高性能共轭梯度基准在GraphBLAS上的有效实现

Efficient FPGA Implementation of Conjugate Gradient Methods for Laplacian System using HLS

In this paper, we study FPGA based pipelined and superscalar design of two variants of conjugate gradient methods for solving Laplacian equation on a discrete grid; the first version corresponds to the original conjugate gradient algorithm, and the second version corresponds to a slightly modified version of the same. In conjugate gradient method to solve partial differential equations, matrix vector operations are required in each iteration; these operations can be implemented as 5 point stencil operations on the grid without explicitely constructing the matrix. We show that a pipelined and s…

Improving strong scaling of the Conjugate Gradient method for solving large linear systems using global reduction pipelining

This paper presents performance results comparing MPI-based implementations of the popular Conjugate Gradient (CG) method and several of its communication hiding (or 'pipelined') variants. Pipelined CG methods are designed to efficiently solve SPD linear systems on massively parallel distributed memory hardware, and typically display significantly improved strong scaling compared to classic CG. This increase in parallel performance is achieved by overlapping the global reduction phase (MPI_Iallreduce) required to compute the inner products in each iteration by (chiefly local) computational wor…

Optimized implementation of the conjugate gradient algorithm for FPGA-based platforms using the Dirac-Wilson operator as an example

It is now a noticeable trend in High Performance Computing that the systems are becoming more and more heterogeneous. Compute nodes with a host CPU are being equipped with accelerators, the latter being a GPU or FPGA cards or both. In many cases at the heart of scientific applications running on such systems are iterative linear solvers. In this work we present a software package which includes an FPGA implementation of the Conjugate Gradient algorithm using a particular problem of the Dirac-Wilson operator as encountered in numerical simulations of Quantum Chromodynamics. The software is writ…

Efficient executions of Pipelined Conjugate Gradient Method on Heterogeneous Architectures

The Preconditioned Conjugate Gradient (PCG) method is widely used for solving linear systems of equations with sparse matrices. A recent version of PCG, Pipelined PCG, eliminates the dependencies in the computations of the PCG algorithm so that the non-dependent computations can be overlapped with communication. In this paper, we propose three methods for efficient execution of the Pipelined PCG algorithm on GPU accelerated heterogeneous architectures. The first two methods achieve task-parallelism using asynchronous executions of different tasks on CPU cores and GPU. The third method achieves…

DCG: Distributed Conjugate Gradient for Efficient Linear Equations Solving

Distributed algorithms to solve linear equations in multi-agent networks have attracted great research attention and many iteration-based distributed algorithms have been developed. The convergence speed is a key factor to be considered for distributed algorithms, and it is shown dependent on the spectral radius of the iteration matrix. However, the iteration matrix is determined by the network structure and is hardly pre-tuned, making the iterative-based distributed algorithms may converge very slowly when the spectral radius is close to 1. In contrast, in centralized optimization, the Conjug…

利用SYCL比较异构共轭梯度和Cholesky求解器在各种硬件上的性能

Comparing the Performance of Heterogeneous Conjugate Gradient and Cholesky Solvers on Various Hardware Using SYCL

本文提出基于SYCL的异构共轭梯度和Cholesky求解器，利用CPU和GPU加速大型矩阵运算，实验证明在GPU-only实现上，CG方法提速32%，Cholesky分解提速29%。

各向异性椭圆偏微分方程的预条件共轭梯度求解器的无矩阵GPU实现

Matrix-free GPU implementation of a preconditioned conjugate gradient solver for anisotropic elliptic PDEs

针对数值天气预报中的各向异性椭圆偏微分方程，提出了一种无矩阵GPU实现的预条件共轭梯度算法，通过避免显式存储矩阵并利用循环融合减少全局内存访问，显著提升了性能。

Algorithm-Based Checkpoint-Recovery for the Conjugate Gradient Method

As computers reach exascale and beyond, the incidence of faults will increase. Solutions to this problem are an active research topic. We focus on strategies to make the preconditioned conjugate gradient (PCG) solver resilient against node failures, specifically, the exact state reconstruction (ESR) method, which exploits redundancies in PCG. Reducing the frequency at which redundant information is stored lessens the runtime overhead. However, after the node failure, the solver must restart from the last iteration for which redundant information was stored, which increases recovery overhead. T…

