For the large and sparse linear systems, we utilize the efficient splittings of the system matrix and introduce an intermediate variable. The main contribution of this paper is that a prediction-correction matrix splitting iteration algorithm is constructed from the view of numerical optimization to solve the derived equation instead, which is inspired by the idea of adaptive parameter update. The novel algorithm adopts the prediction and correction two-step iteration, which uses information with delay to define the iterations. The global convergence results are established and the algorithm enjoys at least a Q-linear convergence rate under some suitable conditions. Further, a preconditioned version is also presented. Compared with some well-known algorithms, numerical experiments show the efficiency and effectiveness of the new proposal with application to the three-dimensional convection-diffusion equation and the image restoration problems.

对于大型和稀疏的线性系统，我们利用系统矩阵的有效分裂并引入一个中间变量。本文的主要贡献是从数值优化的角度构造了一种预测-校正矩阵分裂迭代算法来求解导出方程，其灵感来自于自适应参数更新的思想。新算法采用预测和校正两步迭代，它使用具有延迟的信息来定义迭代。全局收敛结果成立，算法至少享有Q-在一些合适的条件下线性收敛速度。此外，还提供了一个预处理版本。与一些众所周知的算法相比，数值实验表明了新方案在三维对流扩散方程和图像恢复问题上的效率和有效性。