A GENERAL ALTERNATING-DIRECTION IMPLICIT FRAMEWORK WITH GAUSSIAN PROCESS REGRESSION PARAMETER PREDICTION FOR LARGE SPARSE LINEAR SYSTEMS

Kai Jiang

doi:10.1137/21M1450197

2022-07-07

DOI：10.1137/21M1450197

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摘要

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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 existing ADI methods can be unified in 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 only requires a small training data set to learn the regression prediction mapping, can predict accurate splitting parameters, and has high generalization capability. It allows us to efficiently solve linear systems with a one-shot computation, and does not require any repeated computations. Finally, we use the three-dimensional convection-diffusion equation, two-dimensional parabolic 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. As a result, our proposed methods can solve much larger linear systems which these existing ADI methods have not reached.

Keyword:

general alternating-direction implicit framework

large sparse linear systems

convergent analysis

Gaussian process regression

data-driven method