流水线双共轭梯度稳定法中的鲁棒性与准确性：一项比较研究

Robustness and Accuracy in Pipelined Bi-Conjugate Gradient Stabilized Method: A Comparative Study

本文提出基于ExBLAS的可复现技术，以提升流水线BiCGStab方法求解非对称线性系统时的精度，并在SuiteSparse矩阵集合上验证其有效性。

GE-SpMM: General-purpose Sparse Matrix-Matrix Multiplication on GPUs for Graph Neural Networks

Graph Neural Networks (GNNs) have achieved significant improvements in various domains. Sparse Matrix-Matrix multiplication (SpMM) is a fundamental operator in GNNs, which performs a multiplication between a sparse matrix and a dense matrix. Accelerating SpMM on parallel hardware like GPUs can face the following challenges: From the GNN application perspective, the compatibility needs to be considered. General GNN algorithms require SpMM-like operations (e.g., pooling) between matrices, which are not supported in current high-performance GPU libraries (e.g., Nvidia cuSPARSE). Moreover, the sop…

Reproducibility of Parallel Preconditioned Conjugate Gradient in Hybrid Programming Environments

The Preconditioned Conjugate Gradient method is often employed for the solution of linear systems of equations arising in numerical simulations of physical phenomena. While being widely used, the solver is also known for its lack of accuracy while computing the residual. In this article, we propose two algorithmic solutions that originate from the ExBLAS project to enhance the accuracy of the solver as well as to ensure its reproducibility in a hybrid MPI + OpenMP tasks programming environment. One is based on ExBLAS and preserves every bit of information until the final rounding, while the ot…

MPCGPU：基于GPU上预条件共轭梯度法的实时非线性模型预测控制

MPCGPU: Real-Time Nonlinear Model Predictive Control through Preconditioned Conjugate Gradient on the GPU

本文提出GPU加速的实时NMPC求解器MPCGPU，其核心为预条件共轭梯度线性系统求解器，在Kuka IIWA机械臂跟踪任务中实现千赫兹控制率并支持512个轨迹节点，求解性能平均比CPU方法快3.6倍。

一种并行且高度可移植的HPC泊松求解器：基于alpaka的预条件Bi-CGSTAB

A Parallel and Highly-Portable HPC Poisson Solver: Preconditioned Bi-CGSTAB with alpaka

本文提出一种基于alpaka和MPI的并行GPU泊松求解器，采用预条件Bi-CGSTAB方法，其中切比雪夫迭代预条件器可加速6倍以上，并在64设备上实现超90%的强扩展效率。

PageRank Pipeline Benchmark: Proposal for a Holistic System Benchmark for Big-Data Platforms

The rise of big data systems has created a need for benchmarks to measure and compare the capabilities of these systems. Big data benchmarks present unique scalability challenges. The supercomputing community has wrestled with these challenges for decades and developed methodologies for creating rigorous scalable benchmarks (e.g., HPC Challenge). The proposed PageRank pipeline benchmark employs supercomputing benchmarking methodologies to create a scalable benchmark that is reflective of many real-world big data processing systems. The PageRank pipeline benchmark builds on existing prior scala…

用于畴壁费米子的多右端项多重网格：一种多重网格预处理块共轭梯度算法

Multiple right hand side multigrid for domain wall fermions with a multigrid preconditioned block conjugate gradient algorithm

提出面向畴壁费米子的多重网格预处理块共轭梯度多右端项算法，利用 batched GEMM 和 GPU 张量硬件降低粗网格代价，在 Frontier 上最高实现二十倍加速。