## Abstract

This paper presents two methods based on domain decomposition concepts for determining the diagonal of the inverse of specific matrices. The first uses a divide-and-conquer principle and the Sherman-Morrison-Woodbury formula and assumes that the matrix can be decomposed into a 2×2 block-diagonal matrix and a low-rank matrix. The second method is a standard domain decomposition approach in which local solves are combined with a global correction. Both methods can be successfully combined with iterative solvers and sparse approximation techniques. The efficiency of the methods usually depends on the specific implementation, which should be fine-tuned for different test problems. Preliminary results for some two-dimensional (2D) problems are reported to illustrate the proposed methods.

Original language | English (US) |
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Pages (from-to) | 2823-2847 |

Number of pages | 25 |

Journal | SIAM Journal on Scientific Computing |

Volume | 33 |

Issue number | 5 |

DOIs | |

State | Published - Nov 24 2011 |

## Keywords

- Divide-and-conquer method
- Domain decomposition methods
- Iterative methods
- Matrix diagonal extraction
- Schur complement
- Sherman-Morrison-Woodbury formula
- Sparse approximate inverse