  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.
