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SPAI Preconditioners for HPC Applications
Author(s) -
Sawyer William,
Vanini Carlo,
Fourestey Gilles,
Popescu Radu
Publication year - 2012
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201210314
Subject(s) - preconditioner , solver , linear system , linear algebra , computer science , conjugate gradient method , generalized minimal residual method , parallel computing , porting , sparse matrix , computational science , iterative method , algorithm , mathematics , software , mathematical analysis , physics , programming language , geometry , quantum mechanics , gaussian
Iterative methods to solve systems of linear equations Ax = b usually require preconditioners M to speed convergence. But the calculation of many preconditioners can be notoriously sequential. The sparse approximate inverse preconditioner (SPAI) has particular potential for high performance computing [1]. We have ported the SPAI algorithm to graphical processing units (GPUs) within NVIDIA's CUSP library [2] for sparse linear algebra. GPUs perform well on dense linear algebra problems where data resides for long periods on the device. Since the underlying minimization problems are independent, they are mapped to separate thread‐blocks, and an optimized QR algorithm implemented using NVIDIA's CUBLAS library is employed on each. Traditionally the challenge has been to determine a sparsity pattern Sp ( M ) of the preconditioner dynamically [3], which reduces the condition number of MA to a degree where a preconditioned iterative solver such as GMRES becomes computationally viable. Due to the extremely high performance of the GPU, it is possible to consider initial sparsity patterns much denser than have been previously considered. We therefore consider a fixed sparsity patterns to simplify the GPU implementation. We evaluate the performance of the resulting preconditioner on a standard set of sparse matrices, and compare SPAI to other preconditioners. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)