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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 199 (28 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.
An Implementation of the LookAhead Lanczos Algorithm for NonHermitian Matrices Part I
, 1991
"... ..."
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 118 (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.
Iterative Solution of Linear Systems
 Acta Numerica
, 1992
"... this paper is as follows. In Section 2, we present some background material on general Krylov subspace methods, of which CGtype algorithms are a special case. We recall the outstanding properties of CG and discuss the issue of optimal extensions of CG to nonHermitian matrices. We also review GMRES ..."
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Cited by 108 (8 self)
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this paper is as follows. In Section 2, we present some background material on general Krylov subspace methods, of which CGtype algorithms are a special case. We recall the outstanding properties of CG and discuss the issue of optimal extensions of CG to nonHermitian matrices. We also review GMRES and related methods, as well as CGlike algorithms for the special case of Hermitian indefinite linear systems. Finally, we briefly discuss the basic idea of preconditioning. In Section 3, we turn to Lanczosbased iterative methods for general nonHermitian linear systems. First, we consider the nonsymmetric Lanczos process, with particular emphasis on the possible breakdowns and potential instabilities in the classical algorithm. Then we describe recent advances in understanding these problems and overcoming them by using lookahead techniques. Moreover, we describe the quasiminimal residual algorithm (QMR) proposed by Freund and Nachtigal (1990), which uses the lookahead Lanczos process to obtain quasioptimal approximate solutions. Next, a survey of transposefree Lanczosbased methods is given. We conclude this section with comments on other related work and some historical remarks. In Section 4, we elaborate on CGNR and CGNE and we point out situations where these approaches are optimal. The general class of Krylov subspace methods also contains parameterdependent algorithms that, unlike CGtype schemes, require explicit information on the spectrum of the coefficient matrix. In Section 5, we discuss recent insights in obtaining appropriate spectral information for parameterdependent Krylov subspace methods. After that, 4 R.W. Freund, G.H. Golub and N.M. Nachtigal
Homomorphic factorization of brdfs for highperformance rendering
, 2001
"... Figure 1: A model rendered at realtime rates (approximately half the performance of the standard pervertex lighting model on an NVIDIA GeForce 3) with several BRDFs approximated using the technique in this paper. From left to right: satin (anisotropic PoulinFournier model), krylon blue, garnet re ..."
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Cited by 90 (7 self)
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Figure 1: A model rendered at realtime rates (approximately half the performance of the standard pervertex lighting model on an NVIDIA GeForce 3) with several BRDFs approximated using the technique in this paper. From left to right: satin (anisotropic PoulinFournier model), krylon blue, garnet red, cayman, mystique (Cornell measured data), leather, and velvet (CURET measured data). A bidirectional reflectance distribution function (BRDF) describes how a material reflects light from its surface. To use arbitrary BRDFs in realtime rendering, a compression technique must be used to represent BRDFs using the available texturemapping and computational capabilities of an accelerated graphics pipeline. We present a numerical technique, homomorphic factorization, that can decompose arbitrary BRDFs into products of two or more factors of lower dimensionality, each factor dependent on a different interpolated geometric parameter. Compared to an earlier factorization technique based on the singular value decomposition, this new technique generates a factorization with only positive factors (which makes it more suitable for current graphics hardware accelerators), provides control over the smoothness of the result, minimizes relative rather than absolute error, and can deal with scattered, sparse data without a separate resampling and interpolation algorithm.
A restarted GMRES method augmented with eigenvectors
 SIAM J. Matrix Anal. Appl
, 1995
"... Abstract. The GMRES method for solving nonsymmetric linear equations is generally used with restarting to reduce storage and orthogonalization costs. Restarting slows down the convergence. However, it is possible to save some important information at the time of the restart. It is proposed that appr ..."
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Cited by 86 (11 self)
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Abstract. The GMRES method for solving nonsymmetric linear equations is generally used with restarting to reduce storage and orthogonalization costs. Restarting slows down the convergence. However, it is possible to save some important information at the time of the restart. It is proposed that approximate eigenvectors corresponding to a few of the smallest eigenvalues be formed and added to the subspace for GMRES. The convergence can be much faster, and the minimum residual property is retained. Key words. GMRES, conjugate gradient, Krylov subspaces, iterative methods, nonsymmetric systems AMS subject classifications. 65F15, 15A18
A TwoDimensional Data Distribution Method For Parallel Sparse MatrixVector Multiplication
 SIAM REVIEW
"... A new method is presented for distributing data in sparse matrixvector multiplication. The method is twodimensional, tries to minimise the true communication volume, and also tries to spread the computation and communication work evenly over the processors. The method starts with a recursive bipar ..."
