Solving coupled tensor equations via higher order LSQR methods

Abstract

Tensors have a wide application in control theory, data mining, chemistry, information sciences, documents analysis and medical engineering.
The material here is motivated by the development of the efficient
numerical methods for solving the coupled tensor
equations
$$
\left\{
\begin{array}{ll}
\mathcal{A}_1\ast_M\mathcal{X}\ast_N\mathcal{B}_1+\mathcal{C}_1\ast_M\mathcal{Y}\ast_N\mathcal{D}_1=\mathcal{E}_1,& \hbox{} \\\mathcal{A}_2\ast_M\mathcal{X}\ast_N\mathcal{B}_2+\mathcal{C}_2\ast_M\mathcal{Y}\ast_N\mathcal{D}_2=\mathcal{E}_2, & \hbox{}
\end{array}
\right.
$$
with Einstein product. We propose the tensor form of
the LSQR methods to find the solutions
$\mathcal{X}$ and $\mathcal{Y}$ of the coupled tensor
equations. Finally we give some numerical examples to illustrate that our proposed methods are able to accurately and efficiently find the solutions of tensor equations with Einstein product.