  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.
