Adaptive Precision in Block-Jacobi Preconditioning for Iterative Sparse Linear System Solvers

Anzt, Hartwig and Dongarra, Jack and Flegar, Goran and Higham, Nicholas J. and Quintana-Orti, Enrique S. (2017) Adaptive Precision in Block-Jacobi Preconditioning for Iterative Sparse Linear System Solvers. [MIMS Preprint]

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Abstract

We propose an adaptive scheme to reduce communication overhead caused by data movement by selectively storing the diagonal blocks of a block Jacobi preconditioner in different precision formats (half, single, or double). This specialized preconditioner can then be combined with any Krylov subspace method for the solution of sparse linear systems to perform all arithmetic in double precision. We assess the effects of the adaptive-precision preconditioner on the iteration count and data transfer cost of a preconditioned conjugate gradient solver. A preconditioned conjugate gradient method is, in general, a memory-bound algorithm, and therefore its execution time and energy consumption are largely dominated by the costs of accessing the problem's data in memory. Given this observation, we propose a model that quantifies the time and energy savings of our approach based on the assumption that these two costs depend linearly on the bit length of a floating point number. Furthermore, we use a number of test problems from the SuiteSparse matrix collection to estimate the potential benefits of the adaptive block-Jacobi preconditioning scheme.

Item Type: MIMS Preprint
Uncontrolled Keywords: Sparse linear systems; Krylov subspace methods; conjugate gradient (CG) method; Jacobi preconditioners; adaptive precision; communication reduction; energy efficiency
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Nick Higham
Date Deposited: 22 Sep 2017
Last Modified: 08 Nov 2017 18:18
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2578

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