## A Sparse Approximate Inverse Technique for Parallel Preconditioning of General Sparse Matrices (1998)

Venue: | Appl. Math. Comput |

Citations: | 14 - 6 self |

### BibTeX

@TECHREPORT{Zhang98asparse,

author = {Jun Zhang},

title = {A Sparse Approximate Inverse Technique for Parallel Preconditioning of General Sparse Matrices},

institution = {Appl. Math. Comput},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

A sparse approximate inverse technique is introduced to solve general sparse linear systems. The sparse approximate inverse is computed as a factored form and used as a preconditioner to work with some Krylov subspace methods. The new technique is derived from a matrix decomposition algorithm for inverting dense nonsymmetric matrices. Several strategies and special data structures are proposed to implement the algorithm efficiently. Sparsity patterns of the the factored inverse are exploited to reduce computational cost. The computation of the factored sparse approximate inverse is relatively cheaper than the techniques based on norm minimization techniques. The new preconditioner possesses much greater inherent parallelism than traditional preconditioners based on incomplete LU factorizations. Numerical experiments are used to show the effectiveness and efficiency of the new sparse approximate inverse preconditioner.

### Citations

1529 |
Iterative Methods for Sparse Linear Systems
- Saad
- 2003
(Show Context)
Citation Context ...nly solved by iterative methods. In particular, Krylov subspace methods have become popular choices of iterative solution techniques due to their ability to handle general sparse matrices, see, e.g., =-=[36]-=-, and a recent survey by Golub and van der Vorst [20]. Several numerical experiments and analytical investigations have shown that many of these Krylov subspace methods behave similarly for a large nu... |

256 | SPARSKIT: A basic tool kit for sparse matrix computation
- Saad
- 1990
(Show Context)
Citation Context ...ilar to that used by Saad in ILUT [35]. In detail, after Algorithm 3.3 computed a row of U or a column of L with all elements greater than �� in absolute value, a quick split algorithm (from SPARS=-=KIT [34]) is emplo-=-yed to find the largest lf il elements in absolute value of each row or column. These lf il largest elements are kept and the remaining "small" elements are dropped. Here we report a numeric... |

250 |
der Vorst. An iterative solution method for linear systems of which the coe cient matrix is a symmetric M-matrix
- Meijerink, Van
- 1977
(Show Context)
Citation Context ...rpose" preconditioners that are efficient for a large group of problems. The best-known general-purpose preconditioners are those based on the incomplete LU (ILU) factorization of the original ma=-=trix [31, 36]-=-. The accuracy of the ILU preconditioners can be controlled by the amount of fill-ins retained. In general, high accuracy ILU type preconditioners (with large amount of fill-ins) are more robust but c... |

233 | Users’ guide for the harwell-boeing sparse matrix collection
- Duff, Grimes, et al.
- 1992
(Show Context)
Citation Context ...g resolved. 4 Numerical Experiments Numerical experiments have been conducted to solve several matrices from realistic applications. Most of the matrices are taken from the Harwell-Boring collections =-=[19]-=- and a few from other sources. Table 1 is a brief description of the matrices tested. 2 Some of these matrices are larger than those that are traditionally solved by sparse approximate inverse techniq... |

185 | Parallel preconditioning with sparse approximate inverses - Grote, Huckle - 1997 |

155 | A sparse approximate inverse preconditioner for the conjugate gradient method
- Benzi, Meyer, et al.
- 1996
(Show Context)
Citation Context ... may not be well suitable for general sparse matrices. Yet there has been another approach to computing a factored sparse approximate inverse, which is based on the method of incomplete biconjugation =-=[6, 8]-=-. This method does not require that the sparsity pattern be given in advance. The construction of the factored sparse approximate inverse is based on a matrix direct inverse algorithm. In essence, two... |

117 |
Algebraic multilevel preconditioning methods
- Axelsson, Vassilevski
- 1989
(Show Context)
Citation Context ...itional ILU type preconditioners promotes the current strong interest in searching for alternative preconditioning techniques. There are parallelizable preconditioners based on multi-level techniques =-=[2, 1, 39, 40]-=-. These methods exploit the concept of (block) independent set ordering and construct coarse level system via Schur complement approaches. They are highly efficient and may demonstrate grid-independen... |

