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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 105 (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.
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.
Orderings for incomplete factorization preconditioning of nonsymmetric problems
 SIAM J. SCI. COMPUT
, 1999
"... Numerical experiments are presented whereby the effect of reorderings on the convergence of preconditioned Krylov subspace methods for the solution of nonsymmetric linear systems is shown. The preconditioners used in this study are different variants of incomplete factorizations. It is shown that c ..."
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Cited by 52 (11 self)
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Numerical experiments are presented whereby the effect of reorderings on the convergence of preconditioned Krylov subspace methods for the solution of nonsymmetric linear systems is shown. The preconditioners used in this study are different variants of incomplete factorizations. It is shown that certain reorderings for direct methods, such as reverse Cuthill–McKee, can be very beneficial. The benefit can be seen in the reduction of the number of iterations and also in measuring the deviation of the preconditioned operator from the identity.
Robust approximate inverse preconditioning for the conjugate gradient method
 SIAM J. SCI. COMPUT
, 2000
"... We present a variant of the AINV factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix. The new preconditioner is breakdownfree and, when used in conjunction with the conjugate gradient method, results in a reliable solver for highly illcondit ..."
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Cited by 48 (11 self)
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We present a variant of the AINV factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix. The new preconditioner is breakdownfree and, when used in conjunction with the conjugate gradient method, results in a reliable solver for highly illconditioned linear systems. We also investigate an alternative approach to a stable approximate inverse algorithm, based on the idea of diagonally compensated reduction of matrix entries. The results of numerical tests on challenging linear systems arising from finite element modeling of elasticity and diffusion problems are presented.
Preconditioning highly indefinite and nonsymmetric matrices
 SIAM J. SCI. COMPUT
, 2000
"... Standard preconditioners, like incomplete factorizations, perform well when the coefficient matrix is diagonally dominant, but often fail on general sparse matrices. We experiment with nonsymmetric permutationsand scalingsaimed at placing large entrieson the diagonal in the context of preconditionin ..."
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Cited by 43 (4 self)
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Standard preconditioners, like incomplete factorizations, perform well when the coefficient matrix is diagonally dominant, but often fail on general sparse matrices. We experiment with nonsymmetric permutationsand scalingsaimed at placing large entrieson the diagonal in the context of preconditioning for general sparse matrices. The permutations and scalings are those developed by Olschowka and Neumaier [Linear Algebra Appl., 240 (1996), pp. 131–151] and by Duff and
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 27 (2 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.
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.
Multipoint and Multiobjective Aerodynamic
 Shape Optimization,” AIAA Journal
"... A gradientbased Newton–Krylov algorithm is presented for the aerodynamic shape optimization of single and multielement airfoil configurations. The flow is governed by the compressible Navier–Stokes equations in conjunction with a oneequation transport turbulence model. The preconditioned general ..."
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Cited by 22 (15 self)
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A gradientbased Newton–Krylov algorithm is presented for the aerodynamic shape optimization of single and multielement airfoil configurations. The flow is governed by the compressible Navier–Stokes equations in conjunction with a oneequation transport turbulence model. The preconditioned generalized minimal residual method is applied to solve the discreteadjoint equation, which leads to a fast computation of accurate objective function gradients. Optimization constraints are enforced through a penalty formulation, and the resulting unconstrained problem is solved via a quasiNewton method. The new algorithm is evaluated for several design examples, including the lift enhancement of a takeoff configuration and a liftconstrained drag minimization at multiple transonic operating points. Furthermore, the new algorithm is used to compute a Pareto front based on competing objectives, and the results are validated using a genetic algorithm. Overall, the new algorithm provides an efficient approach for addressing the issues of complex aerodynamic design.
Preconditioned Krylov Subspace Methods for Solving Nonsymmetric Matrices from CFD Applications
 Comput. Methods Appl. Mech. Engrg
, 1999
"... 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 multil ..."
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Cited by 21 (12 self)
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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 (...