The graphics processing unit (GPU) is used to solve large linear systems derived from partial differential equations. The differential equations studied are strongly convection-dominated, of various sizes, and common to many fields, including computational fluid dynamics, heat transfer, and structural mechanics. The paper presents comparisons between GPU and CPU implementations of several well-known iterative methods, including Kaczmarz's, Cimmino's, component averaging, conjugate gradient normal residual (CGNR), symmetric successive overrelaxation-preconditioned conjugate gradient, and conjugate-gradient-accelerated component-averaged row projections (CARP-CG). Computations are preformed with dense as well as general banded systems. The results demonstrate that our GPU implementation outperforms CPU implementations of these algorithms, as well as previously studied parallel implementations on Linux clusters and shared memory systems. While the CGNR method had begun to fall out of favor for solving such problems, for the problems studied in this paper, the CGNR method implemented on the GPU performed better than the other methods, including a cluster implementation of the CARP-CG method.

中文翻译：

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使用 Kaczmarz 和其他线性系统迭代算法的 GPU 计算。

图形处理单元 (GPU) 用于求解从偏微分方程导出的大型线性系统。所研究的微分方程以强对流为主导，大小各异，适用于许多领域，包括计算流体动力学、传热和结构力学。本文介绍了几种著名迭代方法的 GPU 和 CPU 实现之间的比较，包括 Kaczmarz 的、Cimmino 的、分量平均、共轭梯度正态残差 (CGNR)、对称连续过松弛预条件共轭梯度和共轭梯度加速分量平均行投影（CARP-CG）。使用密集和一般带状系统进行计算。结果表明，我们的 GPU 实现优于这些算法的 CPU 实现，以及之前研究过的 Linux 集群和共享内存系统上的并行实现。虽然 CGNR 方法已经开始不利于解决此类问题，但对于本文研究的问题，在 GPU 上实现的 CGNR 方法比其他方法表现更好，包括 CARP-CG 方法的集群实现。