On the eigenvalue distribution of a class of preconditioning methods
Numerische Mathematik
On the rate of convergence of the preconditioned conjugate gradient method
Numerische Mathematik
The rate of convergence of conjugate gradients
Numerische Mathematik
On the multi-level splitting of finite element spaces
Numerische Mathematik
Conditioning analysis of positive definite matrices by approximate factorizations
Journal of Computational and Applied Mathematics
Algebraic multilevel preconditioning methods, II
SIAM Journal on Numerical Analysis
Modified incomplete factorization strategies
Proceedings of a conference on Preconditioned conjugate gradient methods
Solving positive (semi) definite linear systems by preconditioned iterative methods
Proceedings of a conference on Preconditioned conjugate gradient methods
Analysis of a recursive 5-point/9-point factorization method
Proceedings of a conference on Preconditioned conjugate gradient methods
The nested recursive two-level factorization method for nine-point difference matrices
SIAM Journal on Scientific and Statistical Computing
Conditioning analysis of sparse block approximate factorizations
Applied Numerical Mathematics
The method of diagonal compensation of reduced matrix entries and multilevel iteration
ISCM '90 Proceedings of the International Symposium on Computation mathematics
Iterative solution methods
Optimal Order Preconditioning of Finite Difference Matrices
SIAM Journal on Scientific Computing
Finite element solution of boundary value problems: theory and computation
Finite element solution of boundary value problems: theory and computation
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
SIAM Journal on Matrix Analysis and Applications
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This presentation is intended to review the state-of-the-art of iterative methods for solving large sparse linear systems such as arising in finite difference and finite element approximations of boundary value problems. However, in order to keep this review within reasonable bounds, we only review those methods for which an algebraic analysis has been achieved.We first review the basic principles and components of iterative solution methods and describe in more detail the main devices used to design preconditioners, showing how the present day complex preconditioners are built through additive and/or multiplicative composition of simpler ones. We also note that acceleration methods may sometimes be viewed, and thus used, as preconditioners.Next, using approximate factorizations as basic flamework, we show how their development led to the study of so-called modified methods and why attention then shifted to specific orderings, of multilevel type. Finally we show how the successful development of multigrid and hierarchical basis methods prompted the introduction of equivalent algebraic techniques: besides recursive orderings, an additional step called stabilization by polynomial preconditioning that plays here the role of the W-cycles of the multigrid method and an algebraic version of V-cycles with smoothing.