A survey of subspace recycling iterative methods

Abstract

This survey concerns subspace recycling methods, a popular class of iterative methods that enable effective reuse of subspace information in order to speed up convergence and find good initial vectors over a sequence of linear systems with slowly changing coefficient matrices, multiple right-hand sides, or both. The subspace information that is recycled is usually generated during the run of an iterative method (usually a Krylov subspace method) on one or more of the systems. Following introduction of definitions and notation, we examine the history of early augmentation schemes along with deflation preconditioning schemes and their influence on the development of recycling methods. We then discuss a general residual constraint framework through which many augmented Krylov and recycling methods can both be viewed. We review several augmented and recycling methods within this framework. We then discuss some known effective strategies for choosing subspaces to recycle before taking the reader through more recent developments that have generalized recycling for (sequences of) shifted linear systems, some of them with multiple right-hand sides in mind. We round out our survey with a brief review of application areas that have seen benefit from subspace recycling methods.

Going Beyond the Limits of Classical Atomistic Modeling of Plasmonic Nanostructures

“…Therefore, the shift-invariance property of the Krylov subspaces may be exploited by resorting to the so-called subspace recycling techniques. 66 , 67 …”

A Dynamically Adjusted Subspace Gradient Method and Its Application in Image Restoration

“…Case I: If d k+1 = −g k+1 , let c 4 = 1, and thus it is proved. Case II: If d k+1 is given by Equation ( 24), from Assumption 2 and condition (22),…”

“…For related research, readers can refer to [17]. More research on the use of subspace technology to construct different methods is still in progress [18][19][20][21][22]. The outline of this article is as follows: in Section 2, we give preliminary information.…”

An augmented wavelet reconstructor for atmospheric tomography

“…However, Saad uses an Arnoldi or a Lanczos process for computing an approximate solution rather than the CG method. For a survey on subspace recycling techniques for iterative methods we refer to [35]. We want to emphasize that we are using an augmented Krylov subspace method but no recycling.…”

“…We want to emphasize that we are using an augmented Krylov subspace method but no recycling. What is commonly known in the literature as Krylov subspace recycling, in addition to augmentation, changes the Krylov space [35]. In our method, the Krylov subspace is not changed.…”