On Signed Incomplete Cholesky Factorization Preconditioners for Saddle-Point Systems

Limited-memory incomplete Cholesky factorizations can provide robust preconditioners for sparse symmetric positive-definite linear systems. In this paper, the focus is on extending the approach to sparse symmetric indefinite systems in saddle-point form. A limited-memory signed incomplete Cholesky factorization of the form LDL is proposed, where the diagonal matrix D has entries ±1. The main advantage of this approach is its simplicity as it avoids the use of numerical pivoting. Instead, a global shift strategy involving two shifts (one for the (1, 1) block and one for the (2, 2) block of the saddle-point matrix) is used to prevent breakdown and to improve performance. The matrix is optionally prescaled and preordered using a standard sparse matrix ordering scheme that is then post-processed to give a constrained ordering that reduces the likelihood of breakdown and need for shifts. The use of intermediate memory (memory used in the construction of the incomplete factorization but subsequently discarded) is shown to significantly improve the performance of the resulting preconditioner. Some new theoretical results are presented and for problems arising from a range of practical applications, numerical results are given to illustrate the effectiveness of the signed incomplete Cholesky factorization as a preconditioner. Comparisons are made with a recent incomplete LDL code that employs pivoting.