## Preconditioned Krylov Subspace Methods for Solving Nonsymmetric Matrices from CFD Applications (1999)

Venue: | Comput. Methods Appl. Mech. Engrg |

Citations: | 22 - 12 self |

### BibTeX

@ARTICLE{Zhang99preconditionedkrylov,

author = {Jun Zhang},

title = {Preconditioned Krylov Subspace Methods for Solving Nonsymmetric Matrices from CFD Applications},

journal = {Comput. Methods Appl. Mech. Engrg},

year = {1999},

volume = {189},

pages = {825--840}

}

### OpenURL

### Abstract

We conduct experimental study on the behavior of several preconditioned iterative methods to solve nonsymmetric matrices arising from computational fluid dynamics (CFD) applications. The preconditioned iterative methods consist of Krylov subspace accelerators and a powerful general purpose multilevel block ILU (BILUM) preconditioner. The BILUM preconditioner and an enhanced version of it are slightly modified versions of the originally proposed preconditioners. They will be used in combination with different Krylov subspace methods. We choose to test three popular transposefree Krylov subspace methods: BiCGSTAB, GMRES and TFQMR. Numerical experiments, using several sets of test matrices arising from various relevant CFD applications, are reported. Key words: Multilevel preconditioner, Krylov subspace methods, nonsymmetric matrices, CFD applications. AMS subject classifications: 65F10, 65F50, 65N06, 65N55. 1 Introduction A challenging problem in computational fluid dynamics (...

### Citations

1666 |
Iterative Methods for Sparse Linear Systems
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(Show Context)
Citation Context ...ical in making iterative methods practically useful [4]. Thus, in practice, large sparse nonsymmetric linear systems are often solved by Krylov subspace methods coupled with a suitable preconditioner =-=[17]-=-. A preconditioning process consists of some auxiliary operations, which solve a linear system with the matrix A approximately. The preconditioner can itself be a direct solver associated with a nearb... |

775 |
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Citation Context ... [17]. Since we concentrate on solving nonsymmetric problems, a representative group of Krylov subspace methods for such a purpose include: conjugate gradient method applied to normal equations (CGN) =-=[12]-=-; generalized minimum residual method (GMRES) [18]; biconjugate gradient stabilized (BiCGSTAB) [23]; and transpose-free variant of quasi-minimum residual method (TFQMR) [8]. There are other methods th... |

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(Show Context)
Citation Context ...pace methods for such a purpose include: conjugate gradient method applied to normal equations (CGN) [12]; generalized minimum residual method (GMRES) [18]; biconjugate gradient stabilized (BiCGSTAB) =-=[23]-=-; and transpose-free variant of quasi-minimum residual method (TFQMR) [8]. There are other methods that are variants of these methods, but are not discussed here due to the space and due to the fact t... |

300 | A flexible inner-outer preconditioned GMRES algorithm
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- 1993
(Show Context)
Citation Context ...r is changing at each iteration. This feature requires that the iterative accelerator be able to accommodate a variable preconditioner. Such an accelerator is available as a flexible version of GMRES =-=[15]-=-. However, it seems that there have been no similar variants of BiCGSTAB and TFQMR available. On the other hand, seeking an exact solution on the coarsest level is neither necessary no computationally... |

171 | A sparse approximate inverse preconditioner for nonsymmetric linear systems - Benzi, Tuma - 1998 |

102 |
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Citation Context ..., GMRES, and TFQMR, are most promising among the Krylov subspace methods and are representative. Over the past years, efforts have been invested to compare various Krylov subspace methods, see, e.g., =-=[13]-=- for some theoretical discussions. Early comparison of Krylov subspace methods preconditioned by ILU(0) and MILU was reported in [14], where the more elaborate MILU was found much less efficient than ... |

71 |
GMRES: A generalized minimal residual method for solving nonsymmetric linear systems
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Citation Context ...c problems, a representative group of Krylov subspace methods for such a purpose include: conjugate gradient method applied to normal equations (CGN) [12]; generalized minimum residual method (GMRES) =-=[18]-=-; biconjugate gradient stabilized (BiCGSTAB) [23]; and transpose-free variant of quasi-minimum residual method (TFQMR) [8]. There are other methods that are variants of these methods, but are not disc... |

62 | Experimental study of ILU preconditioners for indefinite matrices
- Chow, Saad
- 1997
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Citation Context ...re powerful general purpose preconditioner and more realistic nonsymmetric problems from CFD applications. High accuracy preconditioners are relatively new and they have been mostly tested with GMRES =-=[6, 4]-=-. A comprehensive comparison of Krylov subspace methods with modern preconditioning techniques will undoubtedly offer information that may help researchers and application engineers choose suitable it... |

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44 | Approximate inverse techniques for blockpartitioned matrices
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- 1997
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Citation Context ...e of its singular values are small. We use a regularization approach which consists of perturbing only the smallest singular values to obtain a stabilized inverse. A similar strategy was advocated in =-=[5, 21]-=-. Although the underlying idea is similar, our current implementation uses a slightly different formulation. We replace the smallest singular values of \Sigma by larger values. Specifically, given a t... |

41 | BILUTM: a domain-based multilevel block ILUT preconditioner for general sparse matrices - Saad, Zhang - 1999 |