116 |
Factorized sparse approximate inverse preconditioning I. Theory
- Kolotilina, Yeremin
- 1993
(Show Context)
Citation Context ...econditioners. They can be roughly categorized into three groups [7], i.e., sparse approximate inverses based on Frobenius norm minimization [14, 15, 17, 21, 22], factored sparse approximate inverses =-=[8, 25]-=-, sparse approximate inverses computed from an ILU factorization [36]. Each of these groups contains a variety of different constructions and each of them has its own merits and drawbacks. In other wo... |

100 |
How fast are nonsymmetric matrix iterations
- Nachtigal, Reddy, et al.
- 1992
(Show Context)
Citation Context ...by Golub and van der Vorst [20]. Several numerical experiments and analytical investigations have shown that many of these Krylov subspace methods behave similarly for a large number of test problems =-=[32, 43]-=-. On the other hand, it has been recognized that the performance of these Krylov subspace methods can be remarkably enhanced by coupling them with a suitable preconditioner. Thus, a preconditioned ite... |

88 |
Block Preconditioning for the Conjugate Gradient Method
- Concus, Golub, et al.
- 1985
(Show Context)
Citation Context ...rses have been used as local components in some other types of global preconditioners, e.g., in multigrid [3, 42] and multi-level preconditioners [44], and in block preconditioning methods in general =-=[14, 16, 21]-=-. These applications are aimed at utilizing the better local coupling property of the sparse approximate inverse techniques [41]. 3 New Factored Sparse Approximate Inverse In a series of papers [28, 2... |

79 | Approximate inverse preconditioners via sparse-sparse iterations
- Chow, Saad
(Show Context)
Citation Context ...res have been reported [9, 33, 37, 38]. There are another group of parallelizable preconditioners. In the past few years, several versions of sparse approximate inverse techniques have been developed =-=[8, 11, 14, 15, 21, 22]-=-. These methods are based on approximating the inverse matrix directly, i.e., a sparse matrix M = A \Gamma1 is explicitly computed and stored as a preconditioner for a given sparse matrix A. The preco... |

62 |
Approximate inverse preconditioning for sparse linear systems
- Cosgrove, Dias, et al.
- 1992
(Show Context)
Citation Context ...veral techniques to construct sparse approximate inverse preconditioners. They can be roughly categorized into three groups [7], i.e., sparse approximate inverses based on Frobenius norm minimization =-=[14, 15, 17, 21, 22]-=-, factored sparse approximate inverses [8, 25], sparse approximate inverses computed from an ILU factorization [36]. Each of these groups contains a variety of different constructions and each of them... |

56 | A comparative study of sparse approximate inverse preconditioners
- Benzi, Tuma
- 1999
(Show Context)
Citation Context ... matrix-vector operation with M . In other words, these preconditioners have the property of affording maximum potential parallelism. They have been shown to be efficient for certain type of problems =-=[7, 15, 21]-=-. In some cases, the construction of a sparse approximate inverse is possible even if the matrix does not have a stable ILU factorization. One drawback of many sparse approximate inverse techniques is... |

53 | A priori sparsity patterns for parallel sparse approximate inverse preconditioners
- Chow
(Show Context)
Citation Context ...matrices, the parallel construction of M is straightforward. For general sparse matrices, it is very difficult to prescribe a useful sparsity pattern in advance (but see, e.g., recent results of Chow =-=[13]-=-), thus the algorithms must develop suitable sparsity pattern adaptively. Most techniques developed so far try to locate and capture large entries of the inverse [22], a different strategy with simila... |

53 | BILUM: Block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems
- Saad, Zhang
- 1999
(Show Context)
Citation Context ...itional ILU type preconditioners promotes the current strong interest in searching for alternative preconditioning techniques. There are parallelizable preconditioners based on multi-level techniques =-=[2, 1, 39, 40]-=-. These methods exploit the concept of (block) independent set ordering and construct coarse level system via Schur complement approaches. They are highly efficient and may demonstrate grid-independen... |

