Iterated Gauss-Seidel GMRES

0585518 - MÚ 2025 RIV US eng J - Journal Article
Thomas, S. - Carson, E. - Rozložník, Miroslav - Carr, A. - Świrydowicz, K.
Iterated Gauss-Seidel GMRES.
SIAM Journal on Scientific Computing. Roč. 46, č. 2 (2024), S254-S279. ISSN 1064-8275. E-ISSN 1095-7197
R&D Projects: GA ČR(CZ) GA23-06159S
Institutional support: RVO:67985840
Keywords : GMRES * orthogonal complement
OECD category: Pure mathematics
Impact factor: 3.1, year: 2022
Method of publishing: Limited access
https://doi.org/10.1137/22M1491241

The GMRES algorithm of Saad and Schultz [SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856-869] is an iterative method for approximately solving linear systems Ax = b, with initial guess x0 and residual r0 = b Ax0. The algorithm employs the Arnoldi process to generate the Krylov basis vectors (the columns of Vk). It is well known that this process can be viewed as a QR factorization of the matrix Bk = [ r0,AVk ] at each iteration. Despite an \scrO (\varepsilon )\kappa (Bk) loss of orthogonality, for unit roundoff \varepsilon and condition number \kappa , the modified Gram-Schmidt formulation was shown to be backward stable in the seminal paper by Paige et al. [SIAM J. Matrix Anal. Appl., 28 (2006), pp. 264-284]. We present an iterated Gauss-Seidel formulation of the GMRES algorithm (IGS-GMRES) based on the ideas of Ruhe [Linear Algebra Appl., 52 (1983), pp. 591-601] and Świrydowicz et al. [Numer. Linear Algebra Appl., 28 (2020), pp. 1-20]. IGS-GMRES maintains orthogonality to the level \scrO (\varepsilon )\kappa (Bk) or \scrO (\varepsilon ), depending on the choice of one or two iterations, for two Gauss-Seidel iterations, the computed Krylov basis vectors remain orthogonal to working accuracy and the smallest singular value of Vk remains close to one. The resulting GMRES method is thus backward stable. We show that IGS-GMRES can be implemented with only a single synchronization point per iteration, making it relevant to large-scale parallel computing environments. We also demonstrate that, unlike MGS-GMRES, in IGS-GMRES the relative Arnoldi residual corresponding to the computed approximate solution no longer stagnates above machine precision even for highly nonnormal systems.
Permanent Link: https://hdl.handle.net/11104/0353201