arXiv subjects
Publications and source records attributed to Susanne Bradley.
We offer a classification of a broad and practically relevant class of symmetric double saddle-point system. At the core of the paper is the division of the associated matrices into ``block-arrow'' and ``block-tridiagonal'' forms. We describe relevant applications, invertibility conditions, spectral properties, and block preconditioners. Our discussion is kept within a general framework rather than tailored to specific applications.
We consider the iterative solution of symmetric saddle-point matrices with a singular leading block. We develop a new ideal positive definite block diagonal preconditioner that yields a preconditioned operator with four distinct eigenvalues. We offer a few techniques for making the preconditioner practical, and illustrate the effectiveness of our approach with numerical experiments.
We derive bounds on the eigenvalues of saddle-point matrices with singular leading blocks. The technique of proof is based on augmentation. Our bounds depend on the principal angles between the ranges or kernels of the matrix blocks. Numerical experiments validate our analytical findings.
We derive bounds on the eigenvalues of a generic form of double saddle-point matrices. The bounds are expressed in terms of extremal eigenvalues and singular values of the associated block matrices. Inertia and algebraic multiplicity of eigenvalues are considered as well. The analysis includes bounds for preconditioned matrices based on block diagonal preconditioners using Schur complements, and it is shown that in this case the eigenvalues are clustered within a few intervals bounded away from zero. Analysis for approximations of Schur complements is included. Some numerical experiments validate our analytical findings.
We consider the iterative solution of symmetric saddle point systems with a rank-deficient leading block. We develop two preconditioners that, under certain assumptions on the rank structure of the system, yield a preconditioned matrix with a constant number of eigenvalues. We then derive some properties of the inverse of a particular class of saddle point system and exploit these to develop a third preconditioner, which remains ideal even when the earlier assumptions on rank structure are relaxed.