Factorized sparse approximate inverse preconditioning for singular M-matrices
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917169731731456 |
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| author | Bick, Katherina Nabben, Reinhard |
| author_facet | Bick, Katherina Nabben, Reinhard |
| contents | Here we consider the factorized sparse approximate inverse (FSAI) preconditioner. We apply the FSAI preconditioner to singular irreducible M-matrices. These matrices arise e.g. in discrete Markov chain modeling or as graph Laplacians. We show, that there are some restrictions on the nonzero pattern needed for a stable construction of the FSAI preconditioner in this case. With these restrictions FSAI is well-defined. Moreover, we proved that the FSAI preconditioner shares some important properties with the original system. The lower triangular matrix $L_G$ and the upper triangular matrix $U_G$, generated by FSAI, are non-singular and non-negative. The diagonal entries of $L_GAU_G$ are positive and $L_GAU_G$, the preconditioned matrix, is a singular M-matrix. Even more, we establish that a (1,2)-inverse is computed for the complete nonzero patter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21744 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Factorized sparse approximate inverse preconditioning for singular M-matrices Bick, Katherina Nabben, Reinhard Numerical Analysis 65F10, 65F50, 65N22, 65N55 Here we consider the factorized sparse approximate inverse (FSAI) preconditioner. We apply the FSAI preconditioner to singular irreducible M-matrices. These matrices arise e.g. in discrete Markov chain modeling or as graph Laplacians. We show, that there are some restrictions on the nonzero pattern needed for a stable construction of the FSAI preconditioner in this case. With these restrictions FSAI is well-defined. Moreover, we proved that the FSAI preconditioner shares some important properties with the original system. The lower triangular matrix $L_G$ and the upper triangular matrix $U_G$, generated by FSAI, are non-singular and non-negative. The diagonal entries of $L_GAU_G$ are positive and $L_GAU_G$, the preconditioned matrix, is a singular M-matrix. Even more, we establish that a (1,2)-inverse is computed for the complete nonzero patter. |
| title | Factorized sparse approximate inverse preconditioning for singular M-matrices |
| topic | Numerical Analysis 65F10, 65F50, 65N22, 65N55 |
| url | https://arxiv.org/abs/2512.21744 |