHSS) methods, a generalized preconditioned MHSS (GPMHSS) method for a class of complex symmetric linear systems is presented. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From a practical point of view, we have analyzed and implemented inexact GPMHSS (IGPMHSS) iteration, which employs Krylov subspace methods as its inner processes. Numerical experiments are reported to confirm the efficiency of the proposed methods.

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8

Chen, Shanqin. "Krylov SSP Integrating Factor Runge–Kutta WENO Methods." Mathematics 9, no. 13 (June 24, 2021): 1483. http://dx.doi.org/10.3390/math9131483.

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Weighted essentially non-oscillatory (WENO) methods are especially efficient for numerically solving nonlinear hyperbolic equations. In order to achieve strong stability and large time-steps, strong stability preserving (SSP) integrating factor (IF) methods were designed in the literature, but the methods there were only for one-dimensional (1D) problems that have a stiff linear component and a non-stiff nonlinear component. In this paper, we extend WENO methods with large time-stepping SSP integrating factor Runge–Kutta time discretization to solve general nonlinear two-dimensional (2D) problems by a splitting method. How to evaluate the matrix exponential operator efficiently is a tremendous challenge when we apply IF temporal discretization for PDEs on high spatial dimensions. In this work, the matrix exponential computation is approximated through the Krylov subspace projection method. Numerical examples are shown to demonstrate the accuracy and large time-step size of the present method.

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9

Luo, Wei-Hua, and Ting-Zhu Huang. "A Parameterized Splitting Preconditioner for Generalized Saddle Point Problems." Journal of Applied Mathematics 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/489295.

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By using Sherman-Morrison-Woodbury formula, we introduce a preconditioner based on parameterized splitting idea for generalized saddle point problems which may be singular and nonsymmetric. By analyzing the eigenvalues of the preconditioned matrix, we find that whenαis big enough, it has an eigenvalue at 1 with multiplicity at leastn, and the remaining eigenvalues are all located in a unit circle centered at 1. Particularly, when the preconditioner is used in general saddle point problems, it guarantees eigenvalue at 1 with the same multiplicity, and the remaining eigenvalues will tend to 1 as the parameterα→0. Consequently, this can lead to a good convergence when some GMRES iterative methods are used in Krylov subspace. Numerical results of Stokes problems and Oseen problems are presented to illustrate the behavior of the preconditioner.

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10

Li, Yan-Ran, Xin-Hui Shao, and Shi-Yu Li. "New Preconditioned Iteration Method Solving the Special Linear System from the PDE-Constrained Optimal Control Problem." Mathematics 9, no. 5 (March 2, 2021): 510. http://dx.doi.org/10.3390/math9050510.

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In many fields of science and engineering, partial differential equation (PDE) constrained optimal control problems are widely used. We mainly solve the optimization problem constrained by the time-periodic eddy current equation in this paper. We propose the three-block splitting (TBS) iterative method and proved that it is unconditionally convergent. At the same time, the corresponding TBS preconditioner is derived from the TBS iteration method, and we studied the spectral properties of the preconditioned matrix. Finally, numerical examples in two-dimensions is applied to demonstrate the advantages of the TBS iterative method and TBS preconditioner with the Krylov subspace method.

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Dissertations / Theses on the topic "Multilevel Krylov Subspace Splitting"

1

Ul, Jabbar Absaar [Verfasser], Stefan [Akademischer Betreuer] Turek, and Heribert [Gutachter] Blum. "Efficient and robust monolithic finite element multilevel Krylov subspace solvers for the solution of stationary incompressible Navier-Stokes equations / Absaar Ul Jabbar ; Gutachter: Heribert Blum ; Betreuer: Stefan Turek." Dortmund : Universitätsbibliothek Dortmund, 2018. http://d-nb.info/1175625485/34.

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Conference papers on the topic "Multilevel Krylov Subspace Splitting"

1

Kumar, N., K. J. Vinoy, and S. Gopalakrishnan. "A reduced order model for electromagnetic scattering using multilevel Krylov subspace splitting." In 2015 IEEE International Conference on Computational Electromagnetics (ICCEM). IEEE, 2015. http://dx.doi.org/10.1109/compem.2015.7052655.

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