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23
ILUT: A Dual Threshold Incomplete LU Factorization
, 1994
"... In this paper we describe an Incomplete LU factorization technique based on a strategy which combines two heuristics. This ILUT factorization extends the usual ILU(0) factorization without using the concept of level of fillin. There are two traditional ways of developing incomplete factorization ..."
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Cited by 83 (6 self)
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In this paper we describe an Incomplete LU factorization technique based on a strategy which combines two heuristics. This ILUT factorization extends the usual ILU(0) factorization without using the concept of level of fillin. There are two traditional ways of developing incomplete factorization preconditioners. The first uses a symbolic factorization approach in which a level of fill is attributed to each fillin element using only the graph of the matrix. Then each fillin that is introduced is dropped whenever its level of fill exceeds a certain threshold. The second class of methods consists of techniques derived from modifications of a given direct solver by including a dropoff rule, based on the numerical size of the fillins introduced. traditionally referred to as threshold preconditioners. The first type of approach may not be reliable for indefinite problems, since it does not consider numerical values. The second is often far more expensive than the standard IL...
Experimental Study of ILU Preconditioners for Indefinite Matrices
 J. COMPUT. APPL. MATH
, 1997
"... Incomplete LU factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices. However, their failure rate is still too high for them to be useful as blackbox library software for general matrices. Besides fatal breakdowns due to zer ..."
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Cited by 58 (8 self)
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Incomplete LU factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices. However, their failure rate is still too high for them to be useful as blackbox library software for general matrices. Besides fatal breakdowns due to zero pivots, the major causes of failure are inaccuracy, and instability of the triangular solves. When there are small pivots, both these problems can occur, but these problems can also occur without small pivots. Through examples from actual problems, this paper shows how these problems evince themselves, how these problems can be detected, and how these problems can sometimes be circumvented through pivoting, reordering, scaling, perturbing diagonal elements, and preserving symmetric structure. The goal of this paper is to gain a better practical understanding of ILU preconditioners and help improve their reliability.
ILUM: A MultiElimination ILU Preconditioner For General Sparse Matrices
 SIAM J. Sci. Comput
, 1999
"... Standard preconditioning techniques based on incomplete LU (ILU) factorizations offer a limited degree of parallelism, in general. A few of the alternatives advocated so far consist of either using some form of polynomial preconditioning, or applying the usual ILU factorization to a matrix obtain ..."
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Cited by 54 (11 self)
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Standard preconditioning techniques based on incomplete LU (ILU) factorizations offer a limited degree of parallelism, in general. A few of the alternatives advocated so far consist of either using some form of polynomial preconditioning, or applying the usual ILU factorization to a matrix obtained from a multicolor ordering. In this paper we present an incomplete factorization technique based on independent set orderings and multicoloring. We note that in order to improve robustness, it is necessary to allow the preconditioner to have an arbitrarily high accuracy, as is done with ILUs based on threshold techniques. The ILUM factorization described in this paper is in this category. It can be viewed as a multifrontal version a Gaussian elimination procedure with threshold dropping which has a high degree of potential parallelism. The emphasis is on methods that deal specifically with general unstructured sparse matrices such as those arising from finite element methods on un...
BILUM: Block versions of multielimination and multilevel ILU preconditioner for general sparse linear systems
 SIAM J. Sci. Comput
, 1999
"... Abstract. We introduce block versions of the multielimination incomplete LU (ILUM) factorization preconditioning technique for solving general sparse unstructured linear systems. These preconditioners have a multilevel structure and, for certain types of problems, may exhibit properties that are typ ..."
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Cited by 53 (29 self)
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Abstract. We introduce block versions of the multielimination incomplete LU (ILUM) factorization preconditioning technique for solving general sparse unstructured linear systems. These preconditioners have a multilevel structure and, for certain types of problems, may exhibit properties that are typically enjoyed by multigrid methods. Several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed. The advantages of block ILUM over point ILUM include increased robustness and efficiency. We compare several versions of the block ILUM, point ILUM, and the dualthresholdbased ILUT preconditioners. In particular, tests with some convectiondiffusion problems show that it may be possible to obtain convergence that is nearly independent of the Reynolds number as well as of the grid size.
Incomplete Cholesky Factorizations With Limited Memory
 SIAM J. SCI. COMPUT
, 1999
"... We propose an incomplete Cholesky factorization for the solution of largescale trust region subproblems and positive definite systems of linear equations. This factorization depends on a parameter p that specifies the amount of additional memory (in multiples of n, the dimension of the problem) tha ..."
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Cited by 27 (5 self)
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We propose an incomplete Cholesky factorization for the solution of largescale trust region subproblems and positive definite systems of linear equations. This factorization depends on a parameter p that specifies the amount of additional memory (in multiples of n, the dimension of the problem) that is available; there is no need to specify a drop tolerance. Our numerical results show that the number of conjugate gradient iterations and the computing time are reduced dramatically for small values of p. We also show that in contrast with drop tolerance strategies, the new approach is more stable in terms of number of iterations and memory requirements.