46 |
Iterative solution of large sparse linear systems arising in certain multidimensional approximation problems
- Benson, Frederickson
- 1982
(Show Context)
Citation Context ...of several existing sparse approximate inverse techniques can be found in [7]. The most popular approach to constructing sparse approximate inverse is based on the idea of Frobenius norm minimization =-=[5, 15, 22]-=-. The sparse approximate inverse is computed as the matrix M which minimizes kI \Gamma AMkF , subject to certain sparsity constraint. The Frobenius norm is usually chosen as it can be computed conveni... |

43 | Approximate inverse techniques for block-partitioned matrices
- CHOW, SAAD
- 1997
(Show Context)
Citation Context ...res have been reported [9, 33, 37, 38]. There are another group of parallelizable preconditioners. In the past few years, several versions of sparse approximate inverse techniques have been developed =-=[8, 11, 14, 15, 21, 22]-=-. These methods are based on approximating the inverse matrix directly, i.e., a sparse matrix M = A \Gamma1 is explicitly computed and stored as a preconditioner for a given sparse matrix A. The preco... |

43 |
Parallel preconditioning and approximate inverses on the connection machine
- Grote, Simon
- 1992
(Show Context)
Citation Context ...xist with other sparse approximate inverse techniques. They have been overlooked for some time until very recently when the parallel implementation became an issue, see the discussions and remarks in =-=[4, 13, 18, 21, 23]-=-. Our impression is that, for both the norm-minimization based and incomplete biconjugation based sparse approximate inverse techniques, efficient parallel implementations for solving realistic unstru... |

39 | BILUTM: A domain-based multilevel block ILUT preconditioner for general sparse matrices
- Saad, Zhang
- 1999
(Show Context)
Citation Context ...itional ILU type preconditioners promotes the current strong interest in searching for alternative preconditioning techniques. There are parallelizable preconditioners based on multi-level techniques =-=[2, 1, 39, 40]-=-. These methods exploit the concept of (block) independent set ordering and construct coarse level system via Schur complement approaches. They are highly efficient and may demonstrate grid-independen... |

33 | Wavelet sparse approximate inverse preconditioners
- Chan, Tang, et al.
- 1997
(Show Context)
Citation Context ...res have been reported [9, 33, 37, 38]. There are another group of parallelizable preconditioners. In the past few years, several versions of sparse approximate inverse techniques have been developed =-=[8, 11, 14, 15, 21, 22]-=-. These methods are based on approximating the inverse matrix directly, i.e., a sparse matrix M = A \Gamma1 is explicitly computed and stored as a preconditioner for a given sparse matrix A. The preco... |

33 | Distributed Schur Complement techniques for general sparse linear systems - Saad, Sosonkina - 1999 |

32 |
Sparse approximate-inverse preconditioners using norm-minimization techniques
- Gould, Scott
- 1998
(Show Context)
Citation Context |

24 | Sparse approximate inverse smoother for multigrid
- Tang, Wan
- 2000
(Show Context)
Citation Context ...ns, see Section 3.2. In addition to be used as global preconditioners, sparse approximate inverses have been used as local components in some other types of global preconditioners, e.g., in multigrid =-=[3, 42]-=- and multi-level preconditioners [44], and in block preconditioning methods in general [14, 16, 21]. These applications are aimed at utilizing the better local coupling property of the sparse approxim... |

21 |
High-order ILU preconditioners for CFD problems
- Chapman, Saad, et al.
(Show Context)
Citation Context ...umber of iterations RAEFSKY4 Matrix dashdot line: tau = 0.1 dashed line: tau = 0.05 solid line: tau = 0.01 dotted line: tau = 0.005 Figure 5: Solving the RAEFSKY4 matrix with different values of ���=-=� . [12, 40, 44]-=-. 5 Conclusions and Remarks We have proposed a sparse approximate inverse technique for computing parallelizable preconditioners of general sparse matrices. This method is extracted from a dense matri... |

