Preface 1. Background in linear algebra 2. Discretization of partial differential equations 3. Sparse matrices 4. Basic iterative methods 5. Projection methods 6. Krylov subspace methods Part I 7. Krylov subspace methods Part II 8. Methods related to the normal equations 9. Preconditioned iterations 10. Preconditioning techniques 11. Parallel implementations 12. Parallel preconditioners 13. Multigrid methods 14. Domain decomposition methods Bibliography Index.

/pdf/iterative-methods-for-sparse-linear-systems-34ozhpwq89.pdf

Iterative Methods for Sparse Linear Systems

http://www.mathcs.emory.edu/%7Ebenzi/Web_papers/survey.pdf

Preconditioning techniques for large linear systems: a survey

We derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum. We present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function.

On the rate of convergence of the preconditioned conjugate gradient method

The standard incomplete LU (ILU) preconditioners often fail for general sparse indefinite matrices because they give rise to "unstable" factors L and U. In such cases, it may be attractive to approximate the inverse of the matrix directly. This paper focuses on approximate inverse preconditioners based on minimizing ||I-AM||F, where AM is the preconditioned matrix. An iterative descent-type method is used to approximate each column of the inverse. For this approach to be efficient, the iteration must be done in sparse mode, i.e., with "sparse-matrix by sparse-vector" operations. Numerical dropping is applied to maintain sparsity; compared to previous methods, this is a natural way to determine the sparsity pattern of the approximate inverse. This paper describes Newton, "global," and column-oriented algorithms, and discusses options for initial guesses, self-preconditioning, and dropping strategies. Some limited theoretical results on the properties and convergence of approximate inverses are derived. Numerical tests on problems from the Harwell--Boeing collection and the FIDAP fluid dynamics analysis package show the strengths and limitations of approximate inverses. Finally, some ideas and experiments with practical variations and applications are presented.

Approximate Inverse Preconditioners via Sparse-Sparse Iterations

This paper presents a short survey of recent research on Krylov subspace methods with emphasis on implementation on vector and parallel computers. Conjugate gradient methods have proven very useful on traditional scalar computers, and their popularity is likely to increase as three-dimensional models gain importance. A conservative approach to derive effective iterative techniques for supercomputers has been to find efficient parallel/vector implementations of the standard algorithms. The main source of difficulty in the incomplete factorization preconditionings is in the solution of the triangular systems at each step. A few approaches consisting of implementing efficient forward and backward triangular solutions are described in detail. Then polynomial preconditioning as an alternative to standard incomplete factorization techniques is discussed. Another efficient approach is to reorder the equations so as to improve the structure of the matrix to achieve better parallelism or vectorization. An overview of these ideas and others is given in this article, as well as an attempt to comment on their effectiveness or potential for different types of architectures.

/pdf/krylov-subspace-methods-on-supercomputers-5bbepuzurh.pdf

Krylov Subspace Methods on Supercomputers

On some versions of incomplete block-matrix factorization iterative methods

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On some versions of incomplete block-matrix factorization iterative methods