Volume 55, pp. 532-546, 2022.

A note on augmented unprojected Krylov subspace methods

Kirk M. Soodhalter


Subspace recycling iterative methods and other subspace augmentation schemes are a successful extension to Krylov subspace methods in which a Krylov subspace is augmented with a fixed subspace spanned by vectors deemed to be helpful in accelerating convergence or conveying knowledge of the solution. Recently, a survey was published, in which a framework describing the vast majority of such methods was proposed [Soodhalter et al., GAMM-Mitt., 43 (2020), Art. e202000016]. In many of these methods, the Krylov subspace is one generated by the system matrix composed with a projector that depends on the augmentation space. However, it is not a requirement that a projected Krylov subspace be used. There are augmentation methods built on using Krylov subspaces generated by the original system matrix, and these methods also fit into the general framework. In this note, we observe that one gains implementation benefits by considering such augmentation methods with unprojected Krylov subspaces in the general framework. We demonstrate this by applying the idea to the R$^3$GMRES method proposed in [Dong et al., Electron., Trans., Numer., Anal., 42 (2014), pp. 136–146] to obtain a simplified implementation and to connect that algorithm to early augmentation schemes based on flexible preconditioning [Saad, SIAM J. Matrix Anal. Appl., 18 (1997)].

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Key words

Krylov subspaces, augmentation, recycling, discrete ill-posed problems

AMS subject classifications

65F10, 65F50, 65F08

Links to the cited ETNA articles

[7]Vol. 42 (2014), pp. 136-146 Yiqiu Dong, Henrik Garde, and Per Christian Hansen: R$^3$GMRES: Including Prior Information in GMRES-Type Methods for Discrete Inverse Problems
[13]Vol. 39 (2012), pp. 156-185 Martin H. Gutknecht: Spectral deflation in Krylov solvers: a theory of coordinate space based methods
[24]Vol. 54 (2021), pp. 256-275 Ronny Ramlau and Bernadett Stadler: An augmented wavelet reconstructor for atmospheric tomography

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