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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 354 (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 Flexible InnerOuter Preconditioned GMRES Algorithm
, 1993
"... We present a variant of the GMRES lgorithm which l]ows changes in the prcconditioning at every step. There arc many possible applications o the new lgorithm some o which arc briefly discussed. In particular, a result o the flexibility o the new variant is that any iterative method can bc used as a p ..."
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Cited by 300 (31 self)
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We present a variant of the GMRES lgorithm which l]ows changes in the prcconditioning at every step. There arc many possible applications o the new lgorithm some o which arc briefly discussed. In particular, a result o the flexibility o the new variant is that any iterative method can bc used as a prcconditioncr. For example, the standard GMRES lgorithm itself can bc used as a prcconditioncr, as can CGNR (or CGNE) the conjugate gradient method applied to the normal equations. However, the more appealing utilization o the method is in conjunction with relaxation techniques, possibly multilevel techniques. The possibility o changing prcconditioncrs may bc exploited to develop efficient iterative methods and to enhance robustness. A cw numcricM experiments arc reported to illustrate this act.
FastHenry: A MultipoleAccelerated 3D Inductance Extraction Program
, 1994
"... tion based on mesh analysis can be combined with a GMRESstyle iterative matrix solution technique to make a reasonably fast 3D frequency dependent inductance and resistance extraction algorithm. Unfortunately, both the computation time and memory re quired for that approach grow faster than n 2, ..."
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Cited by 192 (39 self)
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tion based on mesh analysis can be combined with a GMRESstyle iterative matrix solution technique to make a reasonably fast 3D frequency dependent inductance and resistance extraction algorithm. Unfortunately, both the computation time and memory re quired for that approach grow faster than n 2, where n is the number of volumefilaments. In this paper, we show that it is possible to use multipoleacceleration to reduce both required memory and computation time to nearly order n. Results from examples are given to demonstrate that the multipole acceleration can reduce required computation time and memory by more than an order of magnitude for realistic packaging problems.
ARPACK Users Guide: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods.
, 1997
"... this document is intended to provide a cursory overview of the Implicitly Restarted Arnoldi/Lanczos Method that this software is based upon. The goal is to provide some understanding of the underlying algorithm, expected behavior, additional references, and capabilities as well as limitations of the ..."
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Cited by 160 (17 self)
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this document is intended to provide a cursory overview of the Implicitly Restarted Arnoldi/Lanczos Method that this software is based upon. The goal is to provide some understanding of the underlying algorithm, expected behavior, additional references, and capabilities as well as limitations of the software. 1.7 Dependence on LAPACK and BLAS
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
Aerodynamic Design Optimization on Unstructured Grids with a Continuous Adjoint Formulation
, 1997
"... A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analyzed. The derivation of the costate equations is presented, and a secondorder accurate discretization method is described. The relationship between the continuous formulation and a discret ..."
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Cited by 108 (4 self)
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A continuous adjoint approach for obtaining sensitivity derivatives on unstructured grids is developed and analyzed. The derivation of the costate equations is presented, and a secondorder accurate discretization method is described. The relationship between the continuous formulation and a discrete formulation is explored for inviscid, as well as for viscous flow. Several limitations in a strict adherence to the continuous approach are uncovered, and an approach that circumvents these difficulties is presented. The issue of grid sensitivities, which do not arise naturally in the continuous formulation, is investigated and is observed to be of importance when dealing with geometric singularities. A method is described for modifying inviscid and viscous meshes during the design cycle to accommodate changes in the surface shape. The accuracy of the sensitivity derivatives is established by comparing with finitedifference gradients and several design examples are presented.
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 105 (19 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.
Implementation of the GMRES method using Householder transformations
 SIAM J. Sci. Statist. Comput
, 1988
"... Abstract. The standard implementation of the GMRES method for solving large nonsymmetric linear systems involves a GramSchmidt process which is a potential source of significant numerical error. An alternative implementation is outlined here in which orthogonalization by Householder transformations ..."
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Cited by 95 (3 self)
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Abstract. The standard implementation of the GMRES method for solving large nonsymmetric linear systems involves a GramSchmidt process which is a potential source of significant numerical error. An alternative implementation is outlined here in which orthogonalization by Householder transformations replaces the GramSchmidt process. This implementation requires slightly less storage but somewhat more arithmetic than the standard one; however, numerical experiments suggest that it is more stable, especially as the limits of residual reduction are reached. The extra arithmetic required may be less significant when products of the coefficient matrix with vectors are expensive or on vector and, in particular, parallel machines. Key words. GMRES method, iterative methods, matrixfree methods, nonsymmetric linear systems, Householder transformations. AMS(MOS) subject classifications. 65F10, 65N20 1. Introduction. Of
Removing the stiffness from interfacial flows with surface tension
 J. Comput. Phys
, 1994
"... A new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in twodimensional, irrotational, and incompressible fluids. Through the LaplaceYoung condition at the interface, surface tension introduces highorder terms, both nonlinear and nonloca ..."
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Cited by 94 (9 self)
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A new formulation and new methods are presented for computing the motion of fluid interfaces with surface tension in twodimensional, irrotational, and incompressible fluids. Through the LaplaceYoung condition at the interface, surface tension introduces highorder terms, both nonlinear and nonlocal, into the dynamics. This leads to severe stability constraints for explicit time integration methods and makes the application of implicit methods difficult. This new formulation has all the nice properties for time integration methods that are associated with having a linear, constant coefficient, highest order term. That is, using this formulation, we give implicit time integration methods that have no high order time step stability constraint associated with surface tension and are explicit in Fourier space. The approach is based on a boundary integral formulation and applies more generally, even to problems beyond the fluid mechanical context. Here they are applied to computing with high resolution the motion of interfaces in HeleShaw flows and the motion of free surfaces in inviscid flows governed by the Euler equations. One HeleShaw computation shows the behavior of an expanding gas bubble over longtime as the interface undergoes successive tipsplittings and finger competition. A second computation shows the formation of a very ramified interface through the interaction of surface tension with an unstable density stratification. In Euler flows, the computation of a vortex sheet shows its rollup through the KelvinHelmholtz instability. This motion culminates in the late time selfintersection of the interface, creating trapped bubbles of fluid. This is, we believe, a type of singularity formation previously unobserved for such flows in 2D. Finally, computations of falling plumes in an unstably stratified Boussinesq fluid show a very similar behavior. © 1994 Academic Press, Inc. 1.