A parallel algorithm for sparse unsymmetric LU factorization

This thesis presents a parallel algorithm for the direct LU factorization of general unsymmetric sparse matrices. The algorithm, D2, is based on a new nondeterministic parallel pivot search that finds a compatible pivot set S of size m, followed by a parallel rank-m update. These two steps alternate until switching to dense matrix code or until the matrix is factored. The algorithm is based on a shared-memory MIMD model and takes advantage of both concurrency and (gather-scatter) vectorization. Experimental comparisons on an Alliant FX/8 show that the method results in shorter elimination trees for matrices with highly asymmetrical nonzero structure than those for previous methods that work with the symmetric structure of A + ${\bf A\sp{T}}$ or ${\bf A\sp{T}A}$ (such as George and Ng's Sparspak-C or Duff and Reid's multifrontal method). The algorithm exploits more parallelism in the pivot search phase than previous algorithms which do not force a symmetric structure onto the matrix during any phase of the factorization. Additional experimental comparisons include fillin, amount of work, numerical stability, memory usage, and run time. The nondeterministic behavior of the D2 algorithm and other performance metrics are analyzed on an Alliant FX/8, a Cray-2, and a Cray-XMP/48. Enhancements to PSolve, a pairwise pivoting algorithm, are discussed, and a software tool for developing sparse matrix algorithms and observing their dynamic behavior on a Sun workstation is presented. The tool was instrumental in the development of the D2 algorithm. Possible extensions to the D2 algorithm are discussed, including the use of dense matrix kernels and replacing the synchronization structure in the pivot search with a software combining tree.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Electrical Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Davis, Timothy Alden

Contributors dc:contributor

• Yew, Pen-Chung

• Mathematics
• Engineering, Electronics and Electrical
• Computer Science

Statement dc:rights

• Copyright 1989 Davis, Timothy Alden

Language dc:language

eng

Identifier

AAI9010841
(UMI)AAI9010841

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/20745