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A multi-GPU parallel optimization model for the preconditioned conjugate gradient algorithm

PARALLEL COMPUTING 2017年 63卷 1-16页

作者： Gao, Jiaquan Zhou, Yuanshen He, Guixia Xia, Yifei Nanjing Normal Univ Sch Comp Sci & Technol Nanjing 210023 Jiangsu Peoples R China Zhejiang Univ Technol Coll Comp Sci & Technol Hangzhou 310023 Zhejiang Peoples R China Zhejiang Univ Technol Zhijiang Coll Hangzhou 310024 Zhejiang Peoples R China Chinese Acad Sci Inst Comp Technol State Key Lab Comp Architecture Beijing 100190 Peoples R China

In this study, we present a novel optimization model that can automatically and rapidly generate an optimally parallel preconditioned conjugate gradient (PCG) algorithm for any given linear system on a specific multi-graphics processing unit (GPU) platform. For our proposed model, there are the following novelties: (1) a profile-based performance model for each one of the main components of the PCG algorithm, including the vector operation, inner product, and sparse matrix-vector multiplication (SpMV), is suggested, and (2) our model is general, independent of the problems, and only dependent on the resources of devices, and (3) our model is extensible. For a vector operation kernel, or inner product kernel, or SpMV kernel that is not included in our framework, once its performance model is successfully constructed, it can be incorporated into our framework. Our model is constructed only once for each type of GPU. The experiments validate the high efficiency of our proposed model. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Optimization model preconditioned conjugate gradient algorithm CUDA Multiple GPUs

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On the complexity of the preconditioned conjugate gradient algorithm for solving Toeplitz systems with a Fisher-Hartwig singularity

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SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS 2005年第3期27卷 638-653页

作者： Lu, Y Hurvich, CM NYU Stern Sch Business New York NY 10012 USA

The Toeplitz matrix T-n with generating function f(omega) = |1 - e(-i omega)|(-2d)h(omega), where d is an element of (- (1)/(2), (1)/(2)) \ {0} and h(omega) is positive, continuous on [- pi, pi], and differentiable on... 详细信息

The Toeplitz matrix T-n with generating function f(omega) = |1 - e(-i omega)|(-2d)h(omega), where d is an element of (- (1)/(2), (1)/(2)) \ preconditioned and h(omega) is positive, continuous on [- pi, pi], and differentiable on [-pi, pi] \ preconditioned, has a Fisher - Hartwig singularity [ M. E. Fisher and R. E. Hartwig (1968), Adv. Chem. Phys., 32, pp. 190 - 225]. The complexity of the preconditioned conjugate gradient (PCG) algorithm is known [R. H. Chan and M. Ng (1996), SIAM Rev., 38, pp. 427 - 482] to be O(nlogn) for Toeplitz systems when d = 0. However, the effect on the PCG algorithm of the Fisher - Hartwig singularity in Tn has not been explored in the literature. We show that the complexity of the conjugate gradient (CG) algorithm for solving T(n)x = b without any preconditioning grows asymptotically as n(1+|d|) log(n). With T. Chan's optimal circulant preconditioner C-n [T. Chan (1988), SIAM J. Sci. Statist. Comput., 9, pp. 766 - 771], the complexity of the PCG algorithm is O(nlog(3)(n)).

关键词： conjugate gradient algorithm preconditioned conjugate gradient algorithm Toeplitz matrix circulant matrix Fisher-Hartwig singularity expected periodogram spectral density

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An improved Dai-Kou conjugate gradient algorithm for unconstrained optimization

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2020年第1期75卷 145-167页

作者： Liu, Zexian Liu, Hongwei Dai, Yu-Hong Xidian Univ Sch Math & Stat Xian 710126 Shaanxi Peoples R China Chinese Acad Sci Acad Math & Syst Sci ICMSEC LSEC Beijing 100190 Peoples R China

