Improved IDR(s) Method for Gaining Very Accurate Solutions

Authors: Yusuke Onoue, Seiji Fujino, Norimasa Nakashima

Abstract:

The IDR(s) method based on an extended IDR theorem was proposed by Sonneveld and van Gijzen. The original IDR(s) method has excellent property compared with the conventional iterative methods in terms of efficiency and small amount of memory. IDR(s) method, however, has unexpected property that relative residual 2-norm stagnates at the level of less than 10-12. In this paper, an effective strategy for stagnation detection, stagnation avoidance using adaptively information of parameter s and improvement of convergence rate itself of IDR(s) method are proposed in order to gain high accuracy of the approximated solution of IDR(s) method. Through numerical experiments, effectiveness of adaptive tuning IDR(s) method is verified and demonstrated.

Keywords: Krylov subspace methods, IDR(s), adaptive tuning, stagnation of relative residual.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1062748

References:

[1] T. Davis : Univ. of Florida Sparse Matrix Collection, http://www.cise.ufl.edu/research/sparse/matrices/index.html
[2] S. Fujino, M. Fujiwara, M. Yoshida : BiCGSafe method based on minimization of associate residual, Trans. of JSCES, Paper No.20050028, 2005. (In Japanese)
[3] P. Sonneveld, M. B. van Gijzen: IDR(s): a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations, SIAM J. Sci. Comput. Vol.31, No.2, pp.1035-1062, 2008.
[4] H. A. van der Vorst: Iterative Krylov preciionditionings for large linear systems, Cambridge University Press, 2003.
[5] P. Wesseling, P. Sonneveld : Numerical Experiments with a Multiple Grid- and a Preconditioned Lanczos Type Methods, Lecture Notes in Math. Springer, No.771, pp.543-562, 1980.
[6] S.-L. Zhang: GPBi-CG: Generalized product-type methods based on Bi- CG for solving nonsymmetric linear systems, SIAM J. Sci. Comput., Vol.18, No.2, pp.537-551, 1997.