On Convergence Speed of Parallel Variants of GPBiCG Method for Solving Linear Equations
On Convergence Speed of Parallel Variants of GPBiCG Method for Solving Linear Equations
复制标题
求解线性方程组GPBiCG方法并行变体的收敛速度
DOI：
10.1088/1742-6596/1391/1/012093
复制
发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Soichiro Ikuno
中科院分区：
文献类型：
--
作者：
Kuniyoshi Abe;Soichiro Ikuno
混合双共轭梯度（Bi-CG）方法如Bi-CG稳定的（Bi-CGSTAB）、基于广义乘积型的Bi-CG（GPBiCG）和BiCGstab（GPIB）是众所周知的有效求解线性方程组的方法。GPBiCG和BiCGstab（BSTAB）在强非对称矩阵问题上比Bi-CGSTAB更有效和鲁棒。在当前千万亿次高性能计算硬件上，Krylov子空间方法的可扩展性问题日益突出。高效并行化的主要瓶颈是需要全局约简的内积。已经提出了Bi-CGSTAB的并行变体，其减少了全局通信阶段的数量并隐藏了通信延迟。然而，据报道，Bi-CGSTAB的平行变体的收敛性比标准品Bi-CGSTAB受舍入误差的影响，并且不如标准品稳健。因此，在本文中，根据[1]，我们设计了GPBiCG的并行变体，它比Bi-CGSTAB收敛得更快，更鲁棒。然后通过数值实验比较了标准GPBiCG算法和并行算法的收敛速度。
The hybrid Bi-Conjugate Gradient (Bi-CG) methods such as Bi-CG stabilized (Bi-CGSTAB), Generalized Product-type based Bi-CG (GPBiCG), and BiCGstab (ℓ) are well-known for efficiently solving linear equations. GPBiCG and BiCGstab (ℓ) are more effective and robust than Bi-CGSTAB on problems with strongly nonsymmetric matrices. On present petascale high-performance computing hardware, the scalability of Krylov subspace methods has recently become increasingly prominent. The main bottleneck for efficient parallelization is the inner products which require a global reduction. The parallel variants of Bi-CGSTAB reducing the number of global communication phases and hiding the communication latency have been proposed. However, it has been reported that the convergence of the parallel variants of Bi-CGSTAB is affected by rounding errors than that of the standard Bi-CGSTAB, and is not as robust as the standard. In this paper, therefore, following [1], we design parallel variants of GPBiCG, which converges faster and is more robust than Bi-CGSTAB. Then we compare the convergence speed between the standard GPBiCG and the parallel variants by numerical experiments.