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A new convergence proof of augmented Lagrangian-based method with full Jacobian decomposition for structured variational inequalities
Deflation by restriction for the inverse-free preconditioned Krylov subspace method
1. | Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, United States, United States |
References:
[1] |
Z. Bai, J. Demmel, J. Dongarra, A. Ruhe and H. van der Vorst, Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, Philadelphia, PA, 2000.
doi: 10.1137/1.9780898719581. |
[2] |
L. Bergamaschi, G. Pini and F. Sartoretto, Approximate inverse preconditioning in the parallel solution of sparse eigenproblems, Numer. Linear Alg. Appl., 7 (2000), 99-116.
doi: 10.1002/(SICI)1099-1506(200004/05)7:3<99::AID-NLA188>3.3.CO;2-X. |
[3] |
J. Bramble, J. Pasciak and A. Knyazev, A subspace preconditioning algorithm for eigenvector/eigenvalue computation, Adv. Comp. Math., 6 (1996), 150-189.
doi: 10.1007/BF02127702. |
[4] |
D. Fokkema, G. Sleijpen and H. Van der Vorst, Jacobi-Davidson style QR and QZ algorithms for the reduction of matrix pencils, SIAM J. Sci. Comput, 20 (1999), 94-125.
doi: 10.1137/S1064827596300073. |
[5] |
G. H. Golub and Q. Ye, An inverse free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems, SIAM Journal on Scientific Computing, 24 (2002), 312-334.
doi: 10.1137/S1064827500382579. |
[6] |
A. V. Knyazev, Preconditioned eigensolvers-an oxymoron, Electronic Transactions on Numerical Analysis, 7 (1998), 104-123. |
[7] |
A. V. Knyazev, Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient, SIAM Journal on Scientific Computing, 23 (2001), 517-541.
doi: 10.1137/S1064827500366124. |
[8] |
Q. Liang and Q. Ye, Computing singular values of large matrices with an inverse-free preconditioned krylov subspace method, Electronic Transactions on Numerical Analysis, 42 (2014), 197-221. |
[9] |
R. B. Lehoucq, Analysis And Implementation of An Implicitly Restarted Arnoldi Iteration, Ph.D Thesis, Rice University, 1995. |
[10] |
R. B. Lehoucq and D. C. Sorensen, Deflation techniques within an implicitly restarted Arnoldi iteration, SIAM Journal on Matrix Analysis and Applications, 17 (1996), 789-821.
doi: 10.1137/S0895479895281484. |
[11] |
R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users' Guides, Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Method, SIAM, Philadelphia, 1998.
doi: 10.1137/1.9780898719628. |
[12] |
R. Lehoucq and K. Meerbergen, The inexact rational Krylov sequence method, SIAM J. Matrix Anal. Appl., 20 (1998), 131-148.
doi: 10.1137/S0895479896311220. |
[13] |
K. Meerbergen and D. Roose, The restarted Arnoldi method applied to iterative linear solvers for the computation of rightmost eigenvalues, SIAM J. Matrix Anal. Appl., 20 (1999), 1-20.
doi: 10.1137/S0895479894274255. |
[14] |
J. H. Money and Q. Ye, Algorithm 845: EIGIFP: A MATLAB program for solving large symmetric generalized eigenvalue problems, ACM Transactions on Mathematical Software, 31 (2005), 270-279.
doi: 10.1145/1067967.1067973. |
[15] |
R. Morgan and D. Scott, Preconditioning the Lanczos algorithm for sparse symmetric eigenvalue problems, SIAM J. Sci. Stat. Comput., 14 (1993), 585-593.
doi: 10.1137/0914037. |
[16] |
Y. Notay, Combination of Jacobi-Davidson and conjugate gradients for the partial symmetric eigenproblem, Numerical Linear Algebra and its Applications, 9 (2002), 21-44.
