The Experts below are selected from a list of 252885 Experts worldwide ranked by ideXlab platform
Zhong-zhi Bai - One of the best experts on this subject based on the ideXlab platform.
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fast matrix splitting Preconditioners for higher dimensional spatial fractional diffusion equations
Journal of Computational Physics, 2020Co-Authors: Zhong-zhi BaiAbstract:Abstract The discretizations of two- and three-dimensional spatial fractional diffusion equations with the shifted finite-difference formulas of the Grunwald-Letnikov type can result in discrete linear systems whose coefficient matrices are of the form D + T , where D is a nonnegative diagonal matrix and T is a block-Toeplitz with Toeplitz-block matrix or a block-Toeplitz with each block being block-Toeplitz with Toeplitz-block matrix. For these discrete spatial fractional diffusion matrices, we construct diagonal and block-circulant with circulant-block splitting Preconditioner for the two-dimensional case, and diagonal and block-circulant with each block being block-circulant with circulant-block splitting Preconditioner for the three-dimensional case, to further accelerate the convergence rates of Krylov subspace iteration methods, and we analyze the eigenvalue distributions for the corresponding preconditioned matrices. Theoretical results show that except for a small number of outliners the eigenvalues of the preconditioned matrices are located within a complex disk centered at 1 with the radius being exactly less than 1, and numerical experiments demonstrate that these structured Preconditioners can significantly improve the convergence behavior of the Krylov subspace iteration methods. Moreover, this approach is superior to the geometric multigrid method and the preconditioned conjugate gradient methods incorporated with the approximate inverse circulant-plus-diagonal Preconditioners in both iteration counts and computing times.
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Restrictive Preconditioners for conjugate gradient methods for symmetric positive definite linear systems
Journal of Computational and Applied Mathematics, 2006Co-Authors: Zhong-zhi Bai, Zeng-qi WangAbstract:The restrictively preconditioned conjugate gradient (RPCG) method for solving large sparse system of linear equations of a symmetric positive definite and block two-by-two coefficient matrix is further studied. In fact, this RPCG method is essentially the classical preconditioned conjugate gradient (PCG) method with a specially structured Preconditioner. Within this setting, we present algorithmic descriptions of two restrictive Preconditioners that, respectively, employ the block Jacobi and the block symmetric Gauss-Seidel matrix splitting matrices as approximations to certain matrices involved in them, and give convergence analyses of the correspondingly induced two PCG methods. Numerical results show that these restrictive Preconditioners can lead to practical and effective PCG methods for solving large sparse systems of linear equations of symmetric positive definite and block two-by-two coefficient matrices.
Jun Zhang - One of the best experts on this subject based on the ideXlab platform.
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incomplete lu preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems
Journal of Computational Physics, 2003Co-Authors: Jun Zhang, Caicheng LuAbstract:We consider the preconditioned iterative solution of large dense linear systems, where the coefficient matrix is a complex valued matrix arising from discretizing the integral equation of electromagnetic scattering. For some scattering structures this matrix can be poorly conditioned. The main purpose of this study is to evaluate the efficiency of a class of incomplete LU (ILU) factorization Preconditioners for solving this type of matrices. We solve the electromagnetic wave equations using the BiCG method with an ILU Preconditioner in the context of a multilevel fast multipole algorithm (MLFMA). The novelty of this work is that the ILU Preconditioner as constructed using the near part block diagonal submatrices generated from the MLFMA. Experimental results show that the ILU Preconditioner reduces the number of BiCG iterations substantially, compared to the block diagonal Preconditioner. The preconditioned iteration scheme also maintains the computational complexity of the MLFMA, and consequently reduces the total CPU time.
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High performance preconditioning techniques for solving general sparse linear systems
2003Co-Authors: Kai Wang, Jun ZhangAbstract:Preconditioned Krylov subspace methods have become standard sparse linear system solvers in many scientific applications. They outperform the Gaussian eliminations based direct solvers in both the CPU time and memory cost for solving large sparse linear systems. Robustness and parallelism are two main research metrics to measure the quality of a high performance Preconditioner. This thesis proposes and investigates several parallel preconditioning strategies that try to increase the preconditioning robustness and to extract inherent parallelism. First, we compare a few parallel preconditioning techniques in the iterative solution of large sparse linear systems arising from the solid Earth simulation. Both global and localized parallel Preconditioners are implemented and their performance are compared. A good sparsity pattern can enhance the convergence performance of the preconditioned iterative solvers. In Chapter 3 we compare two existing sparsity pattern selection strategies with respect to the preconditioning efficiency, effectiveness, and parallelism. This study may help people to decide how to choose a good sparsity pattern for a certain application problem. Then a multistep successive preconditioning strategy is presented to improve the robustness and efficiency of the standard sparse approximate inverse techniques and take advantage of their inherent parallelism. This strategy can be applied to almost any existing sparse approximate inverse techniques and ILU Preconditioners. The introduction of multilevel structure in the ILU preconditioning techniques has successfully improved the preconditioning robustness and parallelism. We first introduce a serial version of the multilevel Preconditioner based on factored sparse approximate inverse techniques, then a parallel version based on the multistep successive preconditioning strategy is proposed. Finally, some theoretical results of the sparse approximate inverse Preconditioners are presented. We give some conditions to justify and guarantee the nonsingularity of the computed preconditioning matrix. Particularly when the original matrix is a strongly diagonally dominant M matrix, we prove that if the preconditioning matrix is nonnegative, it is guaranteed to be nonsingular. The nonnegativeness of the preconditioning matrix is also discussed for dynamic and static sparsity pattern selection strategies.
