A null-space approach for large-scale symmetric saddle point systems with a small and non zero (2,2) block

Scott, J. ORCID: https://orcid.org/0000-0003-2130-1091 and Tuma, M. (2022) A null-space approach for large-scale symmetric saddle point systems with a small and non zero (2,2) block. Numerical Algorithms, 90. pp. 1639-1667. ISSN 1572-9265 doi: 10.1007/s11075-021-01245-z

Abstract/Summary

Null-space methods have long been used to solve large sparse n x n symmetric saddle point systems of equations in which the (2,2) block is zero. This paper focuses on the case where the (1,1) block is ill conditioned or rank deficient and the k x k (2,2) block is non zero and small (k << n). Additionally, the (2,1) block may be rank deficient. Such systems arise in a range of practical applications. A novel null-space approach is proposed that transforms the system matrix into a nicer symmetric saddle point matrix of order n that has a non zero (2,2) block of order at most 2k and, importantly, the(1,1)$ block is symmetric positive definite. Success of any null-space approach depends on constructing a suitable null-space basis. We propose methods for wide matrices having far fewer rows than columns with the aim of balancing stability of the transformed saddle point matrix with preserving sparsity in the (1,1) block. Linear least squares problems that contain a small number of dense rows are an important motivation and are used to illustrate our ideas and to explore their potential for solving large-scale systems.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/102016
Identification Number/DOI	10.1007/s11075-021-01245-z
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	Springer
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