Parallel Symbolic Cholesky Factorization
Pages 1721 - 1727
Abstract
We present a hybrid sequential/parallel symbolic Cholesky factorization algorithm that computes the sparsity pattern of the symbolic factors in parallel. We evaluate the performance on a large subset of the SuiteSparse matrix collection and multicore CPUs as well as flagship GPUs by AMD and NVIDIA, achieving speedups of an order of magnitude compared to a state-of-the-art sequential symbolic Cholesky factorization.
Supplemental Material
MP4 File
Recording of "Parallel Symbolic Cholesky Factorization" presentation at ScalAH'23.
- Download
- 178.75 MB
ZIP File
This artifact contains the code used to produce the results in the paper, as well as an artifact description listing the steps necessary to reproduce them.
- Download
- 42.00 MB
References
[1]
Patrick R. Amestoy, Timothy A. Davis, and Iain S. Duff. 1996. An Approximate Minimum Degree Ordering Algorithm. SIAM J. Matrix Anal. Appl. 17, 4 (Oct. 1996), 886–905. https://doi.org/10.1137/S0895479894278952 Publisher: Society for Industrial and Applied Mathematics.
[2]
Hartwig Anzt, Terry Cojean, Goran Flegar, Fritz Göbel, Thomas Grützmacher, Pratik Nayak, Tobias Ribizel, Yuhsiang Mike Tsai, and Enrique S. Quintana-Ortí. 2022. Ginkgo: A Modern Linear Operator Algebra Framework for High Performance Computing. ACM Trans. Math. Software 48, 1 (Feb. 2022), 2:1–2:33. https://doi.org/10.1145/3480935
[3]
Yanqing Chen, Timothy A. Davis, William W. Hager, and Sivasankaran Rajamanickam. 2008. Algorithm 887: CHOLMOD, Supernodal Sparse Cholesky Factorization and Update/Downdate. ACM Trans. Math. Software 35, 3 (Oct. 2008), 22:1–22:14. https://doi.org/10.1145/1391989.1391995
[4]
Timothy A. Davis. 2006. Direct methods for sparse linear systems. Society for Industrial and Applied Mathematics, Philadelphia.
[5]
Timothy A. Davis and William W. Hager. 2009. Dynamic Supernodes in Sparse Cholesky Update/Downdate and Triangular Solves. ACM Trans. Math. Software 35, 4 (Feb. 2009), 27:1–27:23. https://doi.org/10.1145/1462173.1462176
[6]
Timothy A. Davis and Yifan Hu. 2011. The University of Florida sparse matrix collection. ACM Trans. Math. Software 38, 1 (Dec. 2011), 1:1–1:25. https://doi.org/10.1145/2049662.2049663
[7]
Alan George, Micheal T Heath, Esmond Ng, and Joseph Liu. 1987. Symbolic Cholesky factorization on a local-memory multiprocessor. Parallel Comput. 5, 1 (July 1987), 85–95. https://doi.org/10.1016/0167-8191(87)90009-3
[8]
John R Gilbert and Hjálmtýr Hafsteinsson. 1990. Parallel symbolic factorization of sparse linear systems. Parallel Comput. 14, 2 (June 1990), 151–162. https://doi.org/10.1016/0167-8191(90)90104-H
[9]
George Karypis and Vipin Kumar. 1998. A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs. SIAM J. Sci. Comput. 20, 1 (Dec. 1998), 359–392. https://doi.org/10.1137/S1064827595287997
[10]
Joseph W.H. Liu. 1990. The Role of Elimination Trees in Sparse Factorization. SIAM J. Matrix Anal. Appl. 11, 1 (Jan. 1990), 134–172. https://doi.org/10.1137/0611010 Publisher: Society for Industrial and Applied Mathematics.
[11]
Donald J. Rose and Robert Endre Tarjan. 1978. Algorithmic Aspects of Vertex Elimination on Directed Graphs. SIAM J. Appl. Math. 34, 1 (Jan. 1978), 176–197. https://doi.org/10.1137/0134014 Publisher: Society for Industrial and Applied Mathematics.
[12]
Robert Schreiber. 1982. A New Implementation of Sparse Gaussian Elimination. ACM Trans. Math. Softw. 8, 3 (sep 1982), 256–276. https://doi.org/10.1145/356004.356006
[13]
P Sreenivasa Kumar, M Kishore Kumar, and A Basu. 1992. A parallel algorithm for elimination tree computation and symbolic factorization. Parallel Comput. 18, 8 (Aug. 1992), 849–856. https://doi.org/10.1016/0167-8191(92)90031-2
[14]
Ole Tange. 2020. GNU Parallel 20200522 (’Kraftwerk’). https://doi.org/10.5281/zenodo.3841377 GNU Parallel is a general parallelizer to run multiple serial command line programs in parallel without changing them.
[15]
Earl Zmijewski and John R Gilbert. 1988. A parallel algorithm for sparse symbolic Cholesky factorization on a multiprocessor. Parallel Comput. 7, 2 (June 1988), 199–210. https://doi.org/10.1016/0167-8191(88)90039-7
Index Terms
- Parallel Symbolic Cholesky Factorization
Recommendations
Algorithmic Aspects of Elimination Trees for Sparse Unsymmetric Matrices
The elimination tree of a symmetric matrix plays an important role in sparse elimination. We recently defined a generalization of this structure to the unsymmetric case that retains many of its properties. Here we present an algorithm for constructing ...
Finding Optimal Ordering of Sparse Matrices for Column-Oriented Parallel Cholesky Factorization
In this paper, we consider the problem of finding fill-preserving sparse matrix orderings for parallel factorization. That is, given a large sparse symmetric and positive definite matrix A that has been ordered by some fill-reducing ordering, we want to ...
An Efficient Algorithm to Compute Row and Column Counts for Sparse Cholesky Factorization
Let an undirected graph $G$ be given, along with a specified depth-first spanning tree $T$. Almost-linear-time algorithms are given to solve the following two problems. First, for every vertex $v$, compute the number of descendants $w$ of $v$ for which ...
Comments
Information & Contributors
Information
Published In
November 2023
2180 pages
ISBN:9798400707858
DOI:10.1145/3624062
Copyright © 2023 ACM.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 12 November 2023
Check for updates
Author Tags
Qualifiers
- Research-article
- Research
- Refereed limited
Funding Sources
Conference
SC-W 2023
SC-W 2023: Workshops of The International Conference on High Performance Computing, Network, Storage, and Analysis
November 12 - 17, 2023
CO, Denver, USA
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 43Total Downloads
- Downloads (Last 12 months)43
- Downloads (Last 6 weeks)5
Reflects downloads up to 17 Oct 2024
Other Metrics
Citations
View Options
Get Access
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign inFull Access
View options
View or Download as a PDF file.
PDFeReader
View online with eReader.
eReaderHTML Format
View this article in HTML Format.
HTML Format