A Multilevel Dual Reordering Strategy for Robust \\ Incomplete LU Factorization of Indefinite Matrices

Jun Zhang
Department of Computer Science
University of Kentucky
773 Anderson Hall
Lexington, KY 40506--0046, USA


A dual reordering strategy based on both threshold and graph reorderings is introduced to construct robust incomplete LU (ILU) factorization of indefinite matrices. The ILU matrix is constructed as a preconditioner for the original matrix to be used in a preconditioned iterative scheme. The matrix is first divided into two parts according to a threshold parameter to control diagonal dominance. The first part with large diagonal dominance is reordered using a graph based strategy, followed by an ILU factorization. A partial ILU factorization is applied to the second part to yield an approximate Schur complement matrix. The whole process is repeated on the Schur complement matrix and continues for a few times to yield a multilevel ILU factorization. Analyses are conducted to show how the Schur complement approach removes small diagonal elements of indefinite matrices and how the stability of the LU factor affects the quality of the preconditioner. Numerical results are used to compare the new preconditioning strategy with two popular ILU preconditioning techniques and a multilevel block ILUT preconditioner.

Key words: Reordering strategies, sparse matrices, incomplete LU factorization, multilevel ILU preconditioner.

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This paper has been published in SIAM Journal on Matrix Analysis and Applications, Vol. 22, Num. 3, pp. 925-947 (2001). Technical Report 285-99, Department of Computer Science, University of Kentucky, Lexington, KY, 1999. This work was supported in part by the U.S. National Science Foundation under grants CCR-9902022, CCR-9988165 and CCR-0043861, and in part by the University of Kentucky Center for Computational Sciences and by the University of Kentucky College of Engineering.