Diagonal threshold techniques in robust multi-level ILU preconditioners for general sparse linear systems (Q2760356)

A general sparse linear system \(Au=b\) is considered. Reordering, threshold and Schur complement techniques in developing an efficient multi-elimination incomplete LU factorization preconditioner (ILUM) are studied. The reordering of \(A\) is based on independent sets of vertices. To find them, new heuristic strategies utilizing the adjacency graph and the diagonal values of \(A\) exceeding certain tolerance parameter are introduced. They are combined with a dropping strategy and the Schur complement calculation to factor the transformed \(A\) into the \(LU\) product of block matrices plus an error matrix \(R\) resulting from the dropping rule employed. The complexity of the system is reduced to, roughly speaking, only one block (Schur complement) which is either solved directly (two-level method) or further factored in the same way (multi-level method). A block variant (BILUM) is mentioned too. NEWLINENEWLINENEWLINEAnalytical bounds for \(R\) and \(L^{-1}RU^{-1}\) are obtained for the two-level method. NEWLINENEWLINENEWLINEExtensive numerical experiments are conducted in order to compare the robustness and efficiency of various heuristic strategies with the performance of ILUT, a dual threshold incomplete LU preconditioner. For ILUT, see \textit{Y.~Saad} [Numer.~Linear Algebra Appl.~1, No. 4, 387-402 (1994; Zbl 0838.65026)]. Test matrices come from various collections and originate from computational fluid dynamics mainly.