Deflated and Augmented Krylov Subspace Methods: A Framework for Deflated BiCG and Related Solvers

We present an extension of the framework of Gaul et al. (SIAM J. Matrix Anal. Appl. 34, 495–518 (2013)) for deflated and augmented Krylov subspace methods satisfying a Galerkin condition to more general Petrov–Galerkin conditions. The main goal is to apply the framework to the biconjugate gradient method (BiCG) and some of its generalizations, including BiCGStab and IDR(s). For such applications the assumptions of Gaul et al. were too restrictive. Our abstract approach does not depend on particular recurrences and thus simplifies the derivation of theoretical results. It easily leads to a variety of realizations by specific algorithms. We do not go into algorithmic details, but we show that for every method there are two different approaches for extending it by augmentation and deflation: one that explicitly takes care of the augmentation space in every step, and one that applies the unchanged basic algorithm to a projected problem but requires a correction step at the end. The latter approach typically generates nested sequences of Krylov subspaces for a singular operator that is associated with the projected problem. The deflated BiCG method requires two such sequences, but it also allows us to solve two dual linear systems at the price of one, a property that does no longer hold for the closely related deflated biconjugate residual method (BiCR). Deflated Lanczos-type product methods fit in our new framework too. The question of how to extract the augmentation and deflation subspaces is not addressed here.