Solving symmetric positive semidefinite linear systems is an essential task in many scientific computing problems. While Jacobi-type methods, including the classical Jacobi method and the weighted Jacobi method, exhibit simplicity in their forms and friendliness to parallelization, they are not attractive either because of the potential convergence failure or their slow convergence rate. This paper aims to showcase the possibility of improving classical Jacobi-type methods by employing Nesterov's acceleration technique that results in an accelerated Jacobi-type method with improved convergence properties. Simultaneously, it preserves the appealing features for parallel implementation. In particular, we show that the proposed method has an $O\left(\frac{1}{t^2}\right)$ convergence rate in terms of objective function values of the associated convex quadratic optimization problem, where $t\geq 1$ denotes the iteration counter. To further improve the practical performance of the proposed method, we also develop and analyze a restarted variant of the method, which is shown to have an $O\left(\frac{(\log_2(t))^2}{t^2}\right)$ convergence rate when the coefficient matrix is positive definite. Furthermore, we conduct appropriate numerical experiments to evaluate the efficiency of the proposed method. Our numerical results demonstrate that the proposed method outperforms the classical Jacobi-type methods and the conjugate gradient method and shows a comparable performance as the preconditioned conjugate gradient method with a diagonal preconditioner. Finally, we develop a parallel implementation and conduct speed-up tests on some large-scale systems. Our results indicate that the proposed framework is highly scalable.

翻译：求解对称半正定线性系统是众多科学计算问题中的一项基本任务。Jacobi类方法（包括经典Jacobi方法和加权Jacobi方法）虽然形式简单且易于并行化，但由于可能存在收敛失败或收敛速度缓慢的问题，其应用并不广泛。本文旨在展示通过采用Nesterov加速技术改进经典Jacobi类方法的可能性，从而获得具有更优收敛特性的加速Jacobi类方法，同时保留其适合并行实现的优点。具体而言，我们证明了所提方法在相关凸二次优化问题的目标函数值上具有$O\left(\frac{1}{t^2}\right)$的收敛速率，其中$t\geq 1$表示迭代计数。为进一步提升所提方法的实际性能，我们还开发并分析了该方法的重启变体，当系数矩阵正定时，该变体被证明具有$O\left(\frac{(\log_2(t))^2}{t^2}\right)$的收敛速率。此外，我们通过适当的数值实验评估了所提方法的效率。数值结果表明，所提方法优于经典Jacobi类方法和共轭梯度法，并与采用对角预处理器的预处理共轭梯度法表现出相当的性能。最后，我们开发了并行实现方案，并在一些大规模系统上进行了加速比测试。结果表明，所提框架具有高度的可扩展性。