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223
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 337 (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.
Parametrization and smooth approximation of surface triangulations
 Computer Aided Geometric Design
, 1997
"... Abstract. A method based on graph theory is investigated for creating global parametrizations for surface triangulations for the purpose of smooth surface fitting. The parametrizations, which are planar triangulations, are the solutions of linear systems based on convex combinations. A particular pa ..."
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Cited by 254 (15 self)
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Abstract. A method based on graph theory is investigated for creating global parametrizations for surface triangulations for the purpose of smooth surface fitting. The parametrizations, which are planar triangulations, are the solutions of linear systems based on convex combinations. A particular parametrization, called shapepreserving, is found to lead to visually smooth surface approximations. A standard approach to fitting a smooth parametric curve c(t) through a given sequence of points xi = (xi,yi,zi) ∈ IR 3, i = 1,...,N is to first make a parametrization, a corresponding increasing sequence of parameter values ti. By finding smooth functions x,y,z: [t1,tN] → IR for which x(ti) = xi, y(ti) = yi, z(ti) = zi, an interpolatory curve
Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods
, 1994
"... This document is the electronic version of the 2nd edition of the Templates book, which is available for purchase from the Society for Industrial and Applied ..."
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Cited by 170 (5 self)
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This document is the electronic version of the 2nd edition of the Templates book, which is available for purchase from the Society for Industrial and Applied
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 101 (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
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 80 (10 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
GMRESR: A family of nested GMRES methods
 Num. Lin. Alg. with Appl
, 1991
"... Recently Eirola and Nevanlinna have proposed an iterative solution method for unsymmetric linear systems, in which the preconditioner is updated from step to step. Following their ideas we suggest variants of GMRES, in which a preconditioner is constructed at each iteration step by a suitable approx ..."
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Cited by 58 (16 self)
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Recently Eirola and Nevanlinna have proposed an iterative solution method for unsymmetric linear systems, in which the preconditioner is updated from step to step. Following their ideas we suggest variants of GMRES, in which a preconditioner is constructed at each iteration step by a suitable approximation process, e.g., by GMRES itself. Keywords: GMRES, nonsymmetric linear systems, iterative solver, ENmethod This version is dated June 23, 1992 Introduction The GMRES method, proposed in [13], is a popular method for the iterative solution of sparse linear systems with an unsymmetric nonsingular matrix. In its original form, socalled full GMRES, it is optimal in the sense that it minimizes the residual over the current Krylov subspace. However, it is often too expensive since the required orthogonalization per iteration step grows quadratically with the number of steps. For that reason, one often uses in practice variants of GMRES. The most wellknown variant, already suggested i...
Robust Numerical Methods for PDE Models of Asian Options
 Journal of Computational Finance
, 1998
"... We explore the pricing of Asian options by numerically solving the the associated partial differential equations. We demonstrate that numerical PDE techniques commonly used in finance for standard options are inaccurate in the case of Asian options and illustrate modifications which alleviate this p ..."
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Cited by 46 (14 self)
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We explore the pricing of Asian options by numerically solving the the associated partial differential equations. We demonstrate that numerical PDE techniques commonly used in finance for standard options are inaccurate in the case of Asian options and illustrate modifications which alleviate this problem. In particular, the usual methods generally produce solutions containing spurious oscillations. We adapt flux limiting techniques originally developed in the field of computational fluid dynamics in order to rapidly obtain accurate solutions. We show that flux limiting methods are total variation diminishing (and hence free of spurious oscillations) for nonconservative PDEs such as those typically encountered in finance, for fully explicit, and fully and partially implicit schemes. We also modify the van Leer flux limiter so that the secondorder total variation diminishing property is preserved for nonuniform grid spacing. 1 Introduction Asian options are securities with payoffs...
Preconditioning indefinite systems in interior point methods for optimization
 Computational Optimization and Applications
, 2004
"... Abstract. Every Newton step in an interiorpoint method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today’s codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable il ..."
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Cited by 44 (13 self)
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Abstract. Every Newton step in an interiorpoint method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today’s codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable illconditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used. Keywords: interiorpoint methods, iterative solvers, preconditioners 1.
Globalized NewtonKrylovSchwarz algorithms and software for parallel implicit CFD
 Int. J. High Performance Computing Applications
, 1998
"... Key words. NewtonKrylovSchwarz algorithms, parallel CFD, implicit methods Abstract. Implicit solution methods are important in applications modeled by PDEs with disparate temporal and spatial scales. Because such applications require high resolution with reasonable turnaround, parallelization is e ..."
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Cited by 38 (14 self)
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Key words. NewtonKrylovSchwarz algorithms, parallel CFD, implicit methods Abstract. Implicit solution methods are important in applications modeled by PDEs with disparate temporal and spatial scales. Because such applications require high resolution with reasonable turnaround, parallelization is essential. The pseudotransient matrixfree NewtonKrylovSchwarz (ΨNKS) algorithmic framework is presented as a widely applicable answer. This article shows that, for the classical problem of threedimensional transonic Euler flow about an M6 wing, ΨNKS can simultaneously deliver • globalized, asymptotically rapid convergence through adaptive pseudotransient continuation and Newton’s method; • reasonable parallelizability for an implicit method through deferred synchronization and favorable communicationtocomputation scaling in the Krylov linear solver; and • high perprocessor performance through attention to distributed memory and cache locality, especially through the Schwarz preconditioner. Two discouraging features of ΨNKS methods are their sensitivity to the coding of the underlying PDE discretization and the large number of parameters that must be selected to govern convergence. We therefore distill several recommendations from our experience and from our reading of the literature on various algorithmic components of ΨNKS, and we describe a freely available, MPIbased portable parallel software implementation of the solver employed here. 1. Introduction. Disparate