Preconditioned Methods For Weighted Toeplitz Least Squares Problems

systems with Toeplitz and Toeplitz-related coefficient matrices arise in many different applications. While many efficient algorithms have been developed for solving problems with Toeplitz structure, a few emerging applications for which the available algorithms are not directly applicable. We consider the preconditioners for weighted Toeplitz least squares problems. Applications leading to such least squares problems include image reconstruction and nonlinear image restoration. In many problems the size of the matrices can be very large. Because of the local nature and spatially vari-ant property of weighted Toeplitz matrices, the displacement rank can be very large. Efficient and effective preconditioners need to be investigated to develop fast iterative methods for solving such weighted Toeplitz least squares problems.In this paper, we first rewrite the least square problem to an equivalent sad-dle point problem and then study the preconditioners based on the Hermitian and skew-Hermitian (HS) splitting. We introduce some parameters to construct the pre-conditioner. The spectrum properties of the preconditioned system are investigated in detail. Finally, a few numerical experiments are used to illustrate the effectiveness of the new preconditioners.