Abstract
This paper is concerned with a new approach for preconditioning large sparse least squares problems. Based on the idea of the approximate inverse preconditioner, which was originally developed for square matrices, we construct a generalized approximate inverse (GAINV)M which approximately minimizes ∥/ −M A∥F or ∥I −AM∥F. Then, we also discuss the theoretical issues such as the equivalence between the original least squares problem and the preconditioned problem. Finally, numerical experiments on problems from Matrix Market collection and random matrices show that although the preconditioning is expensive, it pays off in certain cases.
Similar content being viewed by others
References
R.E. Bank and C. Wagner, Multilevel ILU decomposition. Numer. Math.,82 (1999), 543–574.
S.T. Barnard, L.M. Bernardo and H.D. Simon, An MPI implementation of the SPAI preconditioner on the T3E. Int. J. High Perform. Comput. Appl.,13 (1999), 107–128.
M.W. Benson, Iterative Solution of Large Scale Linear Systems. Master’s Thesis, Lakehead Universeity, Thunder Bay, ON, Canada, 1973.
M.W. Benson and P.O. Frederickson, Iterative solution of large sparse linear systems arising in certain multidimensional approximataion problems. Utilitas Math.,22 (1982), 127–140.
M. Benzi, Preconditioning techniques for large linear systems: a survey. J. Comp. Phy.,182 (2002), 418–477.
M. Benzi and M. Tůma, A comparative study of sparse approximate inverse preconditioners. Appl. Numer. Math.,30 (1999), 305–340.
A. Björck, Numerical Methods for Least Squares Problems. SIAM, Philadelphia, 1996.
P.N. Brown and H.F. Walker, GMRES on (nearly) singular system. SIAM J. Matrix Anal. Appl.,18 (1997), 37–51.
E. Chow and Y. Saad, Approximate inverse preconditioners via sparse-sparse iterations. SIAM J. Sci. Comput,19 (1998), 995–1023.
J.D.F. Cosgrove, J.C. Daz and A. Griewank, Approximate inverse preconditioning for sparse linear systems. Int. J. Comput. Math.,44 (1992), 91–110.
I.S. Duff, R.G. Grimes and J.G. Lewis, Sparse matrix test problems. ACM Trans. Math. Software,15 (1989), 1–14.
N.I.M. Gould and J.A. Scott, Sparse approximate-inverse preconditioners using normminimization techniques. SIAM J. Sci. Comput.,19 (1998), 605–625.
M. Grote and T. Huckle, Parallel preconditioning with sparse approximate inverses. SIAM J. Sci. Comput.,18 (1997), 838–853.
K. Hayami, J.-F. Yin and T. Ito, GMRES methods for least squares problems. NII Technical Report, NII-2007-009E, July 2007, 1–28.
C.C. Paige and M.A. Saunders, Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal.,12 (1975), 617–629.
Y. Saad, Iterative Methods for Sparse Linear Systems (2nd edition). SIAM, Philadelphia, 2003.
Y. Saad and M.H. Schultz, GMRES: a generalized minimal residual method for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput.,7 (1986), 856–869.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Cui, X., Hayami, K. Generalized approximate inverse preconditioners for least squares problems. Japan J. Indust. Appl. Math. 26, 1 (2009). https://doi.org/10.1007/BF03167543
Received:
Revised:
DOI: https://doi.org/10.1007/BF03167543