21 | Preconditioned Krylov Subspace Methods for Solving Nonsymmetric Matrices from CFD Applications
- ZHANG
- 2000
(Show Context)
Citation Context ...by Golub and van der Vorst [20]. Several numerical experiments and analytical investigations have shown that many of these Krylov subspace methods behave similarly for a large number of test problems =-=[32, 43]-=-. On the other hand, it has been recognized that the performance of these Krylov subspace methods can be remarkably enhanced by coupling them with a suitable preconditioner. Thus, a preconditioned ite... |

20 |
A Portable MPI Implementation of the SPAI Preconditioner in ISIS
- Barnard, Clay
- 1997
(Show Context)
Citation Context ...truction of the sparse approximate inverses based on norm-minimization is in general very expensive [21], and the parallel implementation involving unstructured sparse matrices is not straightforward =-=[4]-=-. One of the recent developments in this direction is to find more efficient computational strategies to reduce the construction cost [13, 24]. Another important class of methods is to construct facto... |

18 |
der Vorst. Closer to the solution: iterative linear solvers
- Golub, van
- 1997
(Show Context)
Citation Context ...v subspace methods have become popular choices of iterative solution techniques due to their ability to handle general sparse matrices, see, e.g., [36], and a recent survey by Golub and van der Vorst =-=[20]-=-. Several numerical experiments and analytical investigations have shown that many of these Krylov subspace methods behave similarly for a large number of test problems [32, 43]. On the other hand, it... |

18 | Towards an effective sparse approximate inverse preconditioners
- Tang
- 1998
(Show Context)
Citation Context ...ditioners [44], and in block preconditioning methods in general [14, 16, 21]. These applications are aimed at utilizing the better local coupling property of the sparse approximate inverse techniques =-=[41]-=-. 3 New Factored Sparse Approximate Inverse In a series of papers [28, 29, 30, 27], Luo introduced a new method to decompose a matrix into its inverse such that the latter can be used for parallel sol... |

17 | Domain decomposition and multi-level type techniques for general sparse linear systems
- Saad, Sosonkina, et al.
- 1998
(Show Context)
Citation Context ...acted level by level, since operations within each level are block-wise or matrix-vector operations. Certain implementations on distributed and shared-memory parallel architectures have been reported =-=[9, 33, 37, 38]-=-. There are another group of parallelizable preconditioners. In the past few years, several versions of sparse approximate inverse techniques have been developed [8, 11, 14, 15, 21, 22]. These methods... |

13 |
VASSILEVSKI,A survey of multilevel preconditioned iterative methods
- AXELSSON, S
- 1989
(Show Context)
Citation Context |

11 |
ILUT: A dual threshold incomplete ILU preconditioner
- Saad
- 1994
(Show Context)
Citation Context ...ngular part (without the diagonal elements) is stored in compressed sparse row (CSR) format. This can be done conveniently using a pseudo modified compressed sparse row format as that is used in ILUT =-=[35]-=-. The intermediate results are also stored similarly, i.e., the resulting L matrix is stored in CSC format, D in a vector of length n, and U in CSR format. However, careful examination of Line 6 of Al... |

11 | Sparse approximate inverse and multilevel block ILU preconditioning techniques for general sparse matrices
- Zhang
- 1998
(Show Context)
Citation Context ...sed as global preconditioners, sparse approximate inverses have been used as local components in some other types of global preconditioners, e.g., in multigrid [3, 42] and multi-level preconditioners =-=[44]-=-, and in block preconditioning methods in general [14, 16, 21]. These applications are aimed at utilizing the better local coupling property of the sparse approximate inverse techniques [41]. 3 New Fa... |

10 | Parallel implementation of a sparse approximate inverse preconditioner, Workshop on Parallel Algorithms for Irregularly Structured Problems
- Deshpande, Grote, et al.
- 1996
(Show Context)
Citation Context ...xist with other sparse approximate inverse techniques. They have been overlooked for some time until very recently when the parallel implementation became an issue, see the discussions and remarks in =-=[4, 13, 18, 21, 23]-=-. Our impression is that, for both the norm-minimization based and incomplete biconjugation based sparse approximate inverse techniques, efficient parallel implementations for solving realistic unstru... |

