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This paper proposes an efficient general alternating-direction implicit(GADI) framework for solving large sparse linear systems. The convergenceproperty of the GADI framework is discussed. Most of the existing ADI methodscan be viewed as particular schemes of the developed framework. Meanwhile theGADI framework can derive new ADI methods. Moreover, as the algorithmefficiency is sensitive to the splitting parameters, we offer a data-drivenapproach, the Gaussian process regression (GPR) method based on the Bayesianinference, to predict the GADI framework's relatively optimal parameters. TheGPR method requires a small training data set to learn the regressionprediction mapping, which has sufficient accuracy and high generalizationcapability. It allows us to efficiently solve linear systems with a one-shotcomputation, and does not require any repeated computations to obtainrelatively optimal splitting parameters. Finally, we use the three-dimensionalconvection-diffusion equation and continuous Sylvester matrix equation toexamine the performance of our proposed methods. Numerical results demonstratethat the proposed framework is faster tens to thousands of times than theexisting ADI methods, such as (inexact) Hermitian and skew-Hermitian splittingtype methods in which the consumption of obtaining relatively optimal splittingparameters is ignored. Due to the efficiency of the developed methods, we cansolve much larger linear systems which these existing ADI methods have been notreached.

A general alternating-direction implicit framework with Gaussian process regression parameter prediction for large sparse linear systems