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

(In Proceedings of the 31st IEEE International Parallel & Distributed Processing Symposium (IPDPS'17), presented in Orlando, FL, USA, IEEE, May 2017)

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 com-
putations, 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
an asymptotic factor fewer messages, while performing optimal
amounts of communication and computation.

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BibTeX

@inproceedings{comm-avoiding-triang, author={T. Wicky and Edgar Solomonik and Torsten Hoefler}, title={{Communication-Avoiding Parallel Algorithms for Solving Triangular Systems of Linear Equations}}, year={2017}, month={May}, booktitle={Proceedings of the 31st IEEE International Parallel \& Distributed Processing Symposium (IPDPS'17)}, location={Orlando, FL, USA}, publisher={IEEE}, source={http://www.unixer.de/~htor/publications/}, }