It is gradually accepted that the loss of orthogonality of the gradients in a conjugate gradient algorithm may decelerate the convergence rate to some extent. The Dai-Kou conjugate gradient algorithm (SIAM J Optim 23(1):296-320, 2013), called CGOPT, has attracted many researchers' attentions due to its numerical efficiency. In this paper, we present an improved Dai-Kou conjugate gradient algorithm for unconstrained optimization, which only consists of two kinds of iterations. In the improved Dai-Kou conjugate gradient algorithm, we develop a new quasi-Newton method to improve the orthogonality by solving the subproblem in the subspace and design a modified strategy for the choice of the initial stepsize for improving the numerical performance. The global convergence of the improved Dai-Kou conjugate gradient algorithm is established without the strict assumptions in the convergence analysis of other limited memory conjugate gradient methods. Some numerical results suggest that the improved Dai-Kou conjugate gradient algorithm (CGOPT (2.0)) yields a tremendous improvement over the original Dai-Kou CG algorithm (CGOPT (1.0)) and is slightly superior to the latest limited memory conjugate gradient software package CG_DESCENT (6.8) developed by Hager and Zhang (SIAM J Optim 23(4):2150-2168, 2013) for the CUTEr library.

关键词： conjugate gradient algorithm Limited memory Quasi-Newton method preconditioned conjugate gradient algorithm Global convergence

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A fast parallel Poisson solver on irregular domains applied to beam dynamics simulations

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JOURNAL OF COMPUTATIONAL PHYSICS 2010年第12期229卷 4554-4566页

作者： Adelmann, A. Arbenz, P. Ineichen, Y. Paul Scherrer Inst CH-5234 Villigen Switzerland ETH Chair Computat Sci CH-8092 Zurich Switzerland

We discuss the scalable parallel solution of the Poisson equation within a Particle-In-Cell (PIC) code for the simulation of electron beams in particle accelerators of irregular shape. The problem is discretized by Finite Differences. Depending on the treatment of the Dirichlet boundary the resulting system of equations is symmetric or 'mildly' nonsymmetric positive definite. In all cases, the system is solved by the preconditioned conjugate gradient algorithm with smoothed aggregation (SA) based algebraic multigrid (AMG) preconditioning. We investigate variants of the implementation of SA-AMG that lead to considerable improvements in the execution times. We demonstrate good scalability of the solver on distributed memory parallel processor with up to 2048 processors. We also compare our iterative solver with an FFT-based solver that is more commonly used for applications in beam dynamics. (C) 2010 Elsevier Inc. All rights reserved.

关键词： Poisson equation Irregular domains preconditioned conjugate gradient algorithm Algebraic multigrid Beam dynamics Space-charge

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CUDA-based solver for large-scale groundwater flow simulation

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ENGINEERING WITH COMPUTERS 2012年第1期28卷 13-19页

作者： Ji, Xiaohui Cheng, Tangpei Wang, Qun China Univ Geosci Sch Informat Engn Beijing Peoples R China

This article presents a parallel simulation solver for groundwater flow on CUDA. preconditioned conjugate gradient (PCG) algorithm is used to solve the large linear systems arising from the finite-difference discretization of three-dimensional groundwater flow problems. CUDA implementing methods for the two most time-consuming operations in PCG, sparse matrix-vector multiplication and vector inner-product, are given. The experimental results show that CUDA can speed up the solving process of the groundwater simulation significantly. 1.8-3.7 speedup can be achieved with different GPUs for a transient groundwater flow problem.

关键词： Groundwater simulation GPU preconditioned conjugate gradient algorithm CUDA

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Multiplicative Schwarz algorithms for the p-version Galerkin boundary element method in 3D

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APPLIED NUMERICAL MATHEMATICS 2006年第10-11期56卷 1370-1382页

作者： Maischak, Matthias Univ Hannover Inst Angew Math Hannover Germany

We study a 2-level multiplicative Schwarz method for the p version Galerkin boundary element method for a weakly singular integral equation of the first kind in 3D. We prove that the rate of convergence of the multiplicative Schwarz operator for the p version grows only logarithmically in p and is independent of h. (c) 2006 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： p-version boundary integral equation method multiplicative Schwarz operator preconditioned conjugate gradient algorithm

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A dimensionality-reduction genomic prediction method without direct inverse of the genomic relationship matrix for large genomic data

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PLANT CELL REPORTS 2023年第11期42卷 1825-1832页

作者： Liu, Hailan Yu, Shizhou Sichuan Agr Univ Maize Res Inst Chengdu 611130 Sichuan Peoples R China Guizhou Acad Tobacco Sci Mol Genet Key Lab China Tobacco Guiyang 550081 Guizhou Peoples R China

