On large-scale generalized inverses in solving two-by-two block linear systems
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.
scientific article; zbMATH DE number 6141606
scientific article