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Journal articles on the topic 'Jacobi-Davidson Iteration'

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Author: Grafiati

Published: 6 September 2023

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1

Zhao, Jutao, and Pengfei Guo. "A Study on the Convergence Analysis of the Inexact Simplified Jacobi–Davidson Method." Journal of Mathematics 2021 (December 7, 2021): 1–10. http://dx.doi.org/10.1155/2021/2123897.

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The Jacobi–Davidson iteration method is very efficient in solving Hermitian eigenvalue problems. If the correction equation involved in the Jacobi–Davidson iteration is solved accurately, the simplified Jacobi–Davidson iteration is equivalent to the Rayleigh quotient iteration which achieves cubic convergence rate locally. When the involved linear system is solved by an iteration method, these two methods are also equivalent. In this paper, we present the convergence analysis of the simplified Jacobi–Davidson method and present the estimate of iteration numbers of the inner correction equation. Furthermore, based on the convergence factor, we can see how the accuracy of the inner iteration controls the outer iteration.

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2

Kong, Yuan, and Yong Fang. "Behavior of the Correction Equations in the Jacobi–Davidson Method." Mathematical Problems in Engineering 2019 (August 5, 2019): 1–4. http://dx.doi.org/10.1155/2019/5169362.

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The Jacobi–Davidson iteration method is efficient for computing several eigenpairs of Hermitian matrices. Although the involved correction equation in the Jacobi–Davidson method has many developed variants, the behaviors of them are not clear for us. In this paper, we aim to explore, theoretically, the convergence property of the Jacobi–Davidson method influenced by different types of correction equations. As a by-product, we derive the optimal expansion vector, which imposed a shift-and-invert transform on a vector located in the prescribed subspace, to expand the current subspace.

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3

Zhou, Yunkai. "Studies on Jacobi–Davidson, Rayleigh quotient iteration, inverse iteration generalized Davidson and Newton updates." Numerical Linear Algebra with Applications 13, no. 8 (2006): 621–42. http://dx.doi.org/10.1002/nla.490.

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4

Sleijpen, Gerard L. G., and Henk A. Van der Vorst. "A Jacobi--Davidson Iteration Method for Linear Eigenvalue Problems." SIAM Review 42, no. 2 (January 2000): 267–93. http://dx.doi.org/10.1137/s0036144599363084.

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5

G. Sleijpen, Gerard L., and Henk A. Van der Vorst. "A Jacobi–Davidson Iteration Method for Linear Eigenvalue Problems." SIAM Journal on Matrix Analysis and Applications 17, no. 2 (April 1996): 401–25. http://dx.doi.org/10.1137/s0895479894270427.

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6

Freitag, M. A., and A. Spence. "Rayleigh quotient iteration and simplified Jacobi–Davidson method with preconditioned iterative solves." Linear Algebra and its Applications 428, no. 8-9 (April 2008): 2049–60. http://dx.doi.org/10.1016/j.laa.2007.11.013.

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7

Huang, Yin-Liang, Tsung-Ming Huang, Wen-Wei Lin, and Wei-Cheng Wang. "A Null Space Free Jacobi--Davidson Iteration for Maxwell's Operator." SIAM Journal on Scientific Computing 37, no. 1 (January 2015): A1—A29. http://dx.doi.org/10.1137/140954714.

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8

Szyld, Daniel B., and Fei Xue. "Efficient Preconditioned Inner Solves For Inexact Rayleigh Quotient Iteration And Their Connections To The Single-Vector Jacobi–Davidson Method." SIAM Journal on Matrix Analysis and Applications 32, no. 3 (July 2011): 993–1018. http://dx.doi.org/10.1137/100807922.

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9

Jia, ZhongXiao, and Zhen Wang. "A convergence analysis of the inexact Rayleigh quotient iteration and simplified Jacobi-Davidson method for the large Hermitian matrix eigenproblem." Science in China Series A: Mathematics 51, no. 12 (August 26, 2008): 2205–16. http://dx.doi.org/10.1007/s11425-008-0050-y.

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10

Hochstenbach, Michiel E., and Yvan Notay. "Controlling Inner Iterations in the Jacobi–Davidson Method." SIAM Journal on Matrix Analysis and Applications 31, no. 2 (January 2009): 460–77. http://dx.doi.org/10.1137/080732110.

