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29
Preconditioning techniques for large linear systems: A survey
 J. COMPUT. PHYS
, 2002
"... This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization i ..."
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Cited by 188 (5 self)
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This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper.
Efficient largescale power grid analysis based on preconditioned krylovsubspace iterative methods
 In DAC
, 2001
"... 1. ..."
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 42 (6 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.
A scalable parallel algorithm for incomplete factor preconditioning
 SIAM Journal on Scientific Computing
"... Abstract. We describe a parallel algorithm for computing incomplete factor (ILU) preconditioners. The algorithm attains a high degree of parallelism through graph partitioning and a twolevel ordering strategy. Both the subdomains and the nodes within each subdomain are ordered to preserve concurren ..."
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Cited by 37 (3 self)
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Abstract. We describe a parallel algorithm for computing incomplete factor (ILU) preconditioners. The algorithm attains a high degree of parallelism through graph partitioning and a twolevel ordering strategy. Both the subdomains and the nodes within each subdomain are ordered to preserve concurrency. We show through an algorithmic analysis and through computational results that this algorithm is scalable. Experimental results include timings on three parallel platforms for problems with up to 20 million unknowns running on up to 216 processors. The resulting preconditioned Krylov solvers have the desirable property that the number of iterations required for convergence is insensitive to the number of processors.
Crout versions of ILU for general sparse matrices
, 2002
"... This paper presents an e#cient implementation of incomplete LU #ILU# factorizations that are derived from the Crout version of Gaussian elimination #GE#. At step k of the elimination, the kth rowofU and the kth column of L are computed using previously computed rows of U and columns of L. The da ..."
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Cited by 26 (6 self)
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This paper presents an e#cient implementation of incomplete LU #ILU# factorizations that are derived from the Crout version of Gaussian elimination #GE#. At step k of the elimination, the kth rowofU and the kth column of L are computed using previously computed rows of U and columns of L. The data structure and implementation borrow from already known techniques used in developing both sparse direct solution codes and incomplete Cholesky factorizations. It is shown that this version of ILU has many practical advantages. In particular, its data structure allows e#cient implementation of more rigorous and e#ective dropping strategies. Numerical tests show that the method is far more e#cient than standard thresholdbased ILU factorizations computed rowwise or columnwise.
Weighted matchings for preconditioning symmetric indefinite linear systems
 SIAM J. Sci. Comput
, 2006
"... Abstract. Maximum weight matchings have become an important tool for solving highly indefinite unsymmetric linear systems, especially in direct solvers. In this study we investigate the benefit of reorderings and scalings based on symmetrized maximum weight matchings as a preprocessing step for inco ..."
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Cited by 24 (6 self)
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Abstract. Maximum weight matchings have become an important tool for solving highly indefinite unsymmetric linear systems, especially in direct solvers. In this study we investigate the benefit of reorderings and scalings based on symmetrized maximum weight matchings as a preprocessing step for incomplete LDL T factorizations. The reorderings are constructed such that the matched entries form 1 × 1or2 × 2 diagonal blocks in order to increase the diagonal dominance of the system. During the incomplete factorization only tridiagonal pivoting is used. We report results for this approach and comparisons with other solution methods for a diverse set of symmetric indefinite matrices, ranging from nonlinear elasticity to interior point optimization.
Preconditioning Newton's Method
 IN STUDIES IN NUMERICAL ANALYSIS (G.H. GOLUB, ED), THE MATHEMATICAL ASSOCIATION OF AMERICA
, 1998
"... The development of algorithms and software for the solution of largescale optimization problems ... ..."
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Cited by 17 (0 self)
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The development of algorithms and software for the solution of largescale optimization problems ...
Crout versions of the ILU factorization with pivoting for sparse symmetric matrices
 Electronic Transactions on Numerical Analysis
"... Abstract. The Crout variant of ILU preconditioner (ILUC) developed recently has been shown to be generally advantageous over ILU with Threshold (ILUT), a conventional rowbased ILU preconditioner. This paper explores pivoting strategies for sparse symmetric matrices to improve the robustness of ILUC ..."
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Cited by 11 (2 self)
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Abstract. The Crout variant of ILU preconditioner (ILUC) developed recently has been shown to be generally advantageous over ILU with Threshold (ILUT), a conventional rowbased ILU preconditioner. This paper explores pivoting strategies for sparse symmetric matrices to improve the robustness of ILUC. We integrate two symmetrypreserving pivoting strategies, the diagonal pivoting and the BunchKaufman pivoting, into ILUC without significant overheads. The performances of the pivoting methods are compared with ILUC and ILUTP ([20]) on a set of problems, including a few arising from saddlepoint (KKT) problems.
Large Scale Unconstrained Optimization
 The State of the Art in Numerical Analysis
, 1996
"... This paper reviews advances in Newton, quasiNewton and conjugate gradient methods for large scale optimization. It also describes several packages developed during the last ten years, and illustrates their performance on some practical problems. Much attention is given to the concept of partial ..."
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Cited by 8 (0 self)
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This paper reviews advances in Newton, quasiNewton and conjugate gradient methods for large scale optimization. It also describes several packages developed during the last ten years, and illustrates their performance on some practical problems. Much attention is given to the concept of partial separabilitywhich is gaining importance with the arrival of automatic differentiation tools and of optimization software that fully exploits its properties.
BALANCED INCOMPLETE FACTORIZATION
"... In this paper we present a new incomplete factorization of a square matrix into triangular factors in which we get standard LU/LDL T factors (direct factors) and their inverses (inverse factors) at the same time. Algorithmically, we derive this method from the approach based on the ShermanMorrison ..."
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Cited by 8 (0 self)
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In this paper we present a new incomplete factorization of a square matrix into triangular factors in which we get standard LU/LDL T factors (direct factors) and their inverses (inverse factors) at the same time. Algorithmically, we derive this method from the approach based on the ShermanMorrison formula [16]. In contrast to the RIF algorithm [9], the direct and inverse factors here directly influence each other throughout the computation. Consequently, the algorithm to compute the approximate factors may mutually balance dropping in the factors and control their conditioning in this way. Although we describe the theory behind the factorization for general nonsymmetric matrices, in implementation and experiments we restrict for clarity and conciseness only to the case when the system matrix is symmetric and positive definite. In this case, we call the new approximate LDL T factorization Balanced Incomplete Factorization (BIF). Our experimental results confirm that this factorization is very robust and may be useful in solving difficult illconditioned problems by preconditioned iterative methods. Moreover, the internal coupling of computation of direct and inverse factors results in much shorter setup times (times to compute approximate decomposition) than RIF, a method of a similar and very high level of robustness.