Tensor Conjugate-Gradient methods for tensor linear discrete ill-posed problems

Abstract

This paper presents three types of tensor Conjugate-Gradient (tCG) methods for solving large-scale linear discrete ill-posed problems based on the t-product between third-order tensors. An automatic determination strategy of a suitable regularization parameter is proposed for the tCG method in the Fourier domain (A-tCG-FFT). An improved version and a preconditioned version of the tCG method are also presented. The discrepancy principle is employed to determine a suitable regularization parameter. Several numerical examples in image and video restoration are given to show the effectiveness of the proposed tCG methods.

Keywords

• linear discrete ill-posed problems
• conjugate-gradient method
• t-product
• regularization parameter
• discrepancy principle
• tikhonov regularization