# A Multilevel Dual Reordering Strategy for Robust Incomplete LU Factorization of Indefinite Matrices

@article{Zhang2001AMD, title={A Multilevel Dual Reordering Strategy for Robust Incomplete LU Factorization of Indefinite Matrices}, author={Jun Zhang}, journal={SIAM J. Matrix Anal. Appl.}, year={2001}, volume={22}, pages={925-947} }

A dual reordering strategy based on both threshold and graph reorderings is introduced to construct robust incomplete LU (ILU) factorization of indefinite matrices. The ILU matrix is constructed as a preconditioner for the original matrix to be used in a preconditioned iterative scheme. The matrix is first divided into two parts according to a threshold parameter to control diagonal dominance. The first part with large diagonal dominance is reordered using a graph-based strategy, followed by an…

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## References

SHOWING 1-10 OF 68 REFERENCES

### BILUTM: A Domain-Based Multilevel Block ILUT Preconditioner for General Sparse Matrices

- Computer ScienceSIAM J. Matrix Anal. Appl.
- 1999

This implementation is efficient in controlling the fill-in elements during the multilevel block ILU factorization, especially when large size blocks are used in domain decomposition-type implementations.

### ILUM: A Multi-Elimination ILU Preconditioner for General Sparse Matrices

- Computer ScienceSIAM J. Sci. Comput.
- 1996

The ILUM factorization described in this paper can be viewed as a multifrontal version of a Gaussian elimination procedure with threshold dropping which has a high degree of potential parallelism.

### Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems

- Computer ScienceNumer. Linear Algebra Appl.
- 1999

Techniques based on diagonal threshold tolerance when developing multi-elimination and multi-level incomplete LU (ILUM) factorization precondi-tioners for solving general sparse linear systems are introduced.

### An Assessment of Incomplete-LU Preconditioners for Nonsymmetric Linear Systems

- Computer ScienceInformatica
- 2000

A direct method is currently more appropriate than an iterative method for a general-purpose black-box nonsymmetric linear solver and pivoting is even more important for incomplete than for complete factorizations.

### Enhanced multi-level block ILU preconditioning strategies for general sparse linear systems

- Computer Science
- 2001

### A grid-based multilevel incomplete LU factorization preconditioning technique for general sparse matrices

- Computer ScienceAppl. Math. Comput.
- 2001

### Conjugate gradient methods and ILU preconditioning of non‐symmetric matrix systems with arbitrary sparsity patterns

- Computer Science
- 1989

### Comparative Analysis of the Cuthill–McKee and the Reverse Cuthill–McKee Ordering Algorithms for Sparse Matrices

- Computer Science
- 1976

It is proved that for band elimination methods, the two orderings are equivalent and that, surprisingly, the reverse ordering is always at least as good as the original one when envelope elimination techniques are used.