Subspace-by-subspace preconditioners for structured linear systems (Q2760354)

Preconditioners for \(n\) by \(n\) real symmetric positive definite linear systems of equations \(Ax=b\) are constructed under the assumption \(A=\sum _{i=1}^e E_i\), each element \(E_i\) is positive semi-definite. NEWLINENEWLINENEWLINEFirst, element-by-element (EBE) preconditioners are reminded, see, e.g., \textit{T.~J.~R. Hughes, I.~Levit} and \textit{J.~Winget} [Comput. Methods Appl. Mech. Eng. 36, 241-254 (1983; Zbl 0501.73069)]. These are based on a special approximate decomposition of \(E_i\) (the Winget decomposition, see the cited paper). Focusing on non-zero rows and columns of \(E_i\) as well as on an efficient implementation and elaborating the EBE approach, the authors propose a subspace-by-subspace (SBS) preconditioner. The idea is further applied to the case \(E_i=A_i^{\text{T}}A_i\), where \(A_i\) is an \(n\) by \(n_i\) real matrix. To cope with systems, where \(E_i\) can be of very low rank, a combination of these methods with existing techniques is considered. NEWLINENEWLINENEWLINEThe half of the paper deals with numerical experiments. Problems based on various matrices are solved. Different preconditioners an different settings of the SBS preconditiner are tested. Results are presented in numerous tables and discussed from the point of view of efficiency.