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54
Numerical solution of saddle point problems
 ACTA NUMERICA
, 2005
"... Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has b ..."
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Cited by 320 (25 self)
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Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for solving this type of systems. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.
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 189 (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.
SuperLU DIST: A scalable distributedmemory sparse direct solver for unsymmetric linear systems
 ACM Trans. Mathematical Software
, 2003
"... We present the main algorithmic features in the software package SuperLU DIST, a distributedmemory sparse direct solver for large sets of linear equations. We give in detail our parallelization strategies, with a focus on scalability issues, and demonstrate the software’s parallel performance and sc ..."
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Cited by 144 (18 self)
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We present the main algorithmic features in the software package SuperLU DIST, a distributedmemory sparse direct solver for large sets of linear equations. We give in detail our parallelization strategies, with a focus on scalability issues, and demonstrate the software’s parallel performance and scalability on current machines. The solver is based on sparse Gaussian elimination, with an innovative static pivoting strategy proposed earlier by the authors. The main advantage of static pivoting over classical partial pivoting is that it permits a priori determination of data structures and communication patterns, which lets us exploit techniques used in parallel sparse Cholesky algorithms to better parallelize both LU decomposition and triangular solution on largescale distributed machines.
Sweeping preconditioner for the Helmholtz equation: moving perfectly matched layers
, 2010
"... The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variablecoefficient Helmholtz equation including veryhighfrequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Hel ..."
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Cited by 45 (6 self)
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The paper introduces the sweeping preconditioner, which is highly efficient for iterative solutions of the variablecoefficient Helmholtz equation including veryhighfrequency problems. The first central idea of this novel approach is to construct an approximate factorization of the discretized Helmholtz equation by sweeping the domain layer by layer, starting from an absorbing layer or boundary condition. Given this specific order of factorization, the second central idea is to represent the intermediate matrices in the hierarchical matrix framework. In two dimensions, both the construction and the application of the preconditioners are of linear complexity. The generalized minimal residual method (GMRES) solver with the resulting preconditioner converges in an amazingly small number of iterations, which is essentially independent of the number of unknowns. This approach is also extended to the threedimensional case with some success. Numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach. © 2011 Wiley Periodicals, Inc. 1
Multilevel preconditioners constructed from inversebased ILUs
, 2004
"... This paper analyzes dropping strategies in a multilevel incomplete LU decomposition context and presents a few of strategies for obtaining related ILUs with enhanced robustness. The analysis shows that the Incomplete LU factorization resulting from dropping small entries in Gaussian elimination prod ..."
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Cited by 32 (9 self)
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This paper analyzes dropping strategies in a multilevel incomplete LU decomposition context and presents a few of strategies for obtaining related ILUs with enhanced robustness. The analysis shows that the Incomplete LU factorization resulting from dropping small entries in Gaussian elimination produces a good preconditioner when the inverses of these factors have norms that are not too large. As a consequence a few strategies are developed whose goal is to achieve this feature. A number of “templates” for enabling implementations of these factorizations are presented. Numerical experiments show that the resulting ILUs offer a good compromise between robustness and efficiency.
On ApproximateInverse Preconditioners
, 1995
"... We investigate the use of sparse approximateinverse preconditioners for the iterative solution of unsymmetric linear systems of equations. Such methods are of particular interest because of the considerable scope for parallelization. We propose a number of enhancements which may improve their perfo ..."
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Cited by 27 (0 self)
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We investigate the use of sparse approximateinverse preconditioners for the iterative solution of unsymmetric linear systems of equations. Such methods are of particular interest because of the considerable scope for parallelization. We propose a number of enhancements which may improve their performance. When run in a sequential environment, these methods can perform unfavourably when compared with other techniques. However, they can be successful when other methods fail and simulations indicate that they can be competitive when considered in a parallel environment. 1 Current reports available by anonymous ftp from joyousgard.cc.rl.ac.uk (internet 130.246.9.91) in the directory "pub/reports". Computing and Information Systems Department, Atlas Centre, Rutherford Appleton Laboratory, Oxfordshire OX11 0QX, England. June 23, 1995. 1 INTRODUCTION 1 1 Introduction Suppose that A is a real n by n unsymmetric matrix, whose columns are a j , 1 j n. We are principally concerned wit...
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 Helmholtz linear systems
, 2009
"... Linear systems which originate from the simulation of wave propagation phenomena can be very difficult to solve by iterative methods. These systems are typically complex valued and they tend to be highly indefinite, which renders the standard ILUbased preconditioners ineffective. This paper present ..."
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Cited by 22 (1 self)
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Linear systems which originate from the simulation of wave propagation phenomena can be very difficult to solve by iterative methods. These systems are typically complex valued and they tend to be highly indefinite, which renders the standard ILUbased preconditioners ineffective. This paper presents a study of ways to enhance standard preconditioners by altering the diagonal by imaginary shifts. Prior work indicates that modifying the diagonal entries during the incomplete factorization process, by adding to it purely imaginary values can improve the quality of the preconditioner in a substantial way. Here we propose simple algebraic heuristics to perform the shifting and test these techniques with the ARMS and ILUT preconditioners. Comparisons are made with applications stemming from the diffraction of an acoustic wave incident on a bounded obstacle (governed by the Helmholtz Wave Equation).
Multilevel ILU with reorderings for diagonal dominance
 SIAM J. Sci. Comput
, 2005
"... This paper presents a preconditioning method based on combining twosided permutations with a multilevel approach. The nonsymmetric permutation exploits a greedy strategy to put large entries of the matrix in the diagonal of the upper leading submatrix. The method can be regarded as a complete pivot ..."
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Cited by 17 (7 self)
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This paper presents a preconditioning method based on combining twosided permutations with a multilevel approach. The nonsymmetric permutation exploits a greedy strategy to put large entries of the matrix in the diagonal of the upper leading submatrix. The method can be regarded as a complete pivoting version of the incomplete LU factorization. This leads to an effective incomplete factorization preconditioner for general nonsymmetric, irregularly structured, sparse linear systems.
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 15 (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.