Multitask kernel-learning parameter prediction method for solving time-dependent linear systems

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.