Constraint-Style Preconditioners for Regularized Saddle Point Problems

TLDR

The paper tackles preconditioners for indefinite linear systems of regularized saddle point form, extending prior work that used constraint preconditioners when the (2,2) block is zero. The authors aim to generalize the constraint preconditioner framework by allowing the (2,2) block to be symmetric and positive semidefinite. They develop and analyze a 2×2 block preconditioner with a symmetric positive semidefinite (2,2) block for these saddle point systems. Spectral analysis and eigenvector structure are derived, and numerical experiments confirm the theoretical predictions. Citation: Keller, Gould, and Wathen, SIAM J.

Abstract

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.