Kokkwang Phoon - One of the best experts on this subject based on the ideXlab platform.
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a modified ssor Preconditioner for sparse symmetric indefinite linear systems of equations
International Journal for Numerical Methods in Engineering, 2006Co-Authors: Xi Chen, Kimchuan Toh, Kokkwang PhoonAbstract:The standard SSOR Preconditioner is ineffective for the iterative solution of the symmetric indefinite linear systems arising from finite element discretization of the Biot's consolidation equations. In this paper, a modified block SSOR Preconditioner combined with the Eisenstat-trick implementation is proposed. For actual implementation, a pointwise variant of this modified block SSOR Preconditioner is highly recommended to obtain a compromise between simplicity and effectiveness. Numerical experiments show that the proposed modified SSOR preconditioned symmetric QMR solver can achieve faster convergence than several effective Preconditioners published in the recent literature in terms of total runtime. Moreover, the proposed modified SSOR Preconditioners can be generalized to non-symmetric Biot's systems. Copyright © 2005 John Wiley & Sons, Ltd.
Chao Yang - One of the best experts on this subject based on the ideXlab platform.
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elliptic Preconditioner for accelerating the self consistent field iteration in kohn sham density functional theory
SIAM Journal on Scientific Computing, 2013Co-Authors: Lin Lin, Chao YangAbstract:We discuss techniques for accelerating the self-consistent field iteration for solving the Kohn--Sham equations. These techniques are all based on constructing approximations to the inverse of the Jacobian associated with a fixed point map satisfied by the total potential. They can be viewed as Preconditioners for a fixed point iteration. We point out different requirements for constructing Preconditioners for insulating and metallic systems, respectively, and discuss how to construct Preconditioners to keep the convergence rate of the fixed point iteration independent of the size of the atomistic system. We propose a new Preconditioner that can treat insulating and metallic systems in a unified way. The new Preconditioner, which we call an elliptic Preconditioner, is constructed by solving an elliptic partial differential equation. The elliptic Preconditioner is shown to be more effective in accelerating the convergence of a fixed point iteration than the existing approaches for large inhomogeneous syste...
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elliptic Preconditioner for accelerating the self consistent field iteration in kohn sham density functional theory
arXiv: Computational Physics, 2012Co-Authors: Lin Lin, Chao YangAbstract:We discuss techniques for accelerating the self consistent field (SCF) iteration for solving the Kohn-Sham equations. These techniques are all based on constructing approximations to the inverse of the Jacobian associated with a fixed point map satisfied by the total potential. They can be viewed as Preconditioners for a fixed point iteration. We point out different requirements for constructing Preconditioners for insulating and metallic systems respectively, and discuss how to construct Preconditioners to keep the convergence rate of the fixed point iteration independent of the size of the atomistic system. We propose a new Preconditioner that can treat insulating and metallic system in a unified way. The new Preconditioner, which we call an elliptic Preconditioner, is constructed by solving an elliptic partial differential equation. The elliptic Preconditioner is shown to be more effective in accelerating the convergence of a fixed point iteration than the existing approaches for large inhomogeneous systems at low temperature.
Ting-zhu Huang - One of the best experts on this subject based on the ideXlab platform.
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A new Preconditioner for indefinite and asymmetric matrices
Applied Mathematics and Computation, 2013Co-Authors: Chao Wang, Ting-zhu Huang, Chun WenAbstract:We present a novel Preconditioner for numerical solutions of large sparse linear systems with indefinite and asymmetric matrices. This new Preconditioner named as product Preconditioner(PS) is constructed by two fairly simple Preconditioners. The distribution of eigenvalues and the form of the eigenvectors of the preconditioned matrix are analyzed. Moreover, an upper bound on the degree of the minimal polynomial is also studied. Numerical experiments with several examples show that the proposed PS performs better than block diagonal Preconditioner(BD) and block triangular Preconditioner (BT) as well as the constraint Preconditioner(SC) in terms of the number of iteration and computational time.
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A symmetric positive definite Preconditioner for saddle-point problems
International Journal of Computer Mathematics, 2011Co-Authors: Shu-qian Shen, Ting-zhu HuangAbstract:For the classical saddle-point problem, we present precisely two intervals containing the positive and the negative eigenvalues of the preconditioned matrix, respectively, when the inexact version of the symmetric positive definite Preconditioner introduced in Section 2.1 of Gill et al. [Preconditioners for indefinite systems arising in optimization, SIAM J. Matrix Anal. Appl. 13 (1992), pp. 292-311] is employed. The model of Stokes problem is used to test the effectiveness of the presented bounds as well as the quality of the symmetric positive definite Preconditioner.
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the spectral properties of the preconditioned matrix for nonsymmetric saddle point problems
Journal of Computational and Applied Mathematics, 2010Co-Authors: Jianlei Li, Ting-zhu Huang, Liang LiAbstract:In this paper, on the basis of matrix splitting, two Preconditioners are proposed and analyzed, for nonsymmetric saddle point problems. The spectral property of the preconditioned matrix is studied in detail. When the iteration parameter becomes small enough, the eigenvalues of the preconditioned matrices will gather into two clusters-one is near (0,0) and the other is near (2,0)-for the PPSS Preconditioner no matter whether A is Hermitian or non-Hermitian and for the PHSS Preconditioner when A is a Hermitian or real normal matrix. Numerical experiments are given, to illustrate the performances of the two Preconditioners.
