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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 102 (4 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.
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.
Sparse Numerical Linear Algebra: Direct Methods and Preconditioning
, 1996
"... Most of the current techniques for the direct solution of linear equations are based on supernodal or multifrontal approaches. An important feature of these methods is that arithmetic is performed on dense submatrices and Level 2 and Level 3 BLAS (matrixvector and matrixmatrix kernels) can be us ..."
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Cited by 17 (2 self)
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Most of the current techniques for the direct solution of linear equations are based on supernodal or multifrontal approaches. An important feature of these methods is that arithmetic is performed on dense submatrices and Level 2 and Level 3 BLAS (matrixvector and matrixmatrix kernels) can be used. Both sparse LU and QR factorizations can be implemented within this framework. Partitioning and ordering techniques have seen major activity in recent years. We discuss bisection and multisection techniques, extensions to orderings to block triangular form, and recent improvements and modifications to standard orderings such as minimum degree. We also study advances in the solution of indefinite systems and sparse leastsquares problems. The desire to exploit parallelism has been responsible for many of the developments in direct methods for sparse matrices over the last ten years. We examine this aspect in some detail, illustrating how current techniques have been developed or ...
A Robust Incomplete Factorization Preconditioner for Positive Definite Matrices
, 2001
"... this paper we introduce a preconditioner that strikes a compromise between these two extremes ..."
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Cited by 15 (3 self)
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this paper we introduce a preconditioner that strikes a compromise between these two extremes
Vaidya's Preconditioners: Implementation And Experimental Study
, 2001
"... We describe the implementation and performance of a novel class of preconditioners. These preconditioners were proposed and theoretically analyzed by Pravin Vaidya in 1991, but no report on their implementation or performance in practice has ever been published. We show experimentally that these pre ..."
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Cited by 15 (6 self)
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We describe the implementation and performance of a novel class of preconditioners. These preconditioners were proposed and theoretically analyzed by Pravin Vaidya in 1991, but no report on their implementation or performance in practice has ever been published. We show experimentally that these preconditioners have some remarkable properties. We show that within the class of diagonallydominant symmetric matrices, the cost and convergence of these preconditioners depends almost only on the nonzero structure of the matrix, but not on its numerical values. In particular, this property leads to robust convergence behavior on di#cult 3dimensional problems that cause stagnation in incompleteCholesky preconditioners (more specifically, in droptolerance incomplete Cholesky without diagonal modification, with diagonal modification, and with relaxed diagonal modification). On such problems, we have observed cases in which a Vaidyapreconditioned solver is more than 6 times faster than an incompleteCholeskypreconditioned solver, when we allow similar amounts of fill in the factors of both preconditioners. We also show that Vaidya's preconditioners perform and scale similarly or better than droptolerance relaxedmodified incomplete Cholesky preconditioners on a wide range of 2dimensional problems. In particular, on anisotropic 2D problems, Vaidya delivers robust convergence independently of the direction of anisotropy and the ordering of the unknowns. However, on many 3D problems in which incompleteCholeskypreconditioned solvers converge without stagnating, Vaidyapreconditioned solvers are much slower. We also show how the insights gained from this study can be used to design faster and more robust solvers for some di#cult problems. 1.
Preconditioning KKT Systems
, 2002
"... This research presents new preconditioners for linear systems. We proceed from the most general case to the very specific problem area of sparse optimal control. In the first most general approach, we assume only that the coefficient matrix is nonsingular. We target highly indefinite, nonsymmetric p ..."
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Cited by 14 (0 self)
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This research presents new preconditioners for linear systems. We proceed from the most general case to the very specific problem area of sparse optimal control. In the first most general approach, we assume only that the coefficient matrix is nonsingular. We target highly indefinite, nonsymmetric problems that cause difficulties for preconditioned iterative solvers, and where standard preconditioners, like incomplete factorizations, often fail. We experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices. Our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing. Secondly, we present two new preconditioners for KKT systems. KKT systems arise in areas such as quadratic programming, sparse optimal control, and mixed finite element formulations. Our preconditioners approximate a constraint preconditioner with incomplete factorizations for the normal equations. Numerical experiments compare these two preconditioners with exact constraint preconditioning and the approach described above of permuting large entries to the diagonal. Finally, we turn to a specific problem area: sparse optimal control. Many optimal control problems are broken into several phases, and within a phase, most variables and constraints depend only on nearby variables and constraints. However, free initial and final times and timeindependent parameters impact variables and constraints throughout a phase, resulting in dense factored blocks in the KKT matrix. We drop fill due to these variables to reduce density within each phase. The resulting preconditioner is tightly banded and nearly block tridiagonal. Numerical experiments demonstrate that the preconditioners are effective, with very little fill in the factorization.
An Assessment of Some Preconditioning Techniques in Shell Problems
 COMM. NUMER. METHODS ENGRG
, 1998
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Numerical Methods for Quantum Monte Carlo Simulations of the Hubbard Model
, 2009
"... One of the core problems in materials science is how the interactions between electrons in a solid give rise to properties like ..."
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Cited by 7 (3 self)
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One of the core problems in materials science is how the interactions between electrons in a solid give rise to properties like
Iterative Methods for IllConditioned Linear Systems From Optimization
, 1998
"... Preconditioned conjugategradient methods are proposed for solving the illconditioned linear systems which arise in penalty and barrier methods for nonlinear minimization. The preconditioners are chosen so as to isolate the dominant cause of ill conditioning. The methods are stablized using a restr ..."
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Cited by 5 (1 self)
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Preconditioned conjugategradient methods are proposed for solving the illconditioned linear systems which arise in penalty and barrier methods for nonlinear minimization. The preconditioners are chosen so as to isolate the dominant cause of ill conditioning. The methods are stablized using a restricted form of iterative refinement. Numerical results illustrate the approaches considered. 1 Email : n.gould@rl.ac.uk 2 Current reports available from "http://www.rl.ac.uk/departments/ccd/numerical/reports/reports.html". Department for Computation and Information Atlas Centre Rutherford Appleton Laboratory Oxfordshire OX11 0QX August 26, 1998. 1 INTRODUCTION 1 1 Introduction Let A and H be, respectively, fullrank m by n (m n) and symmetric n by n real matrices. Suppose furthermore that any nonzero coefficients in this data are modest, that is the data is O(1). (1) We consider the iterative solution of the linear system (H +A T D \Gamma1 A)x = b (1.1) where b is modest an...