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11

van den Eshof, Jasper. "The convergence of Jacobi-Davidson iterations for Hermitian eigenproblems." Numerical Linear Algebra with Applications 9, no. 2 (2002): 163–79. http://dx.doi.org/10.1002/nla.266.

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12

Ferronato, Massimiliano, Carlo Janna, and Giorgio Pini. "Parallel Jacobi-Davidson with block FSAI preconditioning and controlled inner iterations." Numerical Linear Algebra with Applications 23, no. 3 (January 5, 2016): 394–409. http://dx.doi.org/10.1002/nla.2030.

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13

Huang, Jinzhi, and Zhongxiao Jia. "On Inner Iterations of Jacobi--Davidson Type Methods for Large SVD Computations." SIAM Journal on Scientific Computing 41, no. 3 (January 2019): A1574—A1603. http://dx.doi.org/10.1137/18m1192019.

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14

Jia, ZhongXiao, and Cen Li. "Inner iterations in the shift-invert residual Arnoldi method and the Jacobi-Davidson method." Science China Mathematics 57, no. 8 (February 25, 2014): 1733–52. http://dx.doi.org/10.1007/s11425-014-4791-5.

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15

Zhao, Wenbo, Yingrui Yu, Xiaoming Chai, Zhonghao Ning, Bin Zhang, Yun Cai, Kun Liu, Xingjie Peng, and Junchong Yu. "A SIMPLIFIED TWO-NODE COARSE-MESH FINITE DIFFERENCE METHOD FOR PIN-WISE CALCULATION WITH SP3." EPJ Web of Conferences 247 (2021): 02023. http://dx.doi.org/10.1051/epjconf/202124702023.

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For accurate and efficient pin-by-pin core calculation of SP3 equations, a simplified two-node Coarse Mesh Finite Difference (CMFD) method with the nonlinear iterative strategy is proposed. In this study, the two-node method is only used for discretization of Laplace operator of the 0th moment in the first equation, while the fine mesh finite difference (FMFD) is used for the 2nd moment flux and the second equation. In the two-node problem, transverse flux is expanded to second-order Legendre polynomials. In addition, the associated transverse leakage is approximated with flat distribution. Then the current coupling coefficients are updated in nonlinear iterations. The generalized eigenvalue problem from CMFD is solved using Jacobi-Davidson method. A protype code CORCA-PIN is developed. FMFD scheme is implemented in CORCA-PIN as well. The 2D KAIST 3A benchmark problem and extended 3D problem, which are cell homogenized problems with strong absorber, are tested. Numerical results show that the solution of the simplified two-node method with 1×1 mesh per cell has comparable accuracy of FMFD with 4×4 meshes per cell, but cost less time. The method is suitable for whole core pin-wise calculation.

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16

Windom, Zachary Wayne, and Rodney J. Bartlett. "On the iterative diagonalization of matrices in quantum chemistry: reconciling preconditioner design with Brillouin-Wigner perturbation theory." Journal of Chemical Physics, February 27, 2023. http://dx.doi.org/10.1063/5.0139295.

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Iterative diagonalization of large matrices to search for a subset of eigenvalues that may be of interest has become routine throughout the field of quantum chemistry. The Lanczos and Davidson algorithms hold a monopoly in particular, owing to their excellent performance on diagonally dominant matrices. However, if the eigenvalues happen to be clustered inside overlapping Gershgorin discs the convergence rate of both strategies can be noticeably degraded. <p>In this work, we show how the Davidson, Jacobi-Davidson, Lanczos, and preconditioned Lanczos correction vectors can be formulated using the Reduced Partitioning Procedure (RPP), which takes advantage of the inherent flexibility promoted by Brillouin-Wigner perturbation theory's (BW-PT) resolvent operator. In doing so, we establish a connection between various preconditioning definitions and the BW-PT resolvent operator. Using Natural Localized Molecular Orbitals (NLMOs) to construct Configuration Interaction Singles (CIS) matrices, we study the impact preconditioner choice has on convergence rate for these comparatively dense matrices. We find that an attractive byproduct of preconditioning the Lanczos algorithm is that the preconditioned variant only needs 21-35% and 54-61% of the matrix-vector operations to extract the lowest energy solution of several Hartree-Fock (HF) and NLMO-based CIS matrices, respectively. On the other hand, the standard Davidson preconditioning definition seems to be generally optimal in terms of requisite matrix-vector operations.

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