## Preconditioning highly indefinite and nonsymmetric matrices (2000)

Venue: | SIAM J. SCI. COMPUT |

Citations: | 40 - 4 self |

### BibTeX

@ARTICLE{Benzi00preconditioninghighly,

author = {Michele Benzi and John C. Haws and Miroslav Tůma},

title = {Preconditioning highly indefinite and nonsymmetric matrices},

journal = {SIAM J. SCI. COMPUT},

year = {2000},

volume = {22},

number = {4},

pages = {1333--1353}

}

### Years of Citing Articles

### OpenURL

### Abstract

Standard preconditioners, like incomplete factorizations, perform well when the coefficient matrix is diagonally dominant, but often fail on general sparse matrices. We experiment with nonsymmetric permutationsand scalingsaimed at placing large entrieson the diagonal in the context of preconditioning for general sparse matrices. The permutations and scalings are those developed by Olschowka and Neumaier [Linear Algebra Appl., 240 (1996), pp. 131–151] and by Duff and

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Citation Context ...erically unsymmetric and far from diagonally dominant, it was found in [4] that the performance and robustness of ILU-type preconditioners was generally improved by the reverse Cuthill–McKee ordering =-=[16]-=-, denoted “rcm” in the tables (see column “SO”). Thus, we used rcm as the default ordering for ILU preconditioners. Although it is not always the best ordering, rcm gave good results in a majority of ... |

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Citation Context ...ping rules give rise to different ILU preconditioners. When all fill-ins are discarded and only nonzeros in positions corresponding to the nonzero entries of A are retained, the ILU(0) preconditioner =-=[34]-=- is obtained. This preconditioner is easy to implement and inexpensive to compute, but it is often not good enough, particularly for the kind of challenging problems considered in this paper. More pow... |

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Citation Context ...but it has been reported by several researchers as being quite challenging for sparse approximate inPRECONDITIONING INDEFINITE, NONSYMMETRIC MATRICES 1341 verse preconditioners; see, e.g., [1], [6], =-=[26]-=-, and [27]. Matrix LNS3937 has a zero diagonal block corresponding to the divergence constraint in the Navier–Stokes equations and is challenging for both ILU and approximate inverse techniques; see, ... |

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On algorithms for obtaining a maximum transversal
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Citation Context ... the diagonal. Because our code is row oriented, we limit our discussion to row permutations, i.e., permutations of the form QA. One of the options in MC64 uses the algorithm MC21 implemented by Duff =-=[14]-=-, [15]. MC21 is a depth-first search algorithm with look-ahead; for a sparse matrix with τ nonzero entries, the algorithm has worst-case complexity of O(nτ) but in practice exhibits O(n + τ) behavior.... |

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Citation Context ... process, but in the incomplete factors. Unstable (more accurately, ill-conditioned) ILU factors occur frequently for matrices that are far from symmetric and lack diagonal dominance; see [22], [10], =-=[4]-=-. For general sparse matrices, it is frequently the case that both kinds of instability occur simultaneously, with a crippling effect on the quality of the preconditioner. In general, the complete LU ... |

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Citation Context ...s: The construction phase is guaranteed to be breakdown-free only for special classes of matrices. Sufficient conditions are that A be symmetric positive definite or an H-matrix; see [29], [3], [28], =-=[2]-=-. For general sparse matrices, instabilities due to very small or zero pivots can occur during the construction of the preconditioner, with disastrous effects. This is perfectly analogous to the insta... |

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Citation Context ..., often exhibit a large number of zero diagonal entries and poor spectral distributions, and they represent a challenge for preconditioned Krylov subspace solvers. 1.2. Contributions of the paper. In =-=[36]-=-, Olschowka and Neumaier introduce new permutations and scaling strategies for Gaussian elimination. The goal is to preprocess the coefficient matrix so as to obtain an equivalent system with a matrix... |

