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Pipelined Preconditioned Conjugate Gradient Methods for real and complex linear systems for distributed memory architectures

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JOURNAL OF PARALLEL AND DISTRIBUTED COMPUTING 2022年 163卷 147-155页

作者： Tiwari, Manasi Vadhiyar, Sathish Indian Inst Sci Dept Computat & Data Sci Bangalore 560012 Karnataka India

Preconditioned Conjugate Gradient (PCG) is a popular method for solving large and sparse linear systems of equations. The performance of PCG at scale is affected due to the costly global synchronization steps that arise in dot-products on distributed memory systems. Pipelined PCG (PIPECG) removes the costly global synchronization steps from PCG by only executing a single non-blocking allreduce per iteration and overlapping it with independent *** our previous work, we have developed a novel pipelined PCG algorithm called PIPECG-OATI (One Allreduce per Two Iterations) for real linear systems which executes a single non-blocking allreduce per two iterations and provides a large overlap of global communication with independent computations at higher number of cores. Our method achieves this overlap by using iteration combination and by introducing new recurrence and non-recurrence computations. We implement optimizations in the PIPECG-OATI method to use cache memory efficiently. In this work, we present PIPECG-OATI-c method for linear systems with complex Hermitian positive definite and complex symmetric matrices. We compare our method with various pipelined CG methods on a variety of problems and demonstrate that our method always gives the least run times. We performed experiments with our method using 20M and 30M unknowns on up to 16K cores and obtained up to 2.48X performance improvement over PCG and 2.14X performance improvement over PIPECG methods. We also experimented with up to 1-billion unknowns on 16K cores, the largest problem size explored for the CG problem, to our knowledge, and obtained about 25% improvement over PCG.& nbsp;(c) 2022 Elsevier Inc. All rights reserved.

关键词： Preconditioned Conjugate Gradient Pipelining Complex symmetric systems overlapping communication and computations Complex Hermitian positive definite systems

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communication overlapping Pipelined Conjugate Gradients for Distributed Memory Systems and Heterogeneous Architectures 27th

Communication Overlapping Pipelined Conjugate Gradients for ...

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27th International European Conference on Parallel and Distributed Computing (Euro-Par)

作者： Tiwari, Manasi Vadhiyar, Sathish Indian Inst Sci Dept Computat & Data Sci Bangalore Karnataka India

ISBN: (纸本)9783031061561;9783031061554

Preconditioned Conjugate Gradient (PCG) method has been one of the widely used methods for solving linear systems of equations for sparse problems. Pipelined PCG (PIPECG) attempts to eliminate the dependencies in the computations in the PCG algorithm and overlap non-dependent computations by reorganizing the traditional PCG code and using non-blocking allreduces . We have developed a novel pipelined PCG algorithm called PIPECG-OATI (One Allreduce per Two Iterations) which reduces the number of non-blocking allreduces to one per two iterations and provides large overlap of global communication and computations at higher number of cores in distributed memory CPU systems. PIPECG-OATI gives up to 3x speedup over PCG and 1.73x speedup over PIPECG at large number of cores. For GPU accelerated heterogeneous architectures, we have developed three methods for efficient execution of the PIPECG algorithm. These methods achieve task and data parallelism. Our methods give considerable performance improvements over PCG CPU and GPU implementations of Paralution and PETSc libraries.

关键词： Preconditioned Conjugate Gradient overlapping communication and computations Distributed memory systems Heterogeneous architectures

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Pipelined Preconditioned Conjugate Gradient Methods for Distributed Memory Systems 27

Pipelined Preconditioned Conjugate Gradient Methods for Dist...

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27th IEEE International Conference on High Performance Computing, Data, and Analytics (HiPC)

作者： Tiwari, Manasi Vadhiyar, Sathish Indian Inst Sci Dept Computat & Data Sci Bengaluru India

ISBN: (纸本)9781665422925

Preconditioned Conjugate Gradient (PCG) method has been one of the widely used methods for solving linear systems of equations for sparse problems. Pipelined PCG (PIPECG) attempts to eliminate the dependencies in the computations in the PCG algorithm and overlap non-dependent computations by reorganizing the traditional PCG code and using non-blocking allreduces. We have developed a novel pipelined PCG algorithm called PIPECG-OATI (One Allreduce per Two Iterations) that provides large overlap of global communication and computations at higher number of cores in distributed memory CPU systems. Our method achieves this overlapping by using iteration combination and by introducing new non-recurrence computations. We compare our method with other pipelined CG methods on a variety of problems and demonstrate that our method always gives the least runtimes. Our method gives up to 3x speedup over PCG method and 1.73x speedup over PIPECG method at large number of cores.

关键词： Preconditioned Conjugate Gradient distributed memory systems pipelining overlapping communication and computations

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Pipelined Preconditioned s-step Conjugate Gradient Methods for Distributed Memory Systems

Pipelined Preconditioned s-step Conjugate Gradient Methods f...

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IEEE International Conference on Cluster Computing (Cluster)

作者： Tiwari, Manasi Vadhiyar, Sathish Indian Inst Sci Dept Computat & Data Sci Bengaluru India

ISBN: (纸本)9781728196664

Preconditioned Conjugate Gradient (PCG) method is a widely used iterative method for solving large linear systems of equations. Pipelined variants of PCG present independent computations in the PCG method and overlap these computations with non-blocking allreduces. We have developed a novel pipelined PCG algorithm called PIPE-sCG (Pipelined s-step Conjugate Gradient) that provides a large overlap of global communication and computations at higher number of cores in distributed memory CPU systems. Our method achieves this overlap by introducing new recurrence computations. We have also developed a preconditioned version of PIPE-sCG. The advantages of our methods are that they do not introduce any extra preconditioner or sparse matrix vector product kernels in order to provide the overlap and can work with preconditioned, unpreconditioned and natural norms of the residual, as opposed to the state-of-the-art methods. We compare our method with other pipelined CG methods for Poisson problems and demonstrate that our method gives the least runtimes. Our method gives up to 2.9x speedup over PCG method, 2.15x speedup over PIPECG method and 1.2x speedup over PIPECG-OATI method at large number of cores.

关键词： Preconditioned Conjugate Gradient Distributed Memory Systems Pipelining overlapping communication and computations s-step Methods

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