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15
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 334 (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.
An Implementation of the LookAhead Lanczos Algorithm for NonHermitian Matrices Part I
, 1991
"... ..."
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 100 (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
QMRPACK: a Package of QMR Algorithms
, 1996
"... this paper, we discuss some of the features of the algorithms in the package, with emphasis on the issues related to using the codes. We describe in some detail two routines from the package, one for the solution of linear systems, and the other for the computation of eigenvalue approximations. We p ..."
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Cited by 34 (4 self)
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this paper, we discuss some of the features of the algorithms in the package, with emphasis on the issues related to using the codes. We describe in some detail two routines from the package, one for the solution of linear systems, and the other for the computation of eigenvalue approximations. We present some numerical examples from applications where QMRPACK was used. Categories and Subject Descriptors: F.2.1 [Analysis of Algorithms and Problem Complexity]: Numerical Algorithms and Problemscomputations on matrices; G.1.3 [Numerical Analysis]: Numerical Linear Algebra
Lanczos Methods For The Solution Of Nonsymmetric Systems Of Linear Equations
, 1992
"... . The Lanczos or biconjugate gradient method is often an effective means for solving nonsymmetric systems of linear equations. However, the method sometimes experiences breakdown, a near division by zero which may hinder or preclude convergence. In this paper we present some theoretical results on t ..."
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Cited by 17 (3 self)
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. The Lanczos or biconjugate gradient method is often an effective means for solving nonsymmetric systems of linear equations. However, the method sometimes experiences breakdown, a near division by zero which may hinder or preclude convergence. In this paper we present some theoretical results on the nature and likelihood of the phenomenon of breakdown. We also define several new algorithms which substantially mitigate the problem of breakdown. Numerical comparisons of the new algorithms and the standard algorithms are given. Key words. linear systems, iterative methods, nonsymmetric, Lanczos AMS(MOS) subject classifications. 65F10, 65F15 1. Introduction. In this paper we consider methods for solving the linear system of equations Au = b; (1) where A 2 I C N \ThetaN is a given nonsingular matrix. When A is large and sparse, iterative methods in many cases are effective means for solving (1). In particular, when A is Hermitian and positive definite (HPD), the conjugate gradient (C...
Towards Automatic Multigrid Algorithms For Spd, Nonsymmetric And Indefinite Problems
 SIAM J. Sci. Comput
, 1996
"... . A new multigrid algorithm is constructed for the solution of linear systems of equations which arise from the discretization of elliptic PDEs. It is defined in terms of the difference scheme on the fine grid only, and no rediscretization of the PDE is required. Numerical experiments show that this ..."
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Cited by 14 (7 self)
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. A new multigrid algorithm is constructed for the solution of linear systems of equations which arise from the discretization of elliptic PDEs. It is defined in terms of the difference scheme on the fine grid only, and no rediscretization of the PDE is required. Numerical experiments show that this algorithm gives high convergence rates for several classes of problems: symmetric, nonsymmetric and problems with discontinuous coefficients, nonuniform grids and nonrectangular domains. When supplemented with an acceleration method, good convergence is achieved also for pure convection problems and indefinite Helmholtz equations. Key words. convectiondiffusion equation, discontinuous coefficients, elliptic PDEs, indefinite Helmholtz equation, multigrid method. AMS(MOS) subject classification. 65F10, 65N22, 65N55. 1. Introduction. The multigrid method is a powerful tool for the solution of linear systems which arise from the discretization of elliptic PDEs [4] [5]. In a multigrid itera...
Multigrid Techniques For Highly Indefinite Equations
 8th Copper Mountain Conference on Multigrid Methods
, 1995
"... Definition of a Multi Level (ML) Method We begin with an abstract definition of a multi level (ML) method for the solution of the linear system of equations Ax = b: The notation of this definition will be useful in the sequel. In the following, ~ S is a smoothing procedure and ffl, r, t and o ar ..."
