Numerically Stable Recurrence Relations for the Communication Hiding Pipelined Conjugate Gradient Method

Pipelined Krylov subspace methods (also referred to as communication-hiding methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in parallel. For symmetric and positive definite s...

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Published in:IEEE transactions on parallel and distributed systems Vol. 30; no. 11; pp. 2507 - 2522
Main Authors: Cools, Siegfried, Cornelis, Jeffrey, Vanroose, Wim
Format: Journal Article
Language:English
Published: New York IEEE 01.11.2019
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
attainable accuracy
Communication
Conjugate gradient method
conjugate gradients
Conjugates
Distributed memory
Distributed processing
Global communication
Gradient methods
Hardware
inexact computations
Krylov subspace methods
latency hiding
Mathematical analysis
Matrix algebra
Matrix methods
Methods
Numerical stability
parallel performance
Pipeline processing
pipelining
Pipelining (computers)
Rounding
Stability
Subspace methods
Subspaces
Symmetric matrices
ISSN:1045-9219, 1558-2183
Online Access:Get full text
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Summary:Pipelined Krylov subspace methods (also referred to as communication-hiding methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in parallel. For symmetric and positive definite system matrices the pipelined Conjugate Gradient method, p(ll)-CG, outperforms its classic Conjugate Gradient counterpart on large scale distributed memory hardware by overlapping global communication with essential computations like the matrix-vector product, thus "hiding" global communication. A well-known drawback of the pipelining technique is the (possibly significant) loss of numerical stability. In this work a numerically stable variant of the pipelined Conjugate Gradient algorithm is presented that avoids the propagation of local rounding errors in the finite precision recurrence relations that construct the Krylov subspace basis. The multi-term recurrence relation for the basis vector is replaced by ℓ three-term recurrences, improving stability without increasing the overall computational cost of the algorithm. The proposed modification ensures that the pipelined Conjugate Gradient method is able to attain a highly accurate solution independently of the pipeline length. Numerical experiments demonstrate a combination of excellent parallel performance and improved maximal attainable accuracy for the new pipelined Conjugate Gradient algorithm. This work thus resolves one of the major practical restrictions for the useability of pipelined Krylov subspace methods.
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ISSN:1045-9219
1558-2183
DOI:10.1109/TPDS.2019.2917663