32 |
Sparse approximate-inverse preconditioners using norm-minimization techniques
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- 1998
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Citation Context ...ghtly cheaper than others [3]. Recently, in connection with the development of sparse approximate inverse preconditioners, these Krylov subspace methods have been used as accelerators and compared in =-=[2, 11]-=-. Once again, BiCGSTAB was shown to be faster than the other two in most cases. In spite of these results, there seems to be no comparison of preconditioned Krylov subspace methods involving a more po... |

23 | GMRESR: A family of nested GMRES - Vorst, Vuik - 1994 |

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21 |
High-order ILU preconditioners for CFD problems
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Citation Context ...iability and robustness of iterative methods can be improved dramatically by the use of preconditioning techniques. In fact, preconditioning is critical in making iterative methods practically useful =-=[4]-=-. Thus, in practice, large sparse nonsymmetric linear systems are often solved by Krylov subspace methods coupled with a suitable preconditioner [17]. A preconditioning process consists of some auxili... |

19 |
der Vorst, Closer to solution: Iterative linear solvers, in State of the Art in Numerical Analysis
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Citation Context ...ossible, we do not consider CGN in this paper. An introduction to these and other modern iterative methods can be found in [17], a recent survey of iterative solution methods for linear systems is in =-=[9]-=-, algorithms and availability of computer codes have been outlined in [1]. We believe the three iterative methods, BiCGSTAB, GMRES, and TFQMR, are most promising among the Krylov subspace methods and ... |

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17 | Domain decomposition and multi-level type techniques for general sparse linear systems
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Citation Context ...es a threshold tolerance �� are discarded [20]. This strategy sometimes fails to control the sparsity pattern for large scale applications. A more aggressive double dropping strategy is introduced=-= in [19]-=- to control the amount of fill-ins. First, the single dropping strategy is applied, then only the largest p elements in absolute values of each rows are kept. The advantage of the double dropping stra... |

16 | Enhanced multi-level block ILU preconditioning strategies for general sparse linear systems
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- 1998
(Show Context)
Citation Context ...or indefinite matrices, some blocks may be near-singular and illconditioned and a singular value decomposition (SVD) based regularized-inverse technique may be used to invert the blocks approximately =-=[21]-=-. Here we also give a different version of SVD regularized-inverse. Suppose a matrix B has an SVD of the form [10, p. 16--17] B = U \SigmaV T ; (4) where U and V are two orthogonal matrices and \Sigma... |

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FIDAP: Examples Manual, Revision 6.0
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Citation Context ...rm residual UTM5940 Figure 4: Convergence history for solving UTM5940 with different sparsity. FIDAP Matrices. The FIDAP matrices 2 were extracted from the test problems provided in the FIDAP package =-=[7]-=-. They were generated by I. Hasbani of Fluid Dynamics International and B. Rackner of Minnesota Supercomputer Center. The matrices were resulted from modeling the incompressible Navier-Stokes equation... |

13 |
A transpose-free quasi-minimum residual algorithm for nonHermitian linear systems
- Freund
- 1993
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Citation Context ... to normal equations (CGN) [12]; generalized minimum residual method (GMRES) [18]; biconjugate gradient stabilized (BiCGSTAB) [23]; and transpose-free variant of quasi-minimum residual method (TFQMR) =-=[8]-=-. There are other methods that are variants of these methods, but are not discussed here due to the space and due to the fact that their behaviors are more or less resembling one of the methods under ... |

11 |
ILUT: A dual threshold incomplete ILU preconditioner
- Saad
- 1994
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Citation Context ... On the other hand, seeking an exact solution on the coarsest level is neither necessary no computationally cheap. We decide to choose a compromise by using a high accuracy ILUT(��; p) preconditio=-=ner [16]-=- as an approximate solver on the coarsest level, i.e., AL is approximately factored as LLUL and the Lines 6 and 7 in Algorithm 3.1 are replaced by 7. Solve LLULxL = xL . In this way, although we still... |

6 | A multi-level preconditioner with applications to the numerical simulation of coating problems - Saad, Zhang - 1999 |

4 |
Iterative algorithms for the solution of nonsymmetric systems in the modelling of weak plasma turbulence
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- 1989
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Citation Context ...en invested to compare various Krylov subspace methods, see, e.g., [13] for some theoretical discussions. Early comparison of Krylov subspace methods preconditioned by ILU(0) and MILU was reported in =-=[14]-=-, where the more elaborate MILU was found much less efficient than the standard ILU(0). Comparison of preconditioned nonsymmetric Krylov subspace methods on a large-scale MIMD machine has been reporte... |

4 | A grid based multilevel incomplete LU factorization preconditioning technique for general sparse matrices - Zhang - 1999 |

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1 | Iterative schemes for the neutron diffusion equation
- Bru, Ginestar, et al.
- 1998
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Citation Context ...uniform discretization PDE problems were tested. A very recent application involving the three Krylov subspace methods under current consideration with some specialized preconditioner was reported in =-=[3]-=- for certain problems in neutron diffusion equation. Most of the tests so far have shown that there is no clear winner among the three Krylov subspace methods. However, in terms of operation costs, Bi... |

1 | Iterative schemes for the neutron di€usion equation, in - Bru, Ginestar, et al. - 1998 |