Communication-Avoiding Parallel Algorithms for Solving Triangular Systems of Linear Equations

Tobias Wicky, Edgar Solomonik, Torsten Hoefler

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We present a new parallel algorithm for solving triangular systems with multiple right hand sides (TRSM). TRSM is used extensively in numerical linear algebra computations, both to solve triangular linear systems of equations as well as to compute factorizations with triangular matrices, such as Cholesky, LU, and QR. Our algorithm achieves better theoretical scalability than known alternatives, while maintaining numerical stability, via selective use of triangular matrix inversion. We leverage the fact that triangular inversion and matrix multiplication are more parallelizable than the standard TRSM algorithm. By only inverting triangular blocks along the diagonal of the initial matrix, we generalize the usual way of TRSM computation and the full matrix inversion approach. This flexibility leads to an efficient algorithm for any ratio of the number of right hand sides to the triangular matrix dimension. We provide a detailed communication cost analysis for our algorithm as well as for the recursive triangular matrix inversion. This cost analysis makes it possible to determine optimal block sizes and processor grids a priori. Relative to the best known algorithms for TRSM, our approach can require asymptotically fewer messages, while performing optimal amounts of computation and communication in terms of words sent.

Original languageEnglish (US)
Title of host publicationProceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages678-687
Number of pages10
ISBN (Electronic)9781538639146
DOIs
StatePublished - Jun 30 2017
Event31st IEEE International Parallel and Distributed Processing Symposium, IPDPS 2017 - Orlando, United States
Duration: May 29 2017Jun 2 2017

Publication series

NameProceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017

Other

Other31st IEEE International Parallel and Distributed Processing Symposium, IPDPS 2017
CountryUnited States
CityOrlando
Period5/29/176/2/17

Fingerprint

Linear equations
Parallel algorithms
Computer systems
Communication
Linear algebra
Convergence of numerical methods
Factorization
Linear systems
Scalability
Costs

Keywords

  • 3D algorithms
  • TRSM
  • communication cost

ASJC Scopus subject areas

  • Information Systems
  • Computer Networks and Communications
  • Hardware and Architecture

Cite this

Wicky, T., Solomonik, E., & Hoefler, T. (2017). Communication-Avoiding Parallel Algorithms for Solving Triangular Systems of Linear Equations. In Proceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017 (pp. 678-687). [7967158] (Proceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/IPDPS.2017.104

Communication-Avoiding Parallel Algorithms for Solving Triangular Systems of Linear Equations. / Wicky, Tobias; Solomonik, Edgar; Hoefler, Torsten.

Proceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017. Institute of Electrical and Electronics Engineers Inc., 2017. p. 678-687 7967158 (Proceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Wicky, T, Solomonik, E & Hoefler, T 2017, Communication-Avoiding Parallel Algorithms for Solving Triangular Systems of Linear Equations. in Proceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017., 7967158, Proceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017, Institute of Electrical and Electronics Engineers Inc., pp. 678-687, 31st IEEE International Parallel and Distributed Processing Symposium, IPDPS 2017, Orlando, United States, 5/29/17. https://doi.org/10.1109/IPDPS.2017.104
Wicky T, Solomonik E, Hoefler T. Communication-Avoiding Parallel Algorithms for Solving Triangular Systems of Linear Equations. In Proceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017. Institute of Electrical and Electronics Engineers Inc. 2017. p. 678-687. 7967158. (Proceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017). https://doi.org/10.1109/IPDPS.2017.104
Wicky, Tobias ; Solomonik, Edgar ; Hoefler, Torsten. / Communication-Avoiding Parallel Algorithms for Solving Triangular Systems of Linear Equations. Proceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017. Institute of Electrical and Electronics Engineers Inc., 2017. pp. 678-687 (Proceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017).
@inproceedings{90de727cef90415f85208227e9251b0f,
title = "Communication-Avoiding Parallel Algorithms for Solving Triangular Systems of Linear Equations",
abstract = "We present a new parallel algorithm for solving triangular systems with multiple right hand sides (TRSM). TRSM is used extensively in numerical linear algebra computations, both to solve triangular linear systems of equations as well as to compute factorizations with triangular matrices, such as Cholesky, LU, and QR. Our algorithm achieves better theoretical scalability than known alternatives, while maintaining numerical stability, via selective use of triangular matrix inversion. We leverage the fact that triangular inversion and matrix multiplication are more parallelizable than the standard TRSM algorithm. By only inverting triangular blocks along the diagonal of the initial matrix, we generalize the usual way of TRSM computation and the full matrix inversion approach. This flexibility leads to an efficient algorithm for any ratio of the number of right hand sides to the triangular matrix dimension. We provide a detailed communication cost analysis for our algorithm as well as for the recursive triangular matrix inversion. This cost analysis makes it possible to determine optimal block sizes and processor grids a priori. Relative to the best known algorithms for TRSM, our approach can require asymptotically fewer messages, while performing optimal amounts of computation and communication in terms of words sent.",
keywords = "3D algorithms, TRSM, communication cost",
author = "Tobias Wicky and Edgar Solomonik and Torsten Hoefler",
year = "2017",
month = "6",
day = "30",
doi = "10.1109/IPDPS.2017.104",
language = "English (US)",
series = "Proceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
pages = "678--687",
booktitle = "Proceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017",
address = "United States",

