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Approximate Inverse Preconditioning for Shifted Linear Systems

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Abstract

In this paper we consider the problem of preconditioning symmetric positive definite matrices of the form A α=AI where α>0. We discuss how to cheaply modify an existing sparse approximate inverse preconditioner for A in order to obtain a preconditioner for A α. Numerical experiments illustrating the performance of the proposed approaches are presented.

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REFERENCES

  1. U. R. Ascher, R.M. M. Matteij, and R. D. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, Philadelphia, PA, 1995.

    Google Scholar 

  2. M. Benzi, J. K. Cullum, and M. Tuma, Robust approximate inverse preconditioning for the conjugate gradient method, SIAM J. Sci. Comput. 22 (2000), pp. 1318-1332.

    Google Scholar 

  3. M. Benzi, R. Kouhia, and M. Tuma, Stabilized and block approximate inverse preconditioners for problems in solid and structural mechanics, Comput. Methods Appl. Mech. Engrg., 190 (2001), pp. 6533-6554.

    Google Scholar 

  4. M. Benzi, C. D. Meyer, and M. Tuma, A sparse approximate inverse preconditioner for the conjugate gradient method, SIAM J. Sci. Comput., 17 (1996), pp. 1135-1149.

    Google Scholar 

  5. M. Benzi and M. Tůma, Orderings for factorized sparse approximate inverse preconditioners, SIAM J. Sci. Comput., 21 (2000), pp. 1851-1868.

    Google Scholar 

  6. D. Bertaccini, A circulant preconditioner for the systems of LMF-based ODE codes, SIAM J. Sci. Comput., 22 (2000), pp. 767-786.

    Google Scholar 

  7. D. Bertaccini, Reliable preconditioned iterative linear solvers for some numerical integrators, Numer. Linear Algebra Appl., 8 (2001), pp. 111-125.

    Google Scholar 

  8. M. Bollhöfer and Y. Saad, On the relations between ILUs and factored approximate inverses, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 219-237.

    Google Scholar 

  9. S. Demko, W. F. Moss, and P. W. Smith, Decay rates for inverses of band matrices, Math. Comp., 43 (1984), pp. 491-499.

    Google Scholar 

  10. J. J. Dongarra, I. S. Duff, D. C. Sorensen, and H. A. van der Vorst, Numerical Linear Algebra for High-Performance Computers, SIAM, Philadelphia, PA, 1998.

    Google Scholar 

  11. E. Haber, U. M. Ascher, and D. Oldenburg, On optimization techniques for solving nonlinear inverse problems, Inverse Problems, 16 (2000), pp. 1263-1280.

    Google Scholar 

  12. E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems, Springer-Verlag, Berlin-Heidelberg, 1991.

    Google Scholar 

  13. S. A. Kharchenko, L. Yu. Kolotilina, A. A. Nikishin, and A. Yu. Yeremin, A robust AINV-type method for constructing sparse approximate inverse preconditioners in factored form, Numer. Linear Algebra Appl., 8 (2001), pp. 165-179.

    Google Scholar 

  14. G. Meurant, A review on the inverse of symmetric tridiagonal and block tridiagonal matrices, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 707-728.

    Google Scholar 

  15. G. Meurant, On the incomplete Cholesky decomposition of a class of perturbed matrices, SIAM J. Sci. Comput., 23 (2001), pp. 419-429.

    Google Scholar 

  16. National Institute of Standards, Matrix market, available online at http://math.nist.gov/MatrixMarket.

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Benzi, M., Bertaccini, D. Approximate Inverse Preconditioning for Shifted Linear Systems. BIT Numerical Mathematics 43, 231–244 (2003). https://doi.org/10.1023/A:1026089811044

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