## ARMS: An Algebraic Recursive Multilevel Solver for general sparse linear systems (1999)

Venue: | Numer. Linear Alg. Appl |

Citations: | 46 - 24 self |

### BibTeX

@TECHREPORT{Saad99arms:an,

author = {Yousef Saad and Brian Suchomel},

title = {ARMS: An Algebraic Recursive Multilevel Solver for general sparse linear systems},

institution = {Numer. Linear Alg. Appl},

year = {1999}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper presents a general preconditioning method based on a multilevel partial solution approach. The basic step in constructing the preconditioner is to separate the initial points into two subsets. The first subset which can be termed "coarse" is obtained by using "block" independent sets, or "aggregates". Two aggregates have no coupling between them, but nodes in the same aggregate may be coupled. The nodes not in the coarse set are part of what might be called the "Fringe" set. The idea of the methods is to form the Schur complement related to the fringe set. This leads to a natural block LU factorization which can be used as a preconditioner for the system. This system is then solver recursively using as preconditioner the factorization that could be obtained from the next level. Unlike other multilevel preconditioners available, iterations between levels are allowed. One interesting aspect of the method is that it provides a common framework for many other technique...

### Citations

1594 |
Iterative Methods for Sparse Linear Systems
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- 2003
(Show Context)
Citation Context ... elimination process, then the (2; 2) block C l in the resulting matrix will yield the Schur complement. This is easily seen. The implementation we use follows very closely that of the ILUT algorithm =-=[18]-=-. The process is illustrated in Figure 2 and described in Algorithm 3.1. The notations a i;fi , l i;fi and u i;fi represent the i th rows of A, L and U , respectively. The process is based on the IKJ ... |

790 |
Multigrid methods and applications
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Citation Context ... the next level. The usual reasoning behind the success of this type of technique is that low frequency error is eliminated on the coarse grid, and high frequency error is eliminated on the fine grid =-=[14]-=-. Usually, a direct method or a small number of iterations of a relaxation method, such as Gauss-Seidel, is used for the coarse grid. A number of options exist for the lower level solver. In BILUTM a ... |

512 |
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Citation Context ...still be large and a direct method will tend to be expensive both in terms of computation and memory requirement. However, it is important to note that the method of Nested Dissections (ND) of George =-=[13] is a meth-=-od in this class. The ND reordering exploits independent aggregates obtained by recursively dividing the graph into two disconnected subgraphs using "separators". A number of techniques for ... |

503 |
Iterative Solution Methods
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(Show Context)
Citation Context .... This is not true with the regular ARMS factorization. It is common when analyzing Block-ILU type preconditioners to make assumptions on the approximation to the Schur complement under consideration =-=[1]-=-. Here we make a similar assumption on the smallness of R 22 relative to S l . Specifically, it is assumed that for some vector norm, we have kR 22 xksfl kS l xk ; 8x (10) with jflj ! 1. Then the foll... |

311 | University of Florida Sparse Matrix Collection
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- 1997
(Show Context)
Citation Context ...ARMS preconditioner. The ILUT(0,100) preconditioner takes less time than ILUT(0,200) to factor, but 3 The RAEFSKY matrices are available online from the University of Florida sparse matrix collection =-=[11]-=- at http://www.cise.ufl.edu/~davis/sparse. 23 still more than the total solution time using ARMS. In addition, ILUT(0,100) requires over 600 iterations to reach convergence. 0 200 400 600 800 1000 120... |

256 | Algebraic multigrid
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Citation Context ...sity of Minnesota, 4-192 EE/CS Building, 200 Union Street S.E., Minneapolis, MN 55455. E-mail: fsaad, suchomelg @cs.umn.edu. 1 methods were introduced in the seventies -- initially by Ruge and Stuben =-=[16]-=- -- to remedy these limitations. Their overall success depends on the underlying PDE problem, and has been somewhat mixed. In contrast, preconditioned Krylov methods, using ILU preconditioners, are de... |

196 | A supernodal approach to sparse partial pivoting
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(Show Context)
Citation Context ...e the general observation that the performance is often poorer for tol DD = 0 and then improves -- sometimes substantially. To give a rough idea of a comparison with a direct solver, the SuperLU code =-=[13]-=- with Approximate Minumum degree (APM) required 6.203 M of memory words to factor the BARTH1A matrix, resulting in a fill-factor of 12.89. For the BARTH2A matrix, 13.07M words are needed resulting in ... |

