This paper proposes an efficient general alternating-direction implicit (GADI) framework for solving large sparse linear systems. The convergence property of the GADI framework is discussed. Most of the existing ADI methods can be viewed as particular schemes of the developed framework. Meanwhile the GADI framework can derive new ADI methods. Moreover, as the algorithm efficiency is sensitive to the splitting parameters, we offer a data-driven approach, the Gaussian process regression (GPR) method based on the Bayesian inference, to predict the GADI framework's relatively optimal parameters. The GPR method requires a small training data set to learn the regression prediction mapping, which has sufficient accuracy and high generalization capability. It allows us to efficiently solve linear systems with a one-shot computation, and does not require any repeated computations to obtain relatively optimal splitting parameters. Finally, we use the three-dimensional convection-diffusion equation and continuous Sylvester matrix equation to examine the performance of our proposed methods. Numerical results demonstrate that the proposed framework is faster tens to thousands of times than the existing ADI methods, such as (inexact) Hermitian and skew-Hermitian splitting type methods in which the consumption of obtaining relatively optimal splitting parameters is ignored. Due to the efficiency of the developed methods, we can solve much larger linear systems which these existing ADI methods have been not reached.
翻译:本文提出了解决大型稀散线性系统的有效通用交替隐含(GADI)框架。讨论了GADI框架的趋同特性。现有的ADI方法大多可以被视为发达框架的特定计划。与此同时,GADI框架可以产生新的ADI方法。此外,由于算法效率对分解参数十分敏感,我们提供了一种数据驱动的方法,即基于巴伊西亚推论的Gaussian进程回归法(GPR),以预测GADI框架相对最佳参数。GPR方法需要一套小型的培训数据集来学习回归预测绘图,该数据集具有足够的准确性和高度的通用能力。它使我们能够以一次性计算有效解决线性系统,而不需要再重复计算来获得相对最佳的分解参数。最后,我们用三维调分解方程和连续的Sylvester矩阵公式来检查我们拟议方法的绩效。Numerical结果显示,拟议的框架比现有的ADI方法更快到数千次,因为后者有足够的准确性和高度的概括性能力。它使我们能够以最优化的流化的方法来改变目前的方法。