A class of GADI methods for time-dependent linear systems with multitask kernel-learning parameter prediction

This paper develops a class of general alternating-direction implicit (GADI) iteration methods for solving time-dependent linear systems (TDLS), including linear differential systems and linear matrix systems. We present a GADI Kronecker product splitting (GADI-KP) method and prove the convergence with weak restrictions. The generalized Kronecker product splitting method and Kronecker product splitting method can be unified in the GADI-KP framework. Then, we use the framework to design an effective preconditioner of Krylov subspace methods for solving TDLS. The GADI-KP method is sensitive to the splitting parameters. Different from traditional theoretical estimate methods, we propose multitask kernel learning Gaussian process regression (GPR) method to predict the relative optimal splitting parameters. This method has solved the multi-parameter optimization in GADI framework and kernel selection in GPR method. Finally, we apply our approach to solve a two-dimensional diffusion equation, a two-dimensional convection-diffusion equation, and a differential Sylvester matrix equation. Numerical experiments illustrate that the GADI-KP framework and its preconditioning form have advantage over efficiency and superiority compared with the existing results.