142 |
Modi of the minimum degree algorithm by multiple elimination
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(Show Context)
Citation Context ...ique, -- typically one that is based on a minimum degree-type ordering. The tests shown in the experiments reorder these blocks with the well-known Multiple-Minimum Degree algorithm of George and Liu =-=[17]-=-. We refer to this combination as ND-ARMS. ND-ARMS also incorporates diagonal dominance selection. The weights (17) are computed for all the rows belonging to the leaves of the ND tree. Those rows amo... |

118 |
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Citation Context ...where. In Section 4 additional strategies will be examined. No Coupling Figure 1: Aggregates, groups or blocks. 2.2 Block LU preconditioners In various existing forms of multilevel ILU factorizations =-=[3, 17, 7, 6]-=- the unknowns are reordered, listing the nodes associated with the Independent Set first, followed by the other unknowns. After this reordering, the original matrix A l at the l-th level takes the fol... |

55 | ILUM: A multi-elimination ILU preconditioner for general sparse matrices
- Saad
- 1996
(Show Context)
Citation Context ...of methods developed in the last decade have aspired to combine the good intrinsic properties of multigrid techniques and the generality of preconditioned Krylov subspace methods. Among these we cite =-=[2, 4, 7, 8, 17, 19, 20]-=-. Multigrid methods can be extremely efficient when they work. However, their implementation requires multi-level grids and specialized tuning is often needed. The Algebraic MultiGrid (AMG) This work ... |

53 | BILUM: Block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems
- Saad, Zhang
- 1999
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Citation Context ...of methods developed in the last decade have aspired to combine the good intrinsic properties of multigrid techniques and the generality of preconditioned Krylov subspace methods. Among these we cite =-=[2, 4, 7, 8, 17, 19, 20]-=-. Multigrid methods can be extremely efficient when they work. However, their implementation requires multi-level grids and specialized tuning is often needed. The Algebraic MultiGrid (AMG) This work ... |

44 | Approximate inverse techniques for blockpartitioned matrices
- Chow, Saad
- 1997
(Show Context)
Citation Context ...point is done by level-sets and as soon as the size is above (or equal) bsize, the traversal stops. 2 http://math.nist.gov/MatrixMarket. The FIDAP matrices in this experiment are the same as those in =-=[12]-=-, except that the two smallest matrices have been omitted. 13 bsize nlev f ill inter f ill last droptol I droptol I droptol last tol DD 500 5 60 50 50 0.0001 0.001 0.2 Table 1: Parameters for the test... |

39 | BILUTM: A domain-based multilevel block ILUT preconditioner for general sparse matrices
- Saad, Zhang
- 1999
(Show Context)
Citation Context ... In ILUM [17], B is a diagonal matrix and the above factorization is repeated for A l+1 after an independent set ordering is found. A dropping strategy is used to limit fill-in. In BILUM [19], BILUTM =-=[20]-=-, and in this paper, B is a block diagonal matrix. One motivation behind this type of factorization is that the diagonal elements or blocks may be used as pivots simultaneously, in parallel. The metho... |

34 | Making sparse Gaussian elimination scalable by static pivoting
- Li, Demmel
- 1998
(Show Context)
Citation Context ... independent aggregates obtained by recursively dividing the graph into two disconnected subgraphs using "separators". A number of techniques for stabilizing sparse Gauss Elimination are pre=-=sented in [12]-=-. In traditional Algebraic Multi-Grid (AMG, [16]), grid transfer operators are defined that restrict and interpolate. In a Galerkin formulation, an interpolation matrix is determined algebraically - a... |

21 | Multilevel ILU decomposition
- Bank, Wagner
- 1999
(Show Context)
Citation Context ...he coarse mesh. NGILU method does not work for unstructured grids. In a subsequent method, MRILU [7], two of the authors develop a preconditioner similar to ILUM where B is a diagonal matrix. In MLILU=-=[5]-=-, B is constructed by an algorithm that determines an optimal set of parents, or coarse grid nodes based on generating a small amount of fill upon factorization. Alternatively, in [22], B is made diag... |

21 |
High-order ILU preconditioners for CFD problems
- Chapman, Saad, et al.
(Show Context)
Citation Context ...1A matrix has 481,125 nonzeros and 4 The BARTH matrices are available from the authors. 25 the BARTHT2A matrix investigated in the next section has 1,311,725 nonzeros. These matrices are discussed in =-=[10]-=-. The idea in [10] is to use the ILUT factors of the first order discretiation matrix BARTHT1A to precondition the second order matrix BARTHT2A. This is not done in this paper. BARTHT1A. The first set... |

15 | Schur-complement multigrid – a robust method for groundwater ow and transport problems
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- 1997
(Show Context)
Citation Context ... matrix. In MLILU[5], B is constructed by an algorithm that determines an optimal set of parents, or coarse grid nodes based on generating a small amount of fill upon factorization. Alternatively, in =-=[22]-=-, B is made diagonal by modifying the right-hand-side. The AMLI, MLILU and NGILU preconditioners are recursive in nature. Not only is the factorization defined recursively, but the (l + 1) st level pr... |

