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314
QMR: a QuasiMinimal Residual Method for NonHermitian Linear Systems
, 1991
"... ... In this paper, we present a novel BCGlike approach, the quasiminimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a lookahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from t ..."
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Cited by 395 (26 self)
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... In this paper, we present a novel BCGlike approach, the quasiminimal residual (QMR) method, which overcomes the problems of BCG. An implementation of QMR based on a lookahead version of the nonsymmetric Lanczos algorithm is proposed. It is shown how BCG iterates can be recovered stably from the QMR process. Some further properties of the QMR approach are given and an error bound is presented. Finally, numerical experiments are reported.
A sparse approximate inverse preconditioner for nonsymmetric linear systems
 SIAM J. SCI. COMPUT
, 1998
"... This paper is concerned with a new approach to preconditioning for large, sparse linear systems. A procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed, and the resulting factorized sparse approximate inverse is used as an explicit preconditioner f ..."
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Cited by 197 (22 self)
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This paper is concerned with a new approach to preconditioning for large, sparse linear systems. A procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed, and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient–type methods. Some theoretical properties of the preconditioner are discussed, and numerical experiments on test matrices from the Harwell–Boeing collection and from Tim Davis’s collection are presented. Our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners. Furthermore, the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners.
OSKI: A library of automatically tuned sparse matrix kernels
 Institute of Physics Publishing
, 2005
"... kernels ..."
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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 145 (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.
Matrix Market: A Web Resource for Test Matrix Collections
 The Quality of Numerical Software: Assessment and Enhancement
, 1997
"... We describe a repository of data for the testing of numerical algorithms and mathematical software for matrix computations. The repository is designed to accommodate both dense and sparse matrices, as well as software to generate matrices. It has been seeded with the wellknown HarwellBoeing sparse ..."
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Cited by 85 (6 self)
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We describe a repository of data for the testing of numerical algorithms and mathematical software for matrix computations. The repository is designed to accommodate both dense and sparse matrices, as well as software to generate matrices. It has been seeded with the wellknown HarwellBoeing sparse matrix collection. The raw data files have been augmented with an integrated World Wide Web interface which describes the matrices in the collection quantitatively and visually. For example, each matrix has a Web page which details its attributes, graphically depicts its sparsity pattern, and provides access to the matrix itself in several formats. In addition, a search mechanism is included which allows retrieval of matrices based on a variety of attributes, such as type and size, as well as through freetext search in abstracts. The URL is http://math.nist.gov/MatrixMarket/ .
Exponential integrators
, 2010
"... In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential eq ..."
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Cited by 68 (5 self)
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In this paper we consider the construction, analysis, implementation and application of exponential integrators. The focus will be on two types of stiff problems. The first one is characterized by a Jacobian that possesses eigenvalues with large negative real parts. Parabolic partial differential equations and their spatial discretization are typical examples. The second class consists of highly oscillatory problems with purely imaginary eigenvalues of large modulus. Apart from motivating the construction of exponential integrators for various classes of problems, our main intention in this article is to present the mathematics behind these methods. We will derive error bounds that are independent of stiffness or highest frequencies in the system. Since the implementation of exponential integrators requires the evaluation of the product of a matrix function with a vector, we will briefly discuss some possible approaches as well. The paper concludes with some applications, in
Modeldriven autotuning of sparse matrixvector multiply on GPUs
 In PPoPP
, 2010
"... We present a performance modeldriven framework for automated performance tuning (autotuning) of sparse matrixvector multiply (SpMV) on systems accelerated by graphics processing units (GPU). Our study consists of two parts. First, we describe several carefully handtuned SpMV implementations for G ..."
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Cited by 65 (4 self)
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We present a performance modeldriven framework for automated performance tuning (autotuning) of sparse matrixvector multiply (SpMV) on systems accelerated by graphics processing units (GPU). Our study consists of two parts. First, we describe several carefully handtuned SpMV implementations for GPUs, identifying key GPUspecific performance limitations, enhancements, and tuning opportunities. These implementations, which include variants on classical blocked compressed sparse row (BCSR) and blocked ELLPACK (BELLPACK) storage formats, match or exceed stateoftheart implementations. For instance, our best BELLPACK implementation achieves up to 29.0 Gflop/s in singleprecision and 15.7 Gflop/s in doubleprecision on the NVIDIA T10P multiprocessor (C1060), enhancing prior stateoftheart unblocked implementations (Bell and Garland, 2009) by up to 1.8 × and 1.5 × for single and doubleprecision respectively. However, achieving this level of performance requires input matrixdependent parameter tuning. Thus, in the second part of this study, we develop a performance model that can guide tuning. Like prior autotuning models for CPUs (e.g., Im, Yelick, and Vuduc, 2004), this model requires offline measurements and runtime estimation, but more directly models the structure of multithreaded vector processors like GPUs. We show that our model can identify the implementations that achieve within 15 % of those found through exhaustive search.
Optimizing the performance of sparse matrixvector multiplication
, 2000
"... Copyright 2000 by EunJin Im ..."
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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 60 (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.