On large-scale generalized inverses in solving two-by-two block linear systems
type: Item
instance of: scholarly article
title: On large-scale generalized inverses in solving two-by-two block linear systems
zbMATH Open document ID: 1264.65061
author: Yanyan Li
published in: Linear Algebra and its Applications
publication date: 2013-03-05T00:00:00Z
review text: In large scale nonsymmetric saddle-point problems, one has to solve a linear system whose matrix is of the form \[ \left[\begin{matrix} A&B_1^T\cr B_2&-C\end{matrix}\right] \] with \(A\) and \(C\) square. A projected Schur complement method (PSCM) is used (see [\textit{J.~Haslinger} et al., Numer.\ Linear Algebra Appl. 14, No.~9, 713--739 (2007; Zbl 1199.65102)]). This paper discusses the role of the generalized inverse in the Schur complement of \(A\) when \(A\) is singular. The key point is that a generalized inverse with prescribed range and null space (e.g.\ the Moore-Penrose inverse) can be obtained by choosing appropriate projectors. This is a generalization of \textit{R.~Kučera} et al. [``On the Moore-Penrose inverse in solving saddle-point systems with singular diagonal blocks'', ibid. 19, No.~4, 677--699 (2012)]. Since in practice not the generalized inverse but its action on a vector is needed, an algorithm is described to compute that. Here, also a regularization method is proposed in the case where the null space is known. It causes only little fill-in and avoids the need to check for zero pivots. Numerical examples illustrate the method proposed.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 65F22
Mathematics Subject Classification ID: 65F50
zbMATH DE Number: 6141606
zbMATH Keywords: generalized inverses
zbMATH Keywords: projectors
zbMATH Keywords: block linear systems
zbMATH Keywords: Schur complement
zbMATH Keywords: null-space method
zbMATH Keywords: nonsymmetric saddle-point problem
zbMATH Keywords: Moore-Penrose inverse
zbMATH Keywords: algorithm
zbMATH Keywords: regularization method
zbMATH Keywords: numerical examples
describes a project that uses: MatSol
MaRDI profile type: Publication
full work available at URL: j.laa.2012.09.027
OpenAlex ID: W2015099957
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DOI: 10.1016/J.LAA.2012.09.027
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description: scientific article; zbMATH DE number 6141606