doi: 10.1002/nla.246. |
[17] |
B. N. Parlett, The Symmetric Eigenvalue Problem, Classics in Applied Mathematics, SIAM, Philadelphia, PA, 1998.
doi: 10.1137/1.9781611971163. |
[18] |
P. Quillen and Q. Ye, A block inverse-free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems, J. Comp. Appl. Math, 233 (2010), 1298-1313.
doi: 10.1016/j.cam.2008.10.071. |
[19] |
Y. Saad, Numerical methods for large eigenvalue problems, Revised Edition, Classics in Applied Mathematics, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611970739.ch1. |
[20] |
G. Sleijpen and H. van der Vorst, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM Journal on Matrix Analysis and Applications, 17 (1996), 401-425.
doi: 10.1137/S0895479894270427. |
[21] |
A. Stathopoulos and J. R. McCombs, PRIMME: PReconditioned Iterative MultiMethod Eigensolver: Methods and software description, ACM Transaction on Mathematical Software, 37 (2010), 21:1-21:30. |
[22] |
A. Stathopoulos, Y. Saad and C. Fisher, Robust preconditioning of large sparse symmetric eigenvalue problems, J. Comp. Appl. Math., 64 (1995), 197-215.
doi: 10.1016/0377-0427(95)00141-7. |
[23] |
H. van der Vorst, G. Sleijpen and M. van Gijzen, Efficient expansion of subspaces in the Jacobi-Davison method for standard and generalized eigenproblems, Electronic Transactions on Numerical Analysis, 7 (1998), 75-89. |
[24] |
E. Vecharynski, C. Yang and F. Xue, Generalized preconditioned locally harmonic residual method for non-Hermitian eigenproblems, Preprint. |
[25] |
J. H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University Press, New York, 1965. |
[26] |
C. Yang, Convergence analysis of an inexact truncated RQ iterations, Elec. Trans. Numer. Anal., 7 (1998), 40-55. |
show all references
References:
[1] |
Z. Bai, J. Demmel, J. Dongarra, A. Ruhe and H. van der Vorst, Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide, SIAM, Philadelphia, PA, 2000.
doi: 10.1137/1.9780898719581. |
[2] |
L. Bergamaschi, G. Pini and F. Sartoretto, Approximate inverse preconditioning in the parallel solution of sparse eigenproblems, Numer. Linear Alg. Appl., 7 (2000), 99-116.
doi: 10.1002/(SICI)1099-1506(200004/05)7:3<99::AID-NLA188>3.3.CO;2-X. |
[3] |
J. Bramble, J. Pasciak and A. Knyazev, A subspace preconditioning algorithm for eigenvector/eigenvalue computation, Adv. Comp. Math., 6 (1996), 150-189.
doi: 10.1007/BF02127702. |
[4] |
D. Fokkema, G. Sleijpen and H. Van der Vorst, Jacobi-Davidson style QR and QZ algorithms for the reduction of matrix pencils, SIAM J. Sci. Comput, 20 (1999), 94-125.
doi: 10.1137/S1064827596300073. |
[5] |
G. H. Golub and Q. Ye, An inverse free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems, SIAM Journal on Scientific Computing, 24 (2002), 312-334.
doi: 10.1137/S1064827500382579. |
[6] |
A. V. Knyazev, Preconditioned eigensolvers-an oxymoron, Electronic Transactions on Numerical Analysis, 7 (1998), 104-123. |
[7] |
A. V. Knyazev, Toward the optimal preconditioned eigensolver: Locally optimal block preconditioned conjugate gradient, SIAM Journal on Scientific Computing, 23 (2001), 517-541.
