Matrix Reordering Using Multilevel Graph Coarsening for ILU Preconditioning

Incomplete LU factorization (ILU) techniques are a well-known class of preconditioners, often used in conjunction with Krylov accelerators for the iterative solution of linear systems of equations. However, for certain problems, ILU factorizations can yield factors that are unstable, and in some cases quite dense. Reordering techniques based on permuting the matrix prior to performing the factorization have been shown to improve the quality of the factorization, and the resulting preconditioner. In this paper, we examine the effect of reordering techniques based on multilevel graph coarsening ideas on the level-based ILU(k) and the dual threshold ILUT algorithms. We consider an aggregation-based coarsening idea that implements two main coarsening frameworks a top-down approach, and a bottom-up approach each utilizing one of two different strategies to select the next-level coarse graph. Numerical results are presented to support our findings.