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A generalization of the positive-definite and skew-Hermitian splitting iteration
A new Bramble-Pasciak-like preconditioner for saddle point problems
1. | Nanhai College, South China Normal University, Foshan 528225, China |
2. | School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China |
References:
[1] |
O. Axelsson and G. Lindskog, On the rate of convergence of the preconditioned conjugate gradient method, Numer. Math., 48 (1986), 499-523.
doi: 10.1007/BF01389448. |
[2] |
Z. Z. Bai and G. Q. Li, Restrictively preconditioned conjugate gradient methods for systems of linear equations, IMA J. Numer. Anal., 23 (2003), 561-580.
doi: 10.1093/imanum/23.4.561. |
[3] |
J. H. Bramble and J. E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximations of ellipstic problems, Math. Comp., 50 (1988), 1-17.
doi: 10.1090/S0025-5718-1988-0917816-8. |
[4] |
M. Benzi, G. H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005), 1-137.
doi: 10.1017/S0962492904000212. |
[5] |
M. Benzi and V. Simoncini, On the eigenvalues of a class of saddle point matrices, Numer. Math., 103 (2006), 173-196.
doi: 10.1007/s00211-006-0679-9. |
[6] |
H. S. Dollar, N. I. M Gould, M. Stoll and A. J. Wathen, Preconditioning saddle-point systems with applications in optimization, SIAM J. Sci. Comput., 32 (2010), 249-270.
doi: 10.1137/080727129. |
[7] |
H. C. Elman, D. J. Silvester and Andrew J. Wathen, "Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics," in "Numerical Mathematics and Scientific Computation," Oxford University Press, Oxford, 2005. |
[8] |
H. C. Elman, A. Ramage and D. J. Silvester, Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow, ACM Trans. Math. Softw., 33 (2007), 251-268. |
[9] |
G. N. Gatica and N. Heuer, Conjugate gradient method for dual-dual mixed formulations, Math. Comp., 71 (2002), 1455-1472.
doi: 10.1090/S0025-5718-01-01394-1. |
[10] |
M. R. Hestenes and E. Stiefel, Methods of conjugate gradient for solving linear systems, J. Res. Nat. Bur. Stand., 49 (1952), 409-436. |
[11] |
A. Klawonn, Block-triangular preconditioners for saddle point problems with a penalty term, SIAM J. Sci. Comput., 19 (1998), 172-184.
doi: 10.1137/S1064827596303624. |
[12] |
J. Liesen and B. N. Parlett, On nonsymmetric saddle point matrices that allow conjugate gradient iterations, Numer. Math., 108 (2008), 605-624.
doi: 10.1007/s00211-007-0131-9. |
[13] |
J. A. Meijerink and H. A. Van der vorst, An iterative solution method for linear equation systems of which the coefficient matrix is symmetric M-matrix, Math. Comp., 31 (1977), 148-162. |
[14] |
X. F. Peng, W. Li and S. H. Xiang, New preconditioners based on symmetric-triangular decomposition for saddle point problems, Computing, 93 (2011), 27-46.
doi: 10.1007/s00607-011-0150-3. |
[15] |
J. Schöberl and W. Zulehner, Symmetric indefinite preconditioners for saddle point problems with applications to pde-constrained optimization problems, SIAM J. Matrix Anal. Appl., 29 (2007), 752-773.
doi: 10.1137/060660977. |
[16] |
M. Stoll and A. J. Wathen, Combination preconditioning and the Bramble-Pasciak+ preconditioner, SIAM J. Matrix Anal. Appl., 30 (2008), 582-608.
doi: 10.1137/070688961. |
[17] |
X. N. Wu, G. H. Golub, J. A. Cuminato and J. Y. Yuan, Symmetric-triangular decomposition and its applications - Part Π: Preconditioners for indefinite systems, BIT Numer. Math., 48 (2008), 139-162.
doi: 10.1007/s10543-008-0160-5. |
[18] |
W. Zulehner, Analysis of iterative methods for saddle point problems: A unified approach, Math. Comp., 71 (2002), 479-505.