9 |
W.: MRILU: An effective algebraic multi-level ILU-preconditioner for sparse matrices
- Botta, Wubs
- 1999
(Show Context)
Citation Context ...ck) independent set ordering and construct coarse level system via Schur complement approaches. They are highly efficient and may demonstrate grid-independent convergence for certain type of problems =-=[10, 39]-=-. In addition, parallelism can be extracted level by level, since operations within each level are block-wise or matrix-vector operations. Certain implementations on distributed and shared-memory para... |

8 |
An approximate inverse based multigrid approach to the biharmonic problem
- Banerjee, Benson
- 1991
(Show Context)
Citation Context ...ns, see Section 3.2. In addition to be used as global preconditioners, sparse approximate inverses have been used as local components in some other types of global preconditioners, e.g., in multigrid =-=[3, 42]-=- and multi-level preconditioners [44], and in block preconditioning methods in general [14, 16, 21]. These applications are aimed at utilizing the better local coupling property of the sparse approxim... |

7 | A fast linear-system solver for large unstructured problems on a shared-memory computer - Botta, Ploeg, et al. - 1996 |

7 |
Arithmetic and communication complexity of preconditioning methods
- Neytcheva
- 1995
(Show Context)
Citation Context ...acted level by level, since operations within each level are block-wise or matrix-vector operations. Certain implementations on distributed and shared-memory parallel architectures have been reported =-=[9, 33, 37, 38]-=-. There are another group of parallelizable preconditioners. In the past few years, several versions of sparse approximate inverse techniques have been developed [8, 11, 14, 15, 21, 22]. These methods... |

4 |
An incomplete inverse as a preconditioner for the conjugate gradient method
- Luo
- 1993
(Show Context)
Citation Context ...6, 21]. These applications are aimed at utilizing the better local coupling property of the sparse approximate inverse techniques [41]. 3 New Factored Sparse Approximate Inverse In a series of papers =-=[28, 29, 30, 27]-=-, Luo introduced a new method to decompose a matrix into its inverse such that the latter can be used for parallel solution of dense linear systems. Different algorithms have been designed for banded,... |

4 |
A new class of decomposition for inverting asymmetric and indefinite matrices
- Luo
- 1993
(Show Context)
Citation Context ...6, 21]. These applications are aimed at utilizing the better local coupling property of the sparse approximate inverse techniques [41]. 3 New Factored Sparse Approximate Inverse In a series of papers =-=[28, 29, 30, 27]-=-, Luo introduced a new method to decompose a matrix into its inverse such that the latter can be used for parallel solution of dense linear systems. Different algorithms have been designed for banded,... |

3 |
Extended concept of stair-shape sparsity for the inverse of an asymmetric matrix
- Luo
- 1993
(Show Context)
Citation Context ...6, 21]. These applications are aimed at utilizing the better local coupling property of the sparse approximate inverse techniques [41]. 3 New Factored Sparse Approximate Inverse In a series of papers =-=[28, 29, 30, 27]-=-, Luo introduced a new method to decompose a matrix into its inverse such that the latter can be used for parallel solution of dense linear systems. Different algorithms have been designed for banded,... |

3 |
A sparsity for decomposing a symmetric matrix
- Luo
- 1993
(Show Context)
Citation Context ...s. Such exploitation of sparsity pattern will result in reducing unnecessary computations. The so-called stair-shape sparsity pattern has been discussed by Luo for symmetric and nonsymmetric matrices =-=[30, 27]-=-. Proposition 3.5 Let r j be the half upper bandwidth of the jth row of A, i.e., A ji = 0 for i ? j+r j , then the vector w computed from Line 3 of Algorithm 3.2 has the same sparsity pattern as A j (... |

2 |
Efficient computation of sparse approximate inverse
- Huckel
- 1998
(Show Context)
Citation Context ... involving unstructured sparse matrices is not straightforward [4]. One of the recent developments in this direction is to find more efficient computational strategies to reduce the construction cost =-=[13, 24]-=-. Another important class of methods is to construct factored (or factorized) sparse approximate inverse where the sparse approximate inverse preconditioner is represented as a factored form M = LMUM ... |