Sparse Approximate Inverse Smoother for Multigrid
 SIAM J. Matrix Anal. Appl
, 1999
"... Various forms of sparse approximate inverses (SAI) have been shown to be useful for preconditioning. Their potential usefulness in a parallel environment has motivated much interest in recent years. However, the capability of an approximate inverse in eliminating the local error has not yet been ful ..."
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Cited by 24 (2 self)
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Various forms of sparse approximate inverses (SAI) have been shown to be useful for preconditioning. Their potential usefulness in a parallel environment has motivated much interest in recent years. However, the capability of an approximate inverse in eliminating the local error has not yet been fully exploited in multigrid algorithms. A careful examination of the iteration matrices of these approximate inverses indicates their superiority in smoothing the high frequency error in addition to their inherent parallelism. We propose a new class of sparse approximate inverse smoothers in this paper and present their analytic smoothing factors for constant coecient PDEs. Several distinctive features that make this technique special are: By adjusting the quality of the approximate inverse, the smoothing factor can be improved accordingly. For hard problems, this is useful.
Parallel Thresholdbased ILU Factorization
 Proceedings of the IEEE/ACM SC97 Conference
, 1996
"... Factorization algorithms based on threshold incomplete LU factorization have been found to be quite effective in preconditioning iterative system solvers. However, their parallel formulations have not been well understood and they have been considered to be unsuitable for distributed memory parall ..."
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Cited by 17 (0 self)
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Factorization algorithms based on threshold incomplete LU factorization have been found to be quite effective in preconditioning iterative system solvers. However, their parallel formulations have not been well understood and they have been considered to be unsuitable for distributed memory parallel computers. In this paper we present a highly parallel formulation of such factorization algorithms. Our algorithm utilizes parallel multilevel kway partitioning and independent set computation algorithms to effectively parallelize both the factorization as well as the solution of the resulting triangular systems, used in the application of the preconditioner. Our experiments on Cray T3D show that significant speedup can be achieved in both operations; thus, allowing threshold incomplete factorizations to be successfully used as preconditioners in parallel iterative solvers for sparse linear systems. 1 Introduction The sparse linear systems arising in finite element applications ar...
Weighted Graph Based Ordering Techniques for Preconditioned Conjugate Gradient Methods
 BIT
, 1993
"... We describe the basis for a matrix ordering heuristic for improving incomplete factorization for preconditioned conjugate gradient techniques applied to anisotropic PDE's. Several new matrix ordering techniques, derived from wellknown algorithms in combinatorial graph theory, which attempt to imple ..."
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Cited by 13 (6 self)
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We describe the basis for a matrix ordering heuristic for improving incomplete factorization for preconditioned conjugate gradient techniques applied to anisotropic PDE's. Several new matrix ordering techniques, derived from wellknown algorithms in combinatorial graph theory, which attempt to implement this heuristic, are described. These ordering techniques are tested against a number of matrices arising from linear anisotropic PDE's, and compared with other matrix ordering techniques. A variation of RCM is shown to generally improve the quality of incomplete factorization preconditioners. Keywords: Preconditioned conjugate gradient, preconditioner, matrix ordering, weighted graph Running Title: Weighted Graph Ordering for PCG Methods. AMS Subject Classification: 65F10 This work was supported by by the Natural Sciences and Engineering Research Council of Canada, and by the Information Technology Research Center, which is funded by the Province of Ontario. y Present address Dep...
Ordering Methods For Approximate Factorization Preconditioning
, 1993
"... We investigate the ordering and fillin strategies for approximate factorization preconditioning, focusing on automatic procedures to take care that a user friendly solver should converge equally fast whatever the original numbering of the unknowns. Considering the discrete PDE context, we pay parti ..."
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Cited by 12 (4 self)
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We investigate the ordering and fillin strategies for approximate factorization preconditioning, focusing on automatic procedures to take care that a user friendly solver should converge equally fast whatever the original numbering of the unknowns. Considering the discrete PDE context, we pay particular attention to anisotropic problems for which factorizations without fillin may behave poorly, while the efficiency of the fillin strategies heavily depends on ordering. Both theoretical and numerical results are given, displaying the efficiency of a variant of the reverse CuthillMcKee ordering. keywords: iterative methods for linear systems, acceleration of convergence preconditioning. AMS classification : 65F10, 65B99, 65N20. 1 Introduction We consider here the preconditioned conjugate gradient solution of large sparse linear systems arising from the discretization of second order elliptic PDEs. Incomplete factorization preconditionings are quite popular in this context, and seve...
Iterative Solution of Linear Systems in the 20th Century
 JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
, 2000
"... This paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century. Although iterative methods for solving linear systems find their origin in the early nineteenth century (work by Gauss),the field has seen an explosion of activity ..."
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Cited by 9 (0 self)
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This paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century. Although iterative methods for solving linear systems find their origin in the early nineteenth century (work by Gauss),the field has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences. The past five decades have been particularly rich in new developments,ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientific and industrial computational models. As in any other scientific area,research in iterative methods has been a journey characterized by a chain of contributions building on each other. It is the aim of this paper not only to sketch the most significant of these contributions during the past century,but also to relate them to one another.