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86
Recent computational developments in Krylov subspace methods for linear systems
 NUMER. LINEAR ALGEBRA APPL
, 2007
"... Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are metho ..."
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Cited by 50 (12 self)
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Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.
Encapsulating Multiple CommunicationCost Metrics in Partitioning Sparse Rectangular Matrices for Parallel MatrixVector Multiplies
"... This paper addresses the problem of onedimensional partitioning of structurally unsymmetricsquare and rectangular sparse matrices for parallel matrixvector and matrixtransposevector multiplies. The objective is to minimize the communication cost while maintaining the balance on computational load ..."
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Cited by 35 (22 self)
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This paper addresses the problem of onedimensional partitioning of structurally unsymmetricsquare and rectangular sparse matrices for parallel matrixvector and matrixtransposevector multiplies. The objective is to minimize the communication cost while maintaining the balance on computational loads of processors. Most of the existing partitioning models consider only the total message volume hoping that minimizing this communicationcost metric is likely to reduce other metrics. However, the total message latency (startup time) may be more important than the total message volume. Furthermore, the maximum message volume and latency handled by a single processor are also important metrics. We propose a twophase approach that encapsulates all these four communicationcost metrics. The objective in the first phase is to minimize the total message volume while maintainingthe computationalload balance. The objective in the second phase is to encapsulate the remaining three communicationcost metrics. We propose communicationhypergraph and partitioning models for the second phase. We then present several methods for partitioning communication hypergraphs. Experiments on a wide range of test matrices show that the proposed approach yields very effective partitioning results. A parallel implementation on a PC cluster verifies that the theoretical improvements shown by partitioning results hold in practice.
Randomized Matrix Computations
, 2012
"... We propose new effective randomized algorithms for some fundamental matrix computations such as preconditioning of an ill conditioned matrix that has a small numerical nullity or rank, its 2by2 block triangulation, numerical stabilization of Gaussian elimination with no pivoting, and approximation ..."
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Cited by 34 (2 self)
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We propose new effective randomized algorithms for some fundamental matrix computations such as preconditioning of an ill conditioned matrix that has a small numerical nullity or rank, its 2by2 block triangulation, numerical stabilization of Gaussian elimination with no pivoting, and approximation of a matrix by lowrank matrices and by structured matrices. Our technical advances include estimating the condition number of a random Toeplitz matrix, novel techniques of randomized preprocessing, a proof of their preconditioning power, and a dual version of the Sherman–Morrison–Woodbury formula. According to both our formal study and numerical tests we significantly accelerate the known algorithms and improve their output accuracy.
Block Preconditioners Based on Approximate Commutators
 SIAM J. SCI. COMPUT
, 2006
"... This paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible NavierStokes equations. We consider the "pressure convectiondiffusion preconditioners" proposed by Kay, Loghin, and Wathen [11] and Silvester, Elman, Kay, and Wathen [16]. Numerous t ..."
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Cited by 24 (9 self)
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This paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible NavierStokes equations. We consider the "pressure convectiondiffusion preconditioners" proposed by Kay, Loghin, and Wathen [11] and Silvester, Elman, Kay, and Wathen [16]. Numerous theoretical and numerical studies have demonstrated mesh independent convergence on several problems and the overall e#cacy of this methodology. A drawback, however, is that it requires the construction of a convectiondiffusion operator (denoted Fp ) projected onto the discrete pressure space. This means that integration of this idea into a code that models incompressible flow requires a sophisticated understanding of the discretization and other implementation issues, something often held only by the developers of the model. As an alternative, we consider automatic ways of computing Fp based on purely algebraic considerations. The new methods are closely related to the "BFBt preconditioner" of Elman [6]. We use the fact that the preconditioner is derived from considerations of commutativity between the gradient and convectiondiffusion operators, together with methods for computing sparse approximate inverses, to generate the required matrix Fp automatically. We demonstrate that with this strategy, the favorable convergence properties of the preconditioning methodology are retained.
A comparison of preconditioners for incompressible Navier–Stokes solvers
 International Journal for Numerical Methods in Fluids 2008; 57:1731–1751. DOI: 10.1002/fld.1684
"... We consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier–Stokes equations. These systems are of the socalled saddle point type, which means that there is a large block of zeros on the main diagonal. To solve th ..."
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Cited by 18 (10 self)
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We consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier–Stokes equations. These systems are of the socalled saddle point type, which means that there is a large block of zeros on the main diagonal. To solve these types of systems efficiently, several block preconditioners have been published. These types of preconditioners require adaptation of standard finite element packages. The alternative is to apply a standard ILU preconditioner in combination with a suitable renumbering of unknowns. We introduce a reordering technique for the degrees of freedom that makes the application of ILU relatively fast. We compare the performance of this technique with some block preconditioners. The performance appears to depend on grid size, Reynolds number and quality of the mesh. For mediumsized problems, which are of practical interest, we show that the reordering technique is competitive with the block preconditioners. Its simple implementation makes it worthwhile to implement it in the standard finite element method software. Copyright q 2007
Partitioning sparse matrices for parallel preconditioned iterative methods
 SIAM Journal on Scientific Computing
, 2004
"... Abstract. This paper addresses the parallelization of the preconditioned iterative methods that use explicit preconditioners such as approximate inverses. Parallelizing a full step of these methods requires the coefficient and preconditioner matrices to be well partitioned. We first show that differ ..."
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Cited by 15 (9 self)
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Abstract. This paper addresses the parallelization of the preconditioned iterative methods that use explicit preconditioners such as approximate inverses. Parallelizing a full step of these methods requires the coefficient and preconditioner matrices to be well partitioned. We first show that different methods impose different partitioning requirements for the matrices. Then we develop hypergraph models to meet those requirements. In particular, we develop models that enable us to obtain partitionings on the coefficient and preconditioner matrices simultaneously. Experiments on a set of unsymmetric sparse matrices show that the proposed models yield effective partitioning results. A parallel implementation of the right preconditioned BiCGStab method on a PC cluster verifies that the theoretical gains obtained by the models hold in practice.
On large scale diagonalization techniques for the Anderson model of localization
 SIAM REVIEW
, 2005
"... We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for the largest sparse real and symmetric indefinite matrices of the Anderson model of localization. We compar ..."
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Cited by 8 (6 self)
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We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for the largest sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shiftandinvert techniques in the implicitly restarted Lanczos method and in the JacobiDavidson method. Our preconditioning approaches for the shiftandinvert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete LDL T factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly illconditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerative the computation of a largescale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude.
DECAY PROPERTIES OF SPECTRAL PROJECTORS WITH APPLICATIONS TO ELECTRONIC STRUCTURE
, 2010
"... Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the ..."
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Cited by 6 (1 self)
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Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential offdiagonal decay (‘nearsightedness’) for the density matrix of gapped systems at zero electronic temperature in both orthogonal and nonorthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for nonmetallic systems. Our theory also allows us to treat the case of density matrices for arbitrary systems at finite electronic temperature, including metals. Other possible applications are also discussed.
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.