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Matrix splitting iteration methods play a vital role in solving large sparselinear systems. Their performance heavily depends on the splitting parameters,however, the approach of selecting optimal splitting parameters has not beenwell developed. In this paper, we present a multitask kernel-learning parameterprediction method to automatically obtain relatively optimal splittingparameters, which contains simultaneous multiple parameters prediction and adata-driven kernel learning. For solving time-dependent linear systems,including linear differential systems and linear matrix systems, we give a newmatrix splitting Kronecker product method, as well as its convergence analysisand preconditioning strategy. Numerical results illustrate our methods can savean enormous amount of time in selecting the relatively optimal splittingparameters compared with the exists methods. Moreover, our iteration method asa preconditioner can effectively accelerate GMRES. As the dimension of systemsincreases, all the advantages of our approaches becomes significantly.Especially, for solving the differential Sylvester matrix equation, the speedupratio can reach tens to hundreds of times when the scale of the system islarger than one hundred thousand.

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