Journal ArticleDOI
High Performance Preconditioning
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TLDR
It is shown how a rather high performance can be achieved for the more effective preconditioners, such as successive over-relaxation and incomplete decompositions, on most vector computers if used in a straightforward manner.Abstract:
The discretization of second-order elliptic partial differential equations over three-dimensional rectangular regions, in general, leads to very large sparse linear systems. Because of their huge order and their sparseness, these systems can only be solved by iterative methods using powerful computers, e.g., vector supercomputers. Most of those methods are only attractive when used in combination with a so-called preconditioning matrix. Unfortunately, the more effective preconditioners, such as successive over-relaxation and incomplete decompositions, do not perform very well on most vector computers if used in a straightforward manner. In this paper it is shown how a rather high performance can be achieved for these preconditioners.read more
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Book
An Introduction to Multigrid Methods
TL;DR: These notes were written for an introductory course on the application of multigrid methods to elliptic and hyperbolic partial differential equations for engineers, physicists and applied mathematicians, restricting ourselves to finite volume and finite difference discretization.
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Iterative solution of linear systems in the 20th century
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A Sparse Approximate Inverse Preconditioner for Nonsymmetric Linear Systems
Michele Benzi,Miroslav Tuma +1 more
TL;DR: A procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed, and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient--type methods.
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GMRESR: a family of nested GMRES methods
H.A. van der Vorst,Cornelis Vuik +1 more
TL;DR: In this paper, the authors suggest variants of GMRES, in which a preconditioner is constructed per it<:ration st<:p by a suitable approximation process, e.g., by GMRES itself.
References
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Journal ArticleDOI
GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
Youcef Saad,Martin H. Schultz +1 more
TL;DR: An iterative method for solving linear systems, which has the property of minimizing at every step the norm of the residual vector over a Krylov subspace.
Journal ArticleDOI
LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares
TL;DR: Numerical tests are described comparing I~QR with several other conjugate-gradient algorithms, indicating that I ~QR is the most reliable algorithm when A is ill-conditioned.
Journal ArticleDOI
An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix
TL;DR: A particular class of regular splittings of not necessarily symmetric M-matrices is proposed, if the matrix is symmetric, this splitting is combined with the conjugate-gradient method to provide a fast iterative solution algorithm.
Journal ArticleDOI
CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems
TL;DR: A Lanczos-type method is presented for nonsymmetric sparse linear systems as arising from discretisations of elliptic partial differential equations, based on a polynomial variant of the conjugate gradients algorithm.
Journal ArticleDOI
The incomplete Cholesky—conjugate gradient method for the iterative solution of systems of linear equations
TL;DR: A new iterative method for the solution of systems of linear equations has been recently proposed by Meijerink and van der Vorst and has been applied to real laser fusion problems taken from typical runs of the laser fusion simulation code LASNEX.
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