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Cited by 72 (8 self)
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A new method is presented for distributing data in sparse matrixvector multiplication. The method is twodimensional, tries to minimise the true communication volume, and also tries to spread the computation and communication work evenly over the processors. The method starts with a recursive bipartitioning of the sparse matrix, each time splitting a rectangular matrix into two parts with a nearly equal number of nonzeros. The communication volume caused by the split is minimised. After the matrix partitioning, the input and output vectors are partitioned with the objective of minimising the maximum communication volume per processor. Experimental results of our implementation, Mondriaan, for a set of sparse test matrices show a reduction in communication compared to onedimensional methods, and in general a good balance in the communication work.
An Implementation Of The Qmr Method Based On Coupled TwoTerm Recurrences
, 1992
"... . Recently, the authors have proposed a new Krylov subspace iteration, the quasiminimal residual algorithm (QMR), for solving nonHermitian linear systems. In the original implementation of the QMR method, the Lanczos process with lookahead is used to generate basis vectors for the underlying Kryl ..."
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Cited by 70 (14 self)
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. Recently, the authors have proposed a new Krylov subspace iteration, the quasiminimal residual algorithm (QMR), for solving nonHermitian linear systems. In the original implementation of the QMR method, the Lanczos process with lookahead is used to generate basis vectors for the underlying Krylov subspaces. In the Lanczos algorithm, these basis vectors are computed by means of threeterm recurrences. It has been observed that, in finite precision arithmetic, vector iterations based on threeterm recursions are usually less robust than mathematically equivalent coupled twoterm vector recurrences. This paper presents a lookahead algorithm that constructs the Lanczos basis vectors by means of coupled twoterm recursions. Implementation details are given, and the lookahead strategy is described. A new implementation of the QMR method, based on this coupled twoterm algorithm, is proposed. A simplified version of the QMR algorithm without lookahead is also presented, and the specia...
Objectoriented software for quadratic programming
 ACM Transactions on Mathematical Software
, 2001
"... The objectoriented software package OOQP for solving convex quadratic programming problems (QP) is described. The primaldual interior point algorithms supplied by OOQP are implemented in a way that is largely independent of the problem structure. Users may exploit problem structure by supplying li ..."
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Cited by 68 (2 self)
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The objectoriented software package OOQP for solving convex quadratic programming problems (QP) is described. The primaldual interior point algorithms supplied by OOQP are implemented in a way that is largely independent of the problem structure. Users may exploit problem structure by supplying linear algebra, problem data, and variable classes that are customized to their particular applications. The OOQP distribution contains default implementations that solve several important QP problem types, including general sparse and dense QPs, boundconstrained QPs, and QPs arising from support vector machines and Huber regression. The implementations supplied with the OOQP distribution are based on such well known linear algebra packages as MA27/57, LAPACK, and PETSc. OOQP demonstrates the usefulness of objectoriented design in optimization software development, and establishes standards that can be followed in the design of software packages for other classes of optimization problems. A number of the classes in OOQP may also be reusable directly in other codes.
StructurePreserving Methods for Computing Eigenpairs of Large Sparse SkewHamiltonian/Hamiltonian Pencils
 SIAM J. Sci. Comput
, 2000
"... We study large, sparse generalized eigenvalue problems for matrix pencils, where one of the matrices is Hamiltonian and the other skew Hamiltonian. Problems of this form arise in the numerical simulation of elastic deformation of anisotropic materials, in structural mechanics and in the linearquadr ..."
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Cited by 68 (18 self)
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We study large, sparse generalized eigenvalue problems for matrix pencils, where one of the matrices is Hamiltonian and the other skew Hamiltonian. Problems of this form arise in the numerical simulation of elastic deformation of anisotropic materials, in structural mechanics and in the linearquadratic control problem for partial differential equations. We develop a structurepreserving skewHamiltonian, isotropic, implicitlyrestarted shiftandinvert Arnoldi algorithm (SHIRA). Several numerical examples demonstrate the superiority of SHIRA over a competing unstructured method.