}

TY - GEN

T1 - Communication-Avoiding Parallel Algorithms for Solving Triangular Systems of Linear Equations

AU - Wicky, Tobias

AU - Solomonik, Edgar

AU - Hoefler, Torsten

PY - 2017/6/30

Y1 - 2017/6/30

N2 - We present a new parallel algorithm for solving triangular systems with multiple right hand sides (TRSM). TRSM is used extensively in numerical linear algebra computations, both to solve triangular linear systems of equations as well as to compute factorizations with triangular matrices, such as Cholesky, LU, and QR. Our algorithm achieves better theoretical scalability than known alternatives, while maintaining numerical stability, via selective use of triangular matrix inversion. We leverage the fact that triangular inversion and matrix multiplication are more parallelizable than the standard TRSM algorithm. By only inverting triangular blocks along the diagonal of the initial matrix, we generalize the usual way of TRSM computation and the full matrix inversion approach. This flexibility leads to an efficient algorithm for any ratio of the number of right hand sides to the triangular matrix dimension. We provide a detailed communication cost analysis for our algorithm as well as for the recursive triangular matrix inversion. This cost analysis makes it possible to determine optimal block sizes and processor grids a priori. Relative to the best known algorithms for TRSM, our approach can require asymptotically fewer messages, while performing optimal amounts of computation and communication in terms of words sent.

AB - We present a new parallel algorithm for solving triangular systems with multiple right hand sides (TRSM). TRSM is used extensively in numerical linear algebra computations, both to solve triangular linear systems of equations as well as to compute factorizations with triangular matrices, such as Cholesky, LU, and QR. Our algorithm achieves better theoretical scalability than known alternatives, while maintaining numerical stability, via selective use of triangular matrix inversion. We leverage the fact that triangular inversion and matrix multiplication are more parallelizable than the standard TRSM algorithm. By only inverting triangular blocks along the diagonal of the initial matrix, we generalize the usual way of TRSM computation and the full matrix inversion approach. This flexibility leads to an efficient algorithm for any ratio of the number of right hand sides to the triangular matrix dimension. We provide a detailed communication cost analysis for our algorithm as well as for the recursive triangular matrix inversion. This cost analysis makes it possible to determine optimal block sizes and processor grids a priori. Relative to the best known algorithms for TRSM, our approach can require asymptotically fewer messages, while performing optimal amounts of computation and communication in terms of words sent.

KW - 3D algorithms

KW - TRSM

KW - communication cost

UR - http://www.scopus.com/inward/record.url?scp=85027683884&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85027683884&partnerID=8YFLogxK

U2 - 10.1109/IPDPS.2017.104

DO - 10.1109/IPDPS.2017.104

M3 - Conference contribution

AN - SCOPUS:85027683884

T3 - Proceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017

SP - 678

EP - 687

BT - Proceedings - 2017 IEEE 31st International Parallel and Distributed Processing Symposium, IPDPS 2017

PB - Institute of Electrical and Electronics Engineers Inc.

ER -