doi: 10.1137/S1064827500366124. |
[8] |
Q. Liang and Q. Ye, Computing singular values of large matrices with an inverse-free preconditioned krylov subspace method, Electronic Transactions on Numerical Analysis, 42 (2014), 197-221. |
[9] |
R. B. Lehoucq, Analysis And Implementation of An Implicitly Restarted Arnoldi Iteration, Ph.D Thesis, Rice University, 1995. |
[10] |
R. B. Lehoucq and D. C. Sorensen, Deflation techniques within an implicitly restarted Arnoldi iteration, SIAM Journal on Matrix Analysis and Applications, 17 (1996), 789-821.
doi: 10.1137/S0895479895281484. |
[11] |
R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users' Guides, Solution of Large Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Method, SIAM, Philadelphia, 1998.
doi: 10.1137/1.9780898719628. |
[12] |
R. Lehoucq and K. Meerbergen, The inexact rational Krylov sequence method, SIAM J. Matrix Anal. Appl., 20 (1998), 131-148.
doi: 10.1137/S0895479896311220. |
[13] |
K. Meerbergen and D. Roose, The restarted Arnoldi method applied to iterative linear solvers for the computation of rightmost eigenvalues, SIAM J. Matrix Anal. Appl., 20 (1999), 1-20.
doi: 10.1137/S0895479894274255. |
[14] |
J. H. Money and Q. Ye, Algorithm 845: EIGIFP: A MATLAB program for solving large symmetric generalized eigenvalue problems, ACM Transactions on Mathematical Software, 31 (2005), 270-279.
doi: 10.1145/1067967.1067973. |
[15] |
R. Morgan and D. Scott, Preconditioning the Lanczos algorithm for sparse symmetric eigenvalue problems, SIAM J. Sci. Stat. Comput., 14 (1993), 585-593.
doi: 10.1137/0914037. |
[16] |
Y. Notay, Combination of Jacobi-Davidson and conjugate gradients for the partial symmetric eigenproblem, Numerical Linear Algebra and its Applications, 9 (2002), 21-44.
doi: 10.1002/nla.246. |
[17] |
B. N. Parlett, The Symmetric Eigenvalue Problem, Classics in Applied Mathematics, SIAM, Philadelphia, PA, 1998.
doi: 10.1137/1.9781611971163. |
[18] |
P. Quillen and Q. Ye, A block inverse-free preconditioned Krylov subspace method for symmetric generalized eigenvalue problems, J. Comp. Appl. Math, 233 (2010), 1298-1313.
doi: 10.1016/j.cam.2008.10.071. |
[19] |
Y. Saad, Numerical methods for large eigenvalue problems, Revised Edition, Classics in Applied Mathematics, SIAM, Philadelphia, 2011.
doi: 10.1137/1.9781611970739.ch1. |
[20] |
G. Sleijpen and H. van der Vorst, A Jacobi-Davidson iteration method for linear eigenvalue problems, SIAM Journal on Matrix Analysis and Applications, 17 (1996), 401-425.
doi: 10.1137/S0895479894270427. |
[21] |
A. Stathopoulos and J. R. McCombs, PRIMME: PReconditioned Iterative MultiMethod Eigensolver: Methods and software description, ACM Transaction on Mathematical Software, 37 (2010), 21:1-21:30. |
[22] |
A. Stathopoulos, Y. Saad and C. Fisher, Robust preconditioning of large sparse symmetric eigenvalue problems, J. Comp. Appl. Math., 64 (1995), 197-215.
doi: 10.1016/0377-0427(95)00141-7. |
[23] |
H. van der Vorst, G. Sleijpen and M. van Gijzen, Efficient expansion of subspaces in the Jacobi-Davison method for standard and generalized eigenproblems, Electronic Transactions on Numerical Analysis, 7 (1998), 75-89. |
[24] |
E. Vecharynski, C. Yang and F. Xue, Generalized preconditioned locally harmonic residual method for non-Hermitian eigenproblems, Preprint. |
[25] |
J. H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University Press, New York, 1965. |
[26] |
C. Yang, Convergence analysis of an inexact truncated RQ iterations, Elec. Trans. Numer. Anal., 7 (1998), 40-55. |
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