Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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A SPARSE APPROXIMATE INVERSE PRECONDITIONER FOR NONSYMMETRIC POSITIVE DEFINITE MATRICES

  • Received : 2009.07.31
  • Accepted : 2009.09.22
  • Published : 2010.09.30
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Abstract

We develop an algorithm for computing a sparse approximate inverse for a nonsymmetric positive definite matrix based upon the FFAPINV algorithm. The sparse approximate inverse is computed in the factored form and used to work with some Krylov subspace methods. The preconditioner is breakdown free and, when used in conjunction with Krylov-subspace-based iterative solvers such as the GMRES algorithm, results in reliable solvers. Some numerical experiments are given to show the efficiency of the preconditioner.

Keywords

References

  1. M. Benzi, Preconditioning techniques for large linear systems: A survey, J. of Computational Physics, 182 (2002) 418-477. https://doi.org/10.1006/jcph.2002.7176
  2. M. Benzi, J. K. Cullum, and M. Tuma, and C. D. Meyer, Robust approximate inverse preconditioning for the conjugate gradient Method, SIAM J. Sci. Comput., 22 (2000) 1318-1332. https://doi.org/10.1137/S1064827599356900
  3. M. Benzi, M. Tuma, A comparative study of sparse approximate inverse preconditioners, Appl. Numer. Math., 30 (1999)305-340. https://doi.org/10.1016/S0168-9274(98)00118-4
  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) 1135-1149. https://doi.org/10.1137/S1064827594271421
  5. M. Benzi, M. Tuma, A sparse approximate inverse preconditioner for nonsymmetric linear systems, SIAM J. Sci. Comput., 19 (1998) 968-994. https://doi.org/10.1137/S1064827595294691
  6. T. Davis, University of Florida sparse matrix collection, NA Digest, 92(1994), http://www.cise.ufl.edu/research/sparse/matrices.
  7. S. A. Kharchenko, L. Yu. Kolotilina, A. A. Nikishin, A. Yu. Yeremin, A robust AINV-type method for constructing sparse approximate inverse preconditioners in factored form, Numer. Linear Algebra With Appl., 8 (2001) 165179.
  8. L. Y. Kolotilina and A. Y. Yeremin, Factorized sparse approximate inverse preconditioning I. Theory, SIAM J. Matrix Anal. Appl., 14 (1993) 45-58. https://doi.org/10.1137/0614004
  9. L. Y. Kolotilina and A. Y. Yeremin, Factorized sparse approximate inverse preconditioning II: Solution of 3D FE systems on massively parallel computers, Int. J. High Speed Comput.,7 (1995) 191-215. https://doi.org/10.1142/S0129053395000117
  10. E.-J. Lee and J. Zhang, Fatored approximate inverse preonditioners with dynamic sparsity patterns, Tehnial Report No.488-07,Department of Computer Science,University of Kentuky, Lexington, KY, 2007.
  11. J.-G. Luo, An incomplete inverse as a preconditioner for the conjugate gradient method, Comput. Math. Appl., 25 (1993) 7379.
  12. J.-G. Luo, A new class of decomposition for inverting asymmetric and indefinite matrices, Comput. Math. Appl., 25 (1993) 95104.
  13. J.-G. Luo, A new class of decomposition for symmetric systems, Mechanics Research Communications, 19 (1992) 159166.
  14. Matrix Market page, http://math.nist.gov/MatrixMarket.
  15. A. Rafiei and F. Toutounian, New breakdown-free variant of AINV method for nonsymmetric positive definite matrices, Jornal of Computational and Applied Mathematics, 219 (2008) 72-80. https://doi.org/10.1016/j.cam.2007.07.003
  16. Y. Saad, Iterative Methods for Sparse linear Systems, PWS press, New York, 1995.
  17. Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 7 (1986) 856-869. https://doi.org/10.1137/0907058
  18. H. A. van der Vorst, Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 12 (1992) 631-644.
  19. J. Zhang, A procedure for computing factored approximate inverse, M.Sc. dissertation, Department of Computer Science, University of Kentucky, 1999.
  20. J. Zhang, A sparse approximate inverse technique for parallel preconditioning of general sparse matrices, Appl. Math. Comput., 130 (2002) 63-85. https://doi.org/10.1016/S0096-3003(01)00069-8