Matrix splitting methods play a vital role in solving linear systems, and their efficiency heavily depends on the optimal splitting parameters. However, the splitting parameter selection method has not been well developed. Existing direct traversal method is computationally expensive, and theoretical estimation method is available on a case-by-case basis and usually relies on the upper bound of the spectral radius of iterative matrix. In this paper, we present a multitask kernel-learning parameter prediction method to automatically obtain accurate splitting parameters. For application to solving time-dependent linear systems, we propose a class of matrix splitting Kronecker product methods and give corresponding convergence analysis. Finally, we apply our approaches to solve two-dimensional diffusion and convection-diffusion equations, and the differential Sylvester matrix equation. Numerical results illustrate the efficiency and superiority of our methods.

翻译：矩阵分解方法在解决线性系统方面发挥着关键作用,其效率在很大程度上取决于最佳分解参数。然而,分解参数选择方法尚未充分开发。现有的直接跨轨方法计算成本很高,理论估算方法可以逐案使用,通常依赖迭接矩阵光谱半径的上层。在本文件中,我们提出了一个多任务内核学习参数预测方法,以自动获得准确的分解参数。为了解决基于时间的分解系统,我们建议了一组分解克龙克尔产品方法的矩阵,并进行了相应的趋同分析。最后,我们运用了我们的方法来解决二维扩散和对流扩散方程式,以及不同的Sylvester矩阵方程式。数字结果说明了我们方法的效率和优越性。