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5 - Approximate inverse preconditioners [T2]: direct approximation of An×n−1

Published online by Cambridge University Press:  06 January 2010

Ke Chen
Affiliation:
University of Liverpool
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Summary

In the last few years we have studied preconditioning techniques based on sparse approximate inverses and have found them to be quite effective.

B. Carpentieri, et al. SIAM Journal on Scientific Computing, Vol. 25 (2003)

The objective is to remove the smallest eigenvalues of A which are known to slow down the convergence of GMRES.

Jocelyne Erhel, et al. Journal of Computational and Applied Mathematics, Vol. 69 (1996)

The most successful preconditioning methods in terms of reducing the number of iterations, such as the incomplete LU decomposition or symmetric successive relaxation (SSOR), are notoriously difficult to implement in a parallel architecture, especially for unstructured matrices.

Marcus J. Grote and Thomas Huckle. SIAM Journal on Scientific Computing, Vol. 18 (1997)

This chapter will discuss the construction of Inverse Type preconditioners (or approximate inverse type) i.e. for equation (1.2)

MAx = Mb

and other types as shown on Page 3. Our first concern will be a theoretical one on characterizing A–1. It turns out that answering this concern reveals most underlying ideas of inverse type preconditioners. We shall present the following.

  1. Section 5.1 How to characterize A–1 in terms of A

  2. Section 5.2 Banded preconditioner

  3. Section 5.3 Polynomial pk(A) preconditioners

  4. Section 5.4 General and adaptive SPAI preconditioners

  5. Section 5.5 AINV type preconditioner

  6. Section 5.6 Multi-stage preconditioners

  7. Section 5.7 The dual tolerance self-preconditioning method

  8. Section 5.8 Mesh near neighbour preconditioners

  9. Section 5.9 Numerical experiments

  10. Section 5.10 Discussion of software and Mfile

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Publisher: Cambridge University Press
Print publication year: 2005

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