Two CSCS-based iteration methods for absolute value equations with Toeplitz matrix

Recently, two families of HSS-based iteration methods are constructed for handling the absolute value equation (AVE), which is a class of non-differentiable NP-hard problems. In present paper, we establish the Picard-CSCS iteration method and the nonlinear CSCS-like iteration method for the AVE involving the Toeplitz matrix. Then we analyze the convergence of the Picard-CSCS iteration method for solving the AVE. By using the theory about nonsmooth analysis, we particularly prove the convergence of the nonlinear CSCS-like iteration method for the AVE. The advantage of these methods is that they do not require the storage of coefficient matrices, and the sub-system of linear equations can be solved efficiently via the fast Fourier transform (FFT). Therefore, computational cost and storage may be saved in practical implementations. Numerical experiments including the solution of the fractional diffusion equation with a nonlinear term are presented to show the effectiveness of the proposed methods in comparison with the HSS-based methods.