14 |
On the algebraic multigrid method
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- 1996
(Show Context)
Citation Context ... defined that restrict and interpolate. In a Galerkin formulation, an interpolation matrix is determined algebraically - and then the restriction matrix is its transpose. In other formulations, as in =-=[9]-=- the grid transfer matrices are not transposes of each other. Interpolation weights are typically defined after some coloring scheme determines fine and coarse grid points. This procedure is repeated ... |

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Citation Context ...of methods developed in the last decade have aspired to combine the good intrinsic properties of multigrid techniques and the generality of preconditioned Krylov subspace methods. Among these we cite =-=[3, 5, 9, 7, 19, 22, 23, 26, 27, 28]-=-. Multigrid methods are difficult to surpass when they work. However, their implementation requires multilevel grids and specialized tuning is often needed. The Algebraic Multigrid (AMG) methods were ... |

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10 |
Iterative Algorithms for Large Sparse Linear Systems on Parallel Computers
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Citation Context ...independent sets is "aggregates" [24] 1 . Some simple methods for finding standard and block-independent sets have been considered in [19, 22] and elsewhere. A parallel implementation is des=-=cribed in [1]-=-. In Section 4 we will revisit this issue and other strategies will be examined. No Coupling Figure 1: Aggregates, groups or blocks. In various existing forms of multilevel ILU factorizations [4, 19, ... |

9 |
A Fast and High-Quality Multilevel Scheme for Partitioning Irregular Graphs
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(Show Context)
Citation Context ... the graph into two subgraphs became of prime importance for parallel processing. In particular, a nested dissection technique based on the Metis partitioner is available as part of the Metis package =-=[15]-=-. The leaves of the Nested Dissection tree represent the blocks of the independent set. In more recent fill-reduction strategies, these blocks are also relabeled using a different reordering technique... |

8 |
MRILU: it's the preconditioning that counts
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Citation Context ...of methods developed in the last decade have aspired to combine the good intrinsic properties of multigrid techniques and the generality of preconditioned Krylov subspace methods. Among these we cite =-=[2, 4, 7, 8, 17, 19, 20]-=-. Multigrid methods can be extremely efficient when they work. However, their implementation requires multi-level grids and specialized tuning is often needed. The Algebraic MultiGrid (AMG) This work ... |

7 |
A fast linear-system solver for large unstructured problems on a shared-memory computer
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- 1996
(Show Context)
Citation Context ...ditioners that drew much attention is a collection of ILU factorizations which possess certain features of multigrid techniques. ILUM [17] is one such approach and recent work by Botta and co-workers =-=[6, 7]-=-, and [19, 20], indicates that this type of approach can be fairly robust and scale well with problem size, unlike standard ILU preconditioners. This method combines the generality of Krylov methods a... |

7 |
Independent set orderings for parallel matrix factorizations by Gaussian elimination
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(Show Context)
Citation Context ... 8, 20, 19] exploit the property that a set of unknowns that are not coupled to each other can be eliminated simultaneously in Gaussian elimination. Such sets are termed `independent sets', see e.g., =-=[15]. In [19],-=- the ILUM factorization described in [17] was generalized by resorting to "block independent sets". A block independent set is a set of groups (blocks) of unknowns such that there is no coup... |

4 | A grid based multilevel incomplete LU factorization preconditioning technique for general sparse matrices
- Zhang
- 1999
(Show Context)
Citation Context ...of methods developed in the last decade have aspired to combine the good intrinsic properties of multigrid techniques and the generality of preconditioned Krylov subspace methods. Among these we cite =-=[3, 5, 9, 7, 19, 22, 23, 26, 27, 28]-=-. Multigrid methods are difficult to surpass when they work. However, their implementation requires multilevel grids and specialized tuning is often needed. The Algebraic Multigrid (AMG) methods were ... |

3 |
An algebraic multilevel iteration method for finite element matrices
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- 1997
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2 | RILUM: a general framework for robust multilevel recursive incomplete LU preconditioning techniques
- Zhang
- 1999
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Citation Context ...of methods developed in the last decade have aspired to combine the good intrinsic properties of multigrid techniques and the generality of preconditioned Krylov subspace methods. Among these we cite =-=[3, 5, 9, 7, 19, 22, 23, 26, 27, 28]-=-. Multigrid methods are difficult to surpass when they work. However, their implementation requires multilevel grids and specialized tuning is often needed. The Algebraic Multigrid (AMG) methods were ... |