On Signed Incomplete Cholesky Factorization Preconditioners for Saddle-Point Systems

Abstract

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 $LDL^T$ is proposed, where the diagonal matrix $D$ has entries $\pm 1$. 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 postprocessed 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 d...