Bi-CGSTAB as an induced dimension reduction method

Abstract

The Induced Dimension Reduction method [P. Wesseling, P. Sonneveld, Numerical experiments with a multiple grid- and a preconditioned Lanczos type method, in: Lecture Notes in Mathematics, vol. 771, Springer-Verlag, Berlin, 1980, pp. 543–562] was proposed in 1980 as an iterative method for solving large nonsymmetric linear systems of equations. IDR can be considered as the predecessor of methods like CGS (Conjugate Gradient Squared) [P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 10 (1989) 36–52] and Bi-CGSTAB (Bi-Conjugate Gradients STABilized [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. 13 (2) (1992) 631–644]). All three methods are based on efficient short recurrences. An important similarity between the methods is that they use orthogonalizations with respect to a fixed ‘shadow residual’. Of the three methods, Bi-CGSTAB has gained the most popularity, and is probably still the most widely used short recurrence method for solving nonsymmetric systems. Recently, Sonneveld and van Gijzen revived the interest for IDR. In [P. Sonneveld, M. van Gijzen, IDR( s): a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations, Preprint, Delft Institute of Applied Mathematics, Delft University of Technology, Delft, The Netherlands, March 2007], they demonstrate that a higher dimensional shadow space, defined by the n × s matrix R ˜ 0 , can easily be incorporated into IDR, yielding a highly effective method. The original IDR method is closely related to Bi-CGSTAB. It is therefore natural to ask whether Bi-CGSTAB can be extended in a way similar to IDR. To answer this question we explore the relation between IDR and Bi-CGSTAB and use our findings to derive a variant of Bi-CGSTAB that uses a higher dimensional shadow space.