Modification and Compensation Strategies for Threshold-based Incomplete Factorizations

Abstract

Abstract. Standard (single-level) incomplete factorization preconditioners are known to successfully accelerate Krylov subspace iterations for many linear systems. The classical Modified Incomplete LU (MILU) factorization approach improves the acceleration given by (standard) ILU approaches, by modifying the non-unit diagonal in the factorization to match the action of the system matrix on a given vector, typically the constant vector. Here, we examine the role of similar modifications within the dual-threshold ILUT algorithm. We introduce column and row variants of the modified ILUT algorithm and discuss optimal ways of modifying the columns or rows of the computed factors to improve their accuracy and stability. Modifications are considered for both the diagonal and offdiagonal entries of the factors, based on one or many vectors, chosen a priori or through an Arnoldi iteration. Numerical results are presented to support our findings.

On Positive Semidefinite Modification Schemes for Incomplete Cholesky Factorization

“…In Figure 8, we compare running with and without Jennings-Malik modifications (that is, with and without lines [30][31][32][33][34][35]. We can clearly see that in terms of efficiency, time, and reliability, using modifications for the dropped entries is not, in general, beneficial.…”

“…Although devoted to nonsymmetric incomplete factorizations, the authors point out that a quantification of the norms of the updates could be further developed theoretically. Note that [33] discusses modifications and errors from the local point of view, as we do.…”

“…This removes the requirement that the preconditioner is positive definite. Another recent paper of interest here is that of Maclachlan, OseiKuffuor, and Saad [33]. Although devoted to nonsymmetric incomplete factorizations, the authors point out that a quantification of the norms of the updates could be further developed theoretically.…”

Improving the stability and robustness of incomplete symmetric indefinite factorization preconditioners

“…Note that other approaches to modifying A originated in solving discretized partial differential equations. An example of a general sophisticated modification strategy is given by MacLachlan, Osei‐Kuffuor, and Saad . However, they use a standard incomplete LU factorization that does not employ an intermediate factor R but uses local modifications.…”

Javelin: A Scalable Implementation for Sparse Incomplete LU Factorization

“…In general, these algorithms can be categorized by the method they use to reduce nonzeros due to fill-in: dropping based on numerical value (ILU(τ )) and dropping based on fill-in level (ILU(k)). The design of Javelin allows for any combination of the two to be used (ILU(k,τ )) along with modified ILU [2]. However, most of the current packages that can be used for traditional incomplete factorization are serial because of the difficulty of scaling and the overhead that may exist to reformat the output matrix in order to achieve scalable sparse triangular solves.…”