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Volume 22, Issue 6
Combinative Preconditioners of Modified Incomplete Cholesky Factorization and Sherman-Morrison-Woodbury Update for Self-Adjoint Elliptic Dirichlet-Periodic Boundary Value Problems

Zhongzhi Bai, Guiqing Li & Linzhang Lu

J. Comp. Math., 22 (2004), pp. 833-856.

Published online: 2004-12

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  • Abstract

For the system of linear equations arising from discretization of the second-order self-adjoint elliptic Dirichlet-periodic boundary value problems, by making use of the special structure of the coefficient matrix we present a class of combinative preconditioners which are technical combinations of modified incomplete Cholesky factorizations and Sherman-Morrison-Woodbury update. Theoretical analyses show that the condition numbers of the preconditioned matrices can be reduced to $\mathcal{O}(h^{-1})$, one order smaller than the condition number  $\mathcal{O}(h^{-2})$ of the original matrix. Numerical implementations show that the resulting preconditioned conjugate gradient methods are feasible, robust and efficient for solving this class of linear systems.

  • Keywords

System of linear equations, Conjugate gradient method, Incomplete Cholesky factorization, Sherman-Morrison-Woodbury formula, Conditioning.

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  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-22-833, author = {Bai , ZhongzhiLi , Guiqing and Lu , Linzhang}, title = {Combinative Preconditioners of Modified Incomplete Cholesky Factorization and Sherman-Morrison-Woodbury Update for Self-Adjoint Elliptic Dirichlet-Periodic Boundary Value Problems}, journal = {Journal of Computational Mathematics}, year = {2004}, volume = {22}, number = {6}, pages = {833--856}, abstract = {

For the system of linear equations arising from discretization of the second-order self-adjoint elliptic Dirichlet-periodic boundary value problems, by making use of the special structure of the coefficient matrix we present a class of combinative preconditioners which are technical combinations of modified incomplete Cholesky factorizations and Sherman-Morrison-Woodbury update. Theoretical analyses show that the condition numbers of the preconditioned matrices can be reduced to $\mathcal{O}(h^{-1})$, one order smaller than the condition number  $\mathcal{O}(h^{-2})$ of the original matrix. Numerical implementations show that the resulting preconditioned conjugate gradient methods are feasible, robust and efficient for solving this class of linear systems.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/10288.html} }
TY - JOUR T1 - Combinative Preconditioners of Modified Incomplete Cholesky Factorization and Sherman-Morrison-Woodbury Update for Self-Adjoint Elliptic Dirichlet-Periodic Boundary Value Problems AU - Bai , Zhongzhi AU - Li , Guiqing AU - Lu , Linzhang JO - Journal of Computational Mathematics VL - 6 SP - 833 EP - 856 PY - 2004 DA - 2004/12 SN - 22 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/10288.html KW - System of linear equations, Conjugate gradient method, Incomplete Cholesky factorization, Sherman-Morrison-Woodbury formula, Conditioning. AB -

For the system of linear equations arising from discretization of the second-order self-adjoint elliptic Dirichlet-periodic boundary value problems, by making use of the special structure of the coefficient matrix we present a class of combinative preconditioners which are technical combinations of modified incomplete Cholesky factorizations and Sherman-Morrison-Woodbury update. Theoretical analyses show that the condition numbers of the preconditioned matrices can be reduced to $\mathcal{O}(h^{-1})$, one order smaller than the condition number  $\mathcal{O}(h^{-2})$ of the original matrix. Numerical implementations show that the resulting preconditioned conjugate gradient methods are feasible, robust and efficient for solving this class of linear systems.

Bai , ZhongzhiLi , Guiqing and Lu , Linzhang. (2004). Combinative Preconditioners of Modified Incomplete Cholesky Factorization and Sherman-Morrison-Woodbury Update for Self-Adjoint Elliptic Dirichlet-Periodic Boundary Value Problems. Journal of Computational Mathematics. 22 (6). 833-856. doi:
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