Level-based Incomplete LU Factorization: Graph Model and Algorithms

A graph theoretic process that models level-based, incomplete LU factorization (ILU(`)) of sparse unsymmetric matrices is developed. The model leads to two incomplete fill path theorems that are generalizations of the original fill path theorem of Rose, Tarjan, and Lueker. Our S-level incomplete fill path theorem leads to the development of new, embarrassingly parallel algorithms for computing the structure and storage requirements of ILU(`) factors.

[1]  O. Axelsson.  Iterative solution methods , 1995 .

[2]  D. Rose,et al.  Algorithmic aspects of vertex elimination on directed graphs. , 1975 .

[3]  E. D'Azevedo,et al.  Towards a cost-effective ILU preconditioner with high level fill , 1992 .

[4]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[5]  Alex Pothen,et al.  New Sequential and Scalable Parallel Algorithms for Incomplete Factor Preconditioning , 2001 .

[6]  Joseph W. H. Liu,et al.  Exploiting Structural Symmetry in Unsymmetric Sparse Symbolic Factorization , 1992, SIAM J. Matrix Anal. Appl..

[7]  J. Meijerink,et al.  An iterative solution method for linear systems of which the coefficient matrix is a symmetric -matrix , 1977 .

[8]  Y. Saad,et al.  Experimental study of ILU preconditioners for indefinite matrices , 1997 .

[9]  R. K. Shyamasundar,et al.  Introduction to algorithms , 1996 .

[10]  Joseph W. H. Liu,et al.  Elimination Structures for Unsymmetric Sparse $LU$ Factors , 1993, SIAM J. Matrix Anal. Appl..

[11]  Alex Pothen,et al.  A Scalable Parallel Algorithm for Incomplete Factor Preconditioning , 2000, SIAM J. Sci. Comput..

[12]  S. Parter.  The Use of Linear Graphs in Gauss Elimination , 1961 .

[13]  Joseph W. H. Liu.  The role of elimination trees in sparse factorization , 1990 .

[14]  Robert E. Tarjan,et al.  Algorithmic Aspects of Vertex Elimination on Graphs , 1976, SIAM J. Comput..

[15]  I. Gustafsson.  A class of first order factorization methods , 1978 .