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Citation Context ...nd Koster; see [18] and [19]. Some evidence of the usefulness of these preprocessings in connection with sparse direct solvers and for ILU preconditioning has been provided in [18] and [19]; see also =-=[30]-=-. Our contribution is to carry out a systematic experimental study of the use of these permutation and scaling algorithms in the context of preconditioned iterative methods applied to challenging line... |

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Citation Context ... been reported by several researchers as being quite challenging for sparse approximate inPRECONDITIONING INDEFINITE, NONSYMMETRIC MATRICES 1341 verse preconditioners; see, e.g., [1], [6], [26], and =-=[27]-=-. Matrix LNS3937 has a zero diagonal block corresponding to the divergence constraint in the Navier–Stokes equations and is challenging for both ILU and approximate inverse techniques; see, respective... |

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Algorithm 575: Permutations for a zero-free diagonal
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Citation Context ...iagonal. Because our code is row oriented, we limit our discussion to row permutations, i.e., permutations of the form QA. One of the options in MC64 uses the algorithm MC21 implemented by Duff [14], =-=[15]-=-. MC21 is a depth-first search algorithm with look-ahead; for a sparse matrix with τ nonzero entries, the algorithm has worst-case complexity of O(nτ) but in practice exhibits O(n + τ) behavior. The u... |

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Citation Context ...ve positive real part. 4. Description of test problems. In this section we describe the matrices that were used in the numerical experiments. Most of these matrices are available in the public domain =-=[17]-=-, [12], [35]. They are representative of problems from a variety of applications, but they have in common the fact that they are difficult to solve with iterative methods. The matrices are listed in T... |

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Citation Context ... inverse preconditioners in factorized form is sensitive to the ordering of the matrix. For structurally symmetric (or nearly so) matrices having a stable AINV preconditioner, it was shown in [9] and =-=[7]-=- that symmetric reorderings that reduce fill-in in the inverse factors, like minimum degree or (generalized) nested dissection, can be used to improve the performance of the preconditioner. However, t... |

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A stability analysis of incomplete
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Citation Context ...actorization process, but in the incomplete factors. Unstable (more accurately, ill-conditioned) ILU factors occur frequently for matrices that are far from symmetric and lack diagonal dominance; see =-=[22]-=-, [10], [4]. For general sparse matrices, it is frequently the case that both kinds of instability occur simultaneously, with a crippling effect on the quality of the preconditioner. In general, the c... |

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Der Vorst, “A parallel linear system solver for circuit simulation problems
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Citation Context ...e names to WATSON4a and WATSON5a). The row vector used was e T n =(0,...,0, 1). All diagonal entries are nonzero. The third circuit matrix was kindly provided by Wim Bomhof of Utrecht University; see =-=[8]-=-. This matrix has some zero diagonal entries. All three matrices are very sparse, and they exhibit a good deal of structural symmetry. Finally, we included six matrices from the discretization of PDEs... |

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Citation Context ...roximate inverse preconditioners in factorized form is sensitive to the ordering of the matrix. For structurally symmetric (or nearly so) matrices having a stable AINV preconditioner, it was shown in =-=[9]-=- and [7] that symmetric reorderings that reduce fill-in in the inverse factors, like minimum degree or (generalized) nested dissection, can be used to improve the performance of the preconditioner. Ho... |

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Preconditioning techniques for indefinite and nonsymmetric linear systems
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Citation Context ...en there are zero or small entries on the main diagonal. As long as A is nonsingular it has (in exact arithmetic) an LU factorization with pivoting, but this is not true for incomplete factorizations =-=[38]-=-. On the other hand, ILU factorizations that are both accurate and stable are possible for diagonally dominant matrices. Because of these and other limitations of ILU-type preconditioners, alternative... |

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7 |
Preconditioned iterative methods for sparse linear algebra problems arising in circuit simulation
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Citation Context ...arge number of reported failures) and that realizing the potential of iterative solvers will require improvements in the reordering and/or preconditioning schemes. Similar conclusions were reached in =-=[33]-=- for the use of iterative methods in circuit simulations. The use of preconditioned Krylov subspace methods for the solution of sparse linear systems arising in economic modeling has been investigated... |