Constraint-Style Preconditioners for Regularized Saddle Point Problems

• Published in: SIAM journal on matrix analysis and applications Vol. 29; no. 2; pp. 672 - 684
• Main Author: Dollar, H. S.
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2007
• Subjects: Eigenvalues; Eigenvectors; Iterative methods; Matrix
• ISSN: 0895-4798; 1095-7162
• DOI: 10.1137/050626168

The problem of finding good preconditioners for the numerical solution of an important class of indefinite linear systems is considered. These systems are of a regularized saddle point structure $[\begin{smallmatrix} A &\quad B^T \\ B &\quad -C \end{smallmatrix}] [\begin{smallmatrix} x \\ y \end{smallmatrix}] = [\begin{smallmatrix} c \\ d \end{smallmatrix}], $ where $A\in\mathbb R ^{n\times n}$, $C\in\mathbb R ^{m\times m}$ are symmetric and $B\in\mathbb R ^{m\times n}$. In [SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1300-1317], Keller, Gould, and Wathen analyze the idea of using constraint preconditioners that have a specific 2 by 2 block structure for the case of $C$ being zero. We shall extend this idea by allowing the (2, 2) block to be symmetric and positive semidefinite. Results concerning the spectrum and form of the eigenvectors are presented, as are numerical results to validate our conclusions.