Volume 60, pp. 221-237, 2024.

Augmentation-based preconditioners for saddle-point systems with singular leading blocks

Susanne Bradley and Chen Greif


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. The novelty of the paper lies in the generality of the assumptions made: as long as the saddle-point matrix is nonsingular, there is no assumption on the specific rank of the leading block. Current ideal preconditioners typically rely either on invertibility or a high nullity of the leading block, and the new technique aims to bridge this gap. A spectral analysis is offered, accompanied by numerical experiments.

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Key words

saddle-point systems, preconditioning, augmentation, Schur complement

AMS subject classifications

65F08, 65F10, 65F15

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