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241
GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems
 SIAM J. SCI. STAT. COMPUT
, 1986
"... We present an iterative method for solving linear systems, which has the property ofminimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from the Arnoldi process for constructing an l2orthogonal basis of Krylov subspaces. It can be considered a ..."
Abstract

Cited by 2076 (41 self)
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We present an iterative method for solving linear systems, which has the property ofminimizing at every step the norm of the residual vector over a Krylov subspace. The algorithm is derived from the Arnoldi process for constructing an l2orthogonal basis of Krylov subspaces. It can be considered as a generalization of Paige and Saunders’ MINRES algorithm and is theoretically equivalent to the Generalized Conjugate Residual (GCR) method and to ORTHODIR. The new algorithm presents several advantages over GCR and ORTHODIR.
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 130 (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
Approximate Inverse Preconditioners Via SparseSparse Iterations
, 1998
"... . The standard incomplete LU (ILU) preconditioners often fail for general sparse indefinite matrices because they give rise to `unstable' factors L and U . In such cases, it may be attractive to approximate the inverse of the matrix directly. This paper focuses on approximate inverse preconditi ..."
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Cited by 85 (17 self)
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. The standard incomplete LU (ILU) preconditioners often fail for general sparse indefinite matrices because they give rise to `unstable' factors L and U . In such cases, it may be attractive to approximate the inverse of the matrix directly. This paper focuses on approximate inverse preconditioners based on minimizing kI \Gamma AMkF , where AM is the preconditioned matrix. An iterative descenttype method is used to approximate each column of the inverse. For this approach to be efficient, the iteration must be done in sparse mode, i.e., with `sparsematrix by sparsevector' operations. Numerical dropping is applied to maintain sparsity; compared to previous methods, this is a natural way to determine the sparsity pattern of the approximate inverse. This paper describes Newton, `global' and columnoriented algorithms, and discusses options for initial guesses, selfpreconditioning, and dropping strategies. Some limited theoretical results on the properties and convergence of approxima...
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 85 (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.
Coupling cloud processes with the largescale dynamics using the cloudresolving convective parameterization (CRCP)
 J. ATMOS. SCI
, 2001
"... A formal approach is presented to couple smallscale processes associated with atmospheric moist convection with the largescale dynamics. The approach involves applying a twodimensional cloudresolving model in each column of a threedimensional largescale model. In the spirit of classical convec ..."
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Cited by 79 (0 self)
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A formal approach is presented to couple smallscale processes associated with atmospheric moist convection with the largescale dynamics. The approach involves applying a twodimensional cloudresolving model in each column of a threedimensional largescale model. In the spirit of classical convection parameterization, which assumes scale separation between convection and the largescale flow, the cloudresolving models from neighboring columns interact only through the largescale dynamics. This approach is referred to as CloudResolving Convection Parameterization (CRCP). In short, CRCP involves many twodimensional cloudresolving models interacting in a manner consistent with the largescale dynamics. The approach is first applied to the idealized problem of a convective–radiative equilibrium of a twodimensional nonrotating atmosphere in the presence of SST gradients. This simple dynamical setup allows comparison of CRCP simulations with the cloudresolving model results. In these tests, the largescale model has various horizontal grid spacings, from 20 to 500 km, and the CRCP domains change correspondingly. Comparison between CRCP and cloudresolving simulations shows that the largescale features, such as the mean temperature and moisture profiles and the largescale flow, are reasonably well represented in CRCP simulations. However, the interaction between ascending and descending branches through the gravity wave mechanism, as well as
An Analysis for the DIIS Acceleration Method used in Quantum Chemistry Calculations
, 2010
"... This work features an analysis for the acceleration technique DIIS that is standardly used in most of the important quantum chemistry codes, e.g. in DFT and HartreeFock calculations and in the Coupled Cluster method. Taking up results from [23], we show that for the general nonlinear case, DIIS c ..."
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Cited by 76 (5 self)
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This work features an analysis for the acceleration technique DIIS that is standardly used in most of the important quantum chemistry codes, e.g. in DFT and HartreeFock calculations and in the Coupled Cluster method. Taking up results from [23], we show that for the general nonlinear case, DIIS corresponds to a projected quasiNewton/ secant method. For linear systems, we establish connections to the wellknown GMRES solver and transfer according (positive as well as negative) convergence results to DIIS. In particular, we discuss the circumstances under which DIIS exhibits superlinear convergence behaviour. For the general nonlinear case, we then use these results to show that a DIIS step can be interpreted as step of a quasiNewton method in which the Jacobian used in the Newton step is approximated by finite differences and in which the according linear system is solved by a GMRES procedure, and give according convergence estimates.
Recycling Krylov Subspaces for Sequences of Linear Systems
 SIAM J. Sci. Comput
, 2004
"... Many problems in engineering and physics require the solution of a large sequence of linear systems. We can reduce the cost of solving subsequent systems in the sequence by recycling information from previous systems. We consider two dierent approaches. For several model problems, we demonstrate tha ..."
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Cited by 74 (6 self)
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Many problems in engineering and physics require the solution of a large sequence of linear systems. We can reduce the cost of solving subsequent systems in the sequence by recycling information from previous systems. We consider two dierent approaches. For several model problems, we demonstrate that we can reduce the iteration count required to solve a linear system by a factor of two. We consider both Hermitian and nonHermitian problems, and present numerical experiments to illustrate the eects of subspace recycling.
Design, Implementation and Testing of Extended and Mixed Precision BLAS
, 2001
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An Iterative Method for Nonsymmetric Systems with Multiple RightHand Sides
 SIAM J. Sci. Comput
, 1995
"... . We propose a method for the solution of linear systems AX = B where A is a large, possibly sparse, nonsymmetric matrix of order n, and B is an arbitrary rectangular matrix of order n \Theta s with s of moderate size. The method uses a single Krylov subspace per step as a generator of approximation ..."
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Cited by 67 (3 self)
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. We propose a method for the solution of linear systems AX = B where A is a large, possibly sparse, nonsymmetric matrix of order n, and B is an arbitrary rectangular matrix of order n \Theta s with s of moderate size. The method uses a single Krylov subspace per step as a generator of approximations, a projection process, and a Richardson acceleration technique. It thus combines the advantages of recent hybrid methods with those for solving symmetric systems with multiple righthand sides. Numerical experiments indicate that in several cases the method has better practical performance and significantly lower memory requirements than block versions of nonsymmetric solvers and other proposed methods for the solution of systems with multiple righthand sides. Key words. nonsymmetric systems, iterative method, Krylov subspace, multiple righthand sides, GMRES, hybrid method, Richardson AMS subject classifications. 65F10, 65Y20 1. Introduction. We consider techniques for the solution of ...