Key messageA new genomic prediction method (RHPP) was developed via combining randomized Haseman-Elston regression (RHE-reg), PCR based on genomic information of core population, and preconditioned conjugate gradient (PCG) *** efficiency is becoming a hot issue in the practical application of genomic prediction due to the large number of data generated by the high-throughput genotyping technology. In this study, we developed a fast genomic prediction method RHPP via combining randomized Haseman-Elston regression (RHE-reg), PCR based on genomic information of core population, and preconditioned conjugate gradient (PCG) algorithm. The simulation results demonstrated similar prediction accuracy between RHPP and GBLUP, and significantly higher computational efficiency of the former with the increase of individuals. The results of real datasets of both bread wheat and loblolly pine demonstrated that RHPP had a similar or better predictive accuracy in most cases compared with GBLUP. In the future, RHPP may be an attractive choice for analyzing large-scale and high-dimensional data.

关键词： Genomic prediction Randomized Haseman-Elston regression preconditioned conjugate gradient algorithm Principal component regression GBLUP

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Multilevel additive Schwarz method for the h-p version of the Galerkin boundary element method

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MATHEMATICS OF COMPUTATION 1998年第222期67卷 501-518页

作者： Heuer, N Stephan, EP Tran, T Univ Bremen Inst Wissensch Datenverarbeitung D-28334 Bremen Germany Leibniz Univ Hannover Inst Angew Math D-30167 Hannover Germany Univ New S Wales Sch Math Sydney NSW 2052 Australia

We study a multilevel additive Schwarz method for the h-p version of the Galerkin boundary element method with geometrically graded meshes. Both hypersingular and weakly singular integral equations of the first kind are considered. As it is well known the h-p version with geometric meshes converges exponentially fast in the energy-norm. However, the condition number of the Galerkin matrix in this case blows up exponentially in the number of unknowns M. We prove that the condition number kappa(P) of the multilevel additive Schwarz operator behaves like O(root Mlog(2) M). Asa direct consequence of this we also give the results for the 2-level preconditioner and also for the h-p version with quasi-uniform meshes. Numerical results supporting our theory are presented.

关键词： h-p version boundary integral equation method additive Schwarz operator multilevel method preconditioned conjugate gradient algorithm

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Technical note: A successive over-relaxation preconditioner to solve mixed model equations for genetic evaluation

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JOURNAL OF ANIMAL SCIENCE 2016年第11期94卷 4530-4535页

作者： Meyer, K. Univ New England Anim Genet & Breeding Unit Armidale NSW 2351 Australia

A computationally efficient preconditioned conjugate gradient algorithm with a symmetric successive over-relaxation (SSOR) preconditioner for the iterative solution of set mixed model equations is described. The potential computational savings of this approach are examined for an example of single-step genomic evaluation of Australian sheep. Results show that the SSOR pre-conditioner can substantially reduce the number of iterates required for solutions to converge compared with simpler preconditioners with marked reductions in overall computing time.

关键词： computational requirements genetic evaluation preconditioned conjugate gradient algorithm symmetric successive over-relaxation preconditioner

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Improvements of a Fast Parallel Poisson Solver on Irregular Domains

Improvements of a Fast Parallel Poisson Solver on Irregular ...

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10th Nordic International Conference on Applied Parallel Computing - State of the Art in Scientific and Parallel Computing (PARA)

作者： Adelmann, Andreas Arbenz, Peter Ineichen, Yves ETH Chair Computat Sci Zurich Switzerland Paul Scherrer Inst Villigen Switzerland

ISBN: (纸本)9783642281501;9783642281518

We discuss the scalable parallel solution of the Poisson equation on irregularly shaped domains discretized by finite differences. The symmetric positive definite system is solved by the preconditioned conjugate gradient algorithm with smoothed aggregation (SA) based algebraic multigrid (AMG) preconditioning. We investigate variants of the implementation of SA-AMG that lead to considerable improvements in the execution times. The improvements are due to a better data partitioning and the iterative solution of the coarsest level system in AMG. We demonstrate good scalability of the solver on a distributed memory parallel computer with up to 2048 processors.

关键词： Poisson equation finite differences preconditioned conjugate gradient algorithm algebraic multigrid data partitioning

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