Results 11  20
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47
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 15 (3 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.
Diagonal Threshold Techniques in Robust MultiLevel ILU Preconditioners for General Sparse Linear Systems
 NUMER. LINEAR ALGEBRA APPL
, 1998
"... This paper introduces techniques based on diagonal threshold tolerance when developing multielimination and multilevel incomplete LU (ILUM) factorization preconditioners for solving general sparse linear systems. Existing heuristics solely based on the adjacency graph of the matrices have been ..."
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Cited by 14 (10 self)
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This paper introduces techniques based on diagonal threshold tolerance when developing multielimination and multilevel incomplete LU (ILUM) factorization preconditioners for solving general sparse linear systems. Existing heuristics solely based on the adjacency graph of the matrices have been used to find independent sets and are not robust for matrices arising from certain applications in which the matrices may have small or zero diagonals. New heuristic strategies based on the adjacency graph and the diagonal values of the matrices for finding independent sets are introduced. Analytical bounds for the factorization and preconditioned errors are obtained for the case of a twolevel analysis. These bounds provide useful information in designing robust ILUM preconditioners. Extensive numerical experiments are conducted in order to compare robustness and efficiency of various heuristic strategies.
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.
Numerical Experiments With Parallel Orderings For Ilu Preconditioners
, 1999
"... Incomplete factorization preconditioners such as ILU, ILUT and MILU are wellknown robust generalpurpose techniques for solving linear systems on serial computers. However, they are difficult to parallelize efficiently. Various techniques have been used to parallelize these preconditioners, such as ..."
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Cited by 14 (1 self)
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Incomplete factorization preconditioners such as ILU, ILUT and MILU are wellknown robust generalpurpose techniques for solving linear systems on serial computers. However, they are difficult to parallelize efficiently. Various techniques have been used to parallelize these preconditioners, such as multicolor orderings and subdomain preconditioning. These techniques may degrade the performance and robustness of ILU preconditionings. The purpose of this paper is to perform numerical experiments to compare these techniques in order to assess what are the most effective ways to use ILU preconditioning for practical problems on serial and parallel computers.
A multilevel dual reordering strategy for robust incomplete LU factorization of indefinite matrices
 SIAM J. Matrix Anal. Appl
, 2001
"... Abstract. A dual reordering strategy based on both threshold and graph reorderings is introduced to construct robust incomplete LU (ILU) factorization of indefinite matrices. The ILU matrix is constructed as a preconditioner for the original matrix to be used in a preconditioned iterative scheme. Th ..."
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Cited by 13 (3 self)
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Abstract. A dual reordering strategy based on both threshold and graph reorderings is introduced to construct robust incomplete LU (ILU) factorization of indefinite matrices. The ILU matrix is constructed as a preconditioner for the original matrix to be used in a preconditioned iterative scheme. The matrix is first divided into two parts according to a threshold parameter to control diagonal dominance. The first part with large diagonal dominance is reordered using a graphbased strategy, followed by an ILU factorization. A partial ILU factorization is applied to the second part to yield an approximate Schur complement matrix. The whole process is repeated on the Schur complement matrix and continues for a few times to yield a multilevel ILU factorization. Analyses are conducted to show how the Schur complement approach removes small diagonal elements of indefinite matrices and how the stability of the LU factor affects the quality of the preconditioner. Numerical results are used to compare the new preconditioning strategy with two popular ILU preconditioning techniques and a multilevel block ILU threshold preconditioner.
Sparse Approximate Inverse and MultiLevel Block ILU Preconditioning Techniques for General Sparse Matrices
 Appl. Numer. Math
, 1998
"... We investigate the use of sparse approximate inverse techniques in a multilevel block ILU preconditioner to design a robust and efficient parallelizable preconditioner for solving general sparse matrices. The resulting preconditioner retains robustness of the multilevel block ILU preconditioner (B ..."
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Cited by 11 (7 self)
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We investigate the use of sparse approximate inverse techniques in a multilevel block ILU preconditioner to design a robust and efficient parallelizable preconditioner for solving general sparse matrices. The resulting preconditioner retains robustness of the multilevel block ILU preconditioner (BILUM) and offers a new way to control the fillin elements when large size blocks (subdomains) are used to form block independent set. Moreover, the new preconditioner affords maximum parallelism for operations within each level as well as for the coarsest level solution. Thus it has two advantages over the standard BILUM preconditioner: the ability to control sparsity and increased parallelism. Numerical experiments are used to show the effectiveness and efficiency of the new preconditioner. Key words: Sparse matrices, incomplete LU factorization, multilevel ILU preconditioner, sparse approximate inverse, Krylov subspace methods. AMS subject classifications: 65F10, 65N06. 1 Introduction ...
An Assessment of IncompleteLU Preconditioners for Nonsymmetric Linear Systems
 Informatica
, 1999
"... This paper shows, 1 ..."
A MultiLevel Preconditioner with Applications to the Numerical Simulation of Coating Problems
 Iterative Methods in Scientific Computing II
, 1998
"... A multilevel preconditioned iterative method based on a multilevel block ILU factorization preconditioning technique is introduced and is applied to the solution of unstructured sparse linear systems arising from the numerical simulation of coating problems. The coefficient matrices usually have s ..."
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Cited by 6 (5 self)
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A multilevel preconditioned iterative method based on a multilevel block ILU factorization preconditioning technique is introduced and is applied to the solution of unstructured sparse linear systems arising from the numerical simulation of coating problems. The coefficient matrices usually have several rows with zero diagonal values that may cause stability difficulty in standard ILU factorization techniques. The new preconditioning strategy employs a diagonal threshold tolerance and a local reordering of individual blocks to increase robustness of the multilevel block ILU factorization process. Keywords: sparse matrices, multilevel preconditioning, ILU factorization 1 Introduction In this paper, a multilevel block incomplete LU (ILU) preconditioning technique is designed for solving unstructured sparse linear systems from the numerical simulation of coating problems. Coating is a delicate process of putting a layer of one liquid material (film) over another solid material unifo...
A Fully Coupled NewtonKrylov Solver For Turbulent Aerodynamics Flows
"... A fast NewtonKrylov algorithm is presented for solving the compressible NavierStokes equations on structured multiblock grids with application to turbulent aerodynamic flows. The oneequation SpalartAllmaras model is used to provide the turbulent viscosity. The optimization of the algorithm is di ..."
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Cited by 5 (4 self)
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A fast NewtonKrylov algorithm is presented for solving the compressible NavierStokes equations on structured multiblock grids with application to turbulent aerodynamic flows. The oneequation SpalartAllmaras model is used to provide the turbulent viscosity. The optimization of the algorithm is discussed. ILU(4) is suggested for a preconditioner, operating on a modified Jacobian matrix. An efficient startup method to bring the system into the region of convergence of Newton's method is given. Three test cases are used to demonstrate convergence rates. Singleelement cases are solved in less than 100 seconds on an engineering workstation, while the solution of a multielement case can be found in less than 25 minutes.
Combinatorial problems in solving linear systems
, 2009
"... Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects ..."
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Cited by 5 (3 self)
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Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects. As the core of many of today’s numerical linear algebra computations consists of the solution of sparse linear system by direct or iterative methods, we survey some combinatorial problems, ideas, and algorithms relating to these computations. On the direct methods side, we discuss issues such as matrix ordering; bipartite matching and matrix scaling for better pivoting; task assignment and scheduling for parallel multifrontal solvers. On the iterative method side, we discuss preconditioning techniques including incomplete factorization preconditioners, support graph preconditioners, and algebraic multigrid. In a separate part, we discuss the block triangular form of sparse matrices.