doi: 10.1090/S0025-5718-01-01324-2. |
show all references
References:
[1] |
O. Axelsson and G. Lindskog, On the rate of convergence of the preconditioned conjugate gradient method, Numer. Math., 48 (1986), 499-523.
doi: 10.1007/BF01389448. |
[2] |
Z. Z. Bai and G. Q. Li, Restrictively preconditioned conjugate gradient methods for systems of linear equations, IMA J. Numer. Anal., 23 (2003), 561-580.
doi: 10.1093/imanum/23.4.561. |
[3] |
J. H. Bramble and J. E. Pasciak, A preconditioning technique for indefinite systems resulting from mixed approximations of ellipstic problems, Math. Comp., 50 (1988), 1-17.
doi: 10.1090/S0025-5718-1988-0917816-8. |
[4] |
M. Benzi, G. H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005), 1-137.
doi: 10.1017/S0962492904000212. |
[5] |
M. Benzi and V. Simoncini, On the eigenvalues of a class of saddle point matrices, Numer. Math., 103 (2006), 173-196.
doi: 10.1007/s00211-006-0679-9. |
[6] |
H. S. Dollar, N. I. M Gould, M. Stoll and A. J. Wathen, Preconditioning saddle-point systems with applications in optimization, SIAM J. Sci. Comput., 32 (2010), 249-270.
doi: 10.1137/080727129. |
[7] |
H. C. Elman, D. J. Silvester and Andrew J. Wathen, "Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics," in "Numerical Mathematics and Scientific Computation," Oxford University Press, Oxford, 2005. |
[8] |
H. C. Elman, A. Ramage and D. J. Silvester, Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow, ACM Trans. Math. Softw., 33 (2007), 251-268. |
[9] |
G. N. Gatica and N. Heuer, Conjugate gradient method for dual-dual mixed formulations, Math. Comp., 71 (2002), 1455-1472.
doi: 10.1090/S0025-5718-01-01394-1. |
[10] |
M. R. Hestenes and E. Stiefel, Methods of conjugate gradient for solving linear systems, J. Res. Nat. Bur. Stand., 49 (1952), 409-436. |
[11] |
A. Klawonn, Block-triangular preconditioners for saddle point problems with a penalty term, SIAM J. Sci. Comput., 19 (1998), 172-184.
doi: 10.1137/S1064827596303624. |
[12] |
J. Liesen and B. N. Parlett, On nonsymmetric saddle point matrices that allow conjugate gradient iterations, Numer. Math., 108 (2008), 605-624.
doi: 10.1007/s00211-007-0131-9. |
[13] |
J. A. Meijerink and H. A. Van der vorst, An iterative solution method for linear equation systems of which the coefficient matrix is symmetric M-matrix, Math. Comp., 31 (1977), 148-162. |
[14] |
X. F. Peng, W. Li and S. H. Xiang, New preconditioners based on symmetric-triangular decomposition for saddle point problems, Computing, 93 (2011), 27-46.
doi: 10.1007/s00607-011-0150-3. |
[15] |
J. Schöberl and W. Zulehner, Symmetric indefinite preconditioners for saddle point problems with applications to pde-constrained optimization problems, SIAM J. Matrix Anal. Appl., 29 (2007), 752-773.
doi: 10.1137/060660977. |
[16] |
M. Stoll and A. J. Wathen, Combination preconditioning and the Bramble-Pasciak+ preconditioner, SIAM J. Matrix Anal. Appl., 30 (2008), 582-608.
doi: 10.1137/070688961. |
[17] |
X. N. Wu, G. H. Golub, J. A. Cuminato and J. Y. Yuan, Symmetric-triangular decomposition and its applications - Part Π: Preconditioners for indefinite systems, BIT Numer. Math., 48 (2008), 139-162.
doi: 10.1007/s10543-008-0160-5. |
[18] |
W. Zulehner, Analysis of iterative methods for saddle point problems: A unified approach, Math. Comp., 71 (2002), 479-505.
doi: 10.1090/S0025-5718-01-01324-2. |
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