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Maintaining convergence properties of BiCGstab methods in finite precision arithmetic
 NUMERICAL ALGORITHMS 10(1995)203223
, 1995
"... It is wellknown that BiCG can be adapted so that hybrid methods with computational complexity almost similar to BiCG can be constructed, in which it is attempted to further improve the convergence behavior. In this paper we will study the class of BiCGstab methods. In many applications, the speed ..."
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It is wellknown that BiCG can be adapted so that hybrid methods with computational complexity almost similar to BiCG can be constructed, in which it is attempted to further improve the convergence behavior. In this paper we will study the class of BiCGstab methods. In many applications, the speed of convergence of these methods appears to be determined mainly by the incorporated BiCG process, and the problem is that the BiCG iteration coefficients have to be determined from the BiCGstab process. We will focus our attention to the accuracy of these BiCG coefficients, and how rounding errors may affect the speed of convergence of the BiCGstab methods. We will propose a strategy for a more stable determination of the BiCG iteration coefficients and by experiments we will show that this indeed may lead to faster convergence.
An Overview of Approaches for the Stable Computation of Hybrid BiCG Methods
, 1995
"... It is wellknown that BiCG can be adapted so that the operations with A^T can be avoided, and hybrid methods with computational complexity almost similar to BiCG can be constructed in a further attempt to improve the convergence behavior. Examples of this are CGS, BiCGSTAB, and BiCGstab(`). In man ..."
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It is wellknown that BiCG can be adapted so that the operations with A^T can be avoided, and hybrid methods with computational complexity almost similar to BiCG can be constructed in a further attempt to improve the convergence behavior. Examples of this are CGS, BiCGSTAB, and BiCGstab(`). In many applications, the speed of convergence of these methods is very dependent on the incorporated BiCG process. The accuracy of the iteration coefficients of BiCG depends on the particular choice of the hybrid method. We will discuss the accuracy of these coefficients and how this affects the speed of convergence. We will show that hybrid methods exist which have better accuracy properties. This may lead to faster convergence and more accurate approximations. We also discuss lookahead strategies for the determination of appropriate values for ` in BiCGstab(`). These strategies are easily applied for the hybrid part, in contrast to similar techniques for the BiCG part (but of course they do n...
The Main Effects of Rounding Errors in Krylov Solvers for Symmetric Linear Systems
, 1997
"... The 3term Lanczos process leads, for a symmetric matrix, to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of linear systems, by solving the reduced system in one way or another. This leads to wellknown ..."
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The 3term Lanczos process leads, for a symmetric matrix, to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of linear systems, by solving the reduced system in one way or another. This leads to wellknown methods: MINRES (GMRES), CG, CR, and SYMMLQ. We will discuss in what way and to what extent the various approaches are sensitive to rounding errors. In our analysis we will assume that the Lanczos basis is generated in exactly the same way for the different methods (except CR), and we will not consider the errors in the Lanczos process itself. These errors may lead to large perturbations with respect to the exact process, but convergence takes still place. Our attention is focussed to what happens in the solution phase. We will show that the way of solution may lead, under circumstances, to large additional errors, that are not corrected by continuing the iteration process. Our findings are...
JOURNAL OFATlONAl AND
, 1993
"... New insights in GMRESlike methods with variable preconditioners C. Vuik* ..."
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New insights in GMRESlike methods with variable preconditioners C. Vuik*
Further improvements in nonsymmetric hybrid CG methods
, 1994
"... In the past few years new methods have been proposed that can be seen as combinations of standard Krylov subspave methods, such as Bi CG and GMRES. One of the first hybrid schemes of this type is CGS, actually the BiCG squared method. Other such hybrid schemes include BiCGSTAB (a combination of ..."
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In the past few years new methods have been proposed that can be seen as combinations of standard Krylov subspave methods, such as Bi CG and GMRES. One of the first hybrid schemes of this type is CGS, actually the BiCG squared method. Other such hybrid schemes include BiCGSTAB (a combination of BiCG and GMRES(1)), QMRS, TFQMR, Hybrid GMRES (polynomial preconditioned GMRES) and the nested GMRESR method (GMRES preconditioned by itself or other schemes). These methods have been successful in solving relevant sparse nonsymmetric linear systems, but there is still a need for further improvements. In this paper we will highlight some of the recent advancements in the search for effective iterative solvers.
Mention: Informatique Ecole doctorale Matisse présentée par
, 2013
"... préparée à l’unité de recherche IRISA – UMR6074 ..."
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Deliberate IllConditioning of Krylov Matrices
"... This paper starts off with studying simple extrapolation methods for the classical iteration schemes such as Richardson, Jacobi and GaussSeidel iteration. The extrapolation procedures can be interpreted as approximate minimal residual methods in a Krylov subspace. It seems therefore logical to cons ..."
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This paper starts off with studying simple extrapolation methods for the classical iteration schemes such as Richardson, Jacobi and GaussSeidel iteration. The extrapolation procedures can be interpreted as approximate minimal residual methods in a Krylov subspace. It seems therefore logical to consider, conversely, classical methods as preprocessors for Krylov subspace methods, as was done by Z'itko (1996) for the Conjugate Gradient method. The observation made by Ipsen (1998) that small residuals necessarily imply an illconditioned Krylov matrix, explains the success of such preprocessing schemes: residuals of classical methods are (unscaled) power method iterates, and building a Krylov subspace on such a classical residual will therefore lead to expansion vectors that are at small angle to the previous Krylov vectors. This results in an illconditioned Krylov matrix. In this paper, we present a large number of experiments that support this claim, and give theoretical interpretations of the preprocessing. The results are mainly of interest in Krylov subspace methods for nonHermitian matrices based on long recurrences, and in particular for applications with heavy memory limitations. Also, in applications in which minimal residual methods stagnate due to a lack of illconditioning, the use of a classical preprocessor can be a cheap and easily parallelizable remedy. 1
FEBUI – FINITE ELEMENT BEOWULF USER INTERFACE: A HOMEMADE PACKAGE FOR AUTOMATED PARALLEL FINITE ELEMENT COMPUTATIONS
, 2005
"... Abstract. A computational package, the FEBUI, is introduced for automated parallelization of finite element ..."
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Abstract. A computational package, the FEBUI, is introduced for automated parallelization of finite element