Incomplete factorization preconditioning for linear least squares problems
Wang, Xiaoge
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https://hdl.handle.net/2142/20281
Description
Title
Incomplete factorization preconditioning for linear least squares problems
Author(s)
Wang, Xiaoge
Issue Date
1994
Doctoral Committee Chair(s)
Gallivan, Kyle A.
Department of Study
Computer Science
Discipline
Computer Science
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Computer Science
Language
eng
Abstract
A new family of preconditioners for conjugate gradient-like iterative methods applied to large sparse linear least squares problems, $min\Vert Ax-b\Vert\sb2$, is proposed. The family is based on incomplete Gram-Schmidt (IGS) factorizations of A. Particular attention has been given to the following members of the family: Incomplete Classical Gram-Schmidt (ICGS), Incomplete Modified Gram-Schmidt (IMGS) and Compressed Incomplete Modified Gram-Schmidt (CIMGS) factorizations. The numerical properties of each of these methods have been considered as well as the relationships between the methods concerning the preservation of sparsity, computational efficiency and the quality of the preconditioner. The implementation of these methods has been investigated and all of the important family members have been coded. One of the important topics in this portion of the dissertation is the careful symbolic analysis of the production of the preconditioner and its use during the incomplete factorization phase to avoid excessive unnecessary work.
We have also analyzed some of the implications of viewing the incomplete Cholesky (IC) factorization as a limit case of the IGS family. When certain conditions are imposed on the sparsity pattern of the preconditioning matrix, IC is identical to IMGS. Therefore, a sufficient condition on the sparsity pattern for a stable IC factorization can be derived. Based on this condition, two algorithms for modifying a sparsity pattern for which IC may not succeed have been designed. The altered IC method with the pattern modifications (ICPM) has been studied and its performance evaluated. These methods differ from previously proposed methods to guarantee the existence and improve the stability of the IC factorization since they do not require numerical information. Of course, for a particular matrix we can then use numerical information in addition to the pattern modification techniques to further improve the preconditioner.
Numerical experiments illustrating the capabilities of the preconditioners are also presented. These include matrices from the standard Harwell-Boeing collection and various test matrices from different practical application problems.
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