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Cited by 9 (7 self)
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Definition of a Multi Level (ML) Method We begin with an abstract definition of a multi level (ML) method for the solution of the linear system of equations Ax = b: The notation of this definition will be useful in the sequel. In the following, ~ S is a smoothing procedure and ffl, r, t and o are nonnegative integers denoting, respectively, the cycle index, the number of presmoothings, the number of postsmoothings and the minimal order of A for which ML is called recursively. The operators R (restriction), P (prolongation) and Q (coarse grid coefficient matrix) will be defined later. ML(x in ; A; b; x out ) : if A is of order ! o x out / A \Gamma1 b otherwise: x in / ~ Sx in (repeat r times). e / 0 (1) ML(e; Q; R(Ax in \Gamma b); e out ) e / e out ) repeat ffl times x out / x in \Gamma P e x out / ~ Sx out (repeat t times). An iterative application of ML is given by x 0 = 0; k = 0 while kAx k \Gamma bk 2 threshold \Delta kAx 0 \Gamma bk 2 ML(x k ; A; b; x k+1 ) (2...
Multigrid And Cyclic Reduction Applied To The Helmholtz Equation
 Sixth Copper Mountain Conference on Multigrid Methods
, 1993
"... We consider the Helmholtz equation with a discontinuous complex parameter and inhomogeneous Dirichlet boundary conditions in a rectangular domain. A variant of the direct method of cyclic reduction is employed to facilitate the design of improved multigrid components, resulting in the method of CRM ..."
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Cited by 6 (0 self)
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We consider the Helmholtz equation with a discontinuous complex parameter and inhomogeneous Dirichlet boundary conditions in a rectangular domain. A variant of the direct method of cyclic reduction is employed to facilitate the design of improved multigrid components, resulting in the method of CRMG. We demonstrate the improved convergence properties of this method. 1 INTRODUCTION Microwave heating of foods has revolutionised the food processing industry. Effective and efficient microwave heating depends very much on a detailed knowledge and understanding of the dielectric properties of the food to be processed. This need has given rise to extensive research into the dielectric properties of materials; see, for example, Tinga and Nelson [1]. Microwave heating can be compared to heating by alternating current. The electric field of alternating current changes direction approximately 100 times each second, whereas the microwave field changes direction approximately 5 billion times each ...
Recent Advances in LanczosBased Iterative Methods for Nonsymmetric Linear Systems
 In Algorithmic Trends in Computational Fluid Dynamics, Hussaini MY, Kumar A, Salas MD (eds
, 1992
"... . In recent years, there has been a true revival of the nonsymmetric Lanczos method. On the one hand, the possible breakdowns in the classical algorithm are now better understood, and socalled lookahead variants of the Lanczos process have been developed, which remedy this problem. On the other han ..."
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Cited by 6 (0 self)
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. In recent years, there has been a true revival of the nonsymmetric Lanczos method. On the one hand, the possible breakdowns in the classical algorithm are now better understood, and socalled lookahead variants of the Lanczos process have been developed, which remedy this problem. On the other hand, various new Lanczosbased iterative schemes for solving nonsymmetric linear systems have been proposed. This paper gives a survey of some of these recent developments. 1 Introduction Many numerical computations involve the solution of large nonsingular systems of linear equations Ax = b: (1:1) For example, such systems arise from finite difference or finite element approximations to partial differential equations (PDEs), as intermediate steps in computing the solution of nonlinear problems, or as subproblems in largescale linear and nonlinear programming. Typically, the coefficient The work of these authors was supported by Cooperative Agreement NCC 2387 between NASA and the Univer...
Multigrid Techniques for 3D Definite and Indefinite Problems with Discontinuous Coefficients
, 1994
"... A multigrid method for the solution of finite difference approximations of elliptic PDEs is introduced. A parallelizable version of it, suitable for two and multi level analysis, is also defined, and serves as a theoretical tool for deriving an optimal implementation for the main version. For indefi ..."
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Cited by 4 (4 self)
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A multigrid method for the solution of finite difference approximations of elliptic PDEs is introduced. A parallelizable version of it, suitable for two and multi level analysis, is also defined, and serves as a theoretical tool for deriving an optimal implementation for the main version. For indefinite Helmholtz equations, this analysis provides a prediction of the optimal mesh size for the coarsest grid used. Numerical experiments show the applicability of the method to 3d diffusion problems with discontinuous coefficients and highly indefinite Helmholtz equations. 1 Introduction The multigrid method is a powerful tool for the solution of linear systems which arise from the discretization of elliptic PDEs [3] [4]. In a multigrid iteration, the equation is first relaxed on fine grid in order to smooth the error; then residual equations are transferred to a coarser grids, to be solved consequently and to supply correction terms. In order to implement this procedure, the PDE has to be...