An iterative method for solving the constrained tensor equations with the Einstein product

Abstract

In this paper, we present an iterative method for solving tensor equations, specifically multilinear systems of the form $A{\ast }_{N}X{\ast }_{M}B+C{\ast }_{N}X{\ast }_{M}D=E$ with one of the constraints $X=X^T$, $X=P{\ast }_{N}X{\ast }_{M}Q$, $X=-P{\ast }_{N}X{\ast }_{M}Q$ and $X=P{\ast }_{N}X{\ast }_{N}P$, where $P$ and $Q$ are reflexive tensors. The proposed method is grounded in the generalized least squares method with the Einstein product. To address the constrained tensor equation using the global least squares method, we introduce a multilinear operator and its adjoint. For a more detailed survey, we compare the proposed method for solving the constrained tensor equation with one of the matrix format methods for the associated matrix equation. We also use the new method to solve the image restoration problem with a symmetrical structure, as a special case of constrained tensor equation. Finally, we give some examples to illustrate the effectiveness of the proposed method.