# On the Golub--Kahan bidiagonalization for ill-posed tensor equations with applications to color image restoration

## TL;DR

This paper introduces the Tensor Golub--Kahan bidiagonalization algorithm combined with Tikhonov regularization to effectively solve ill-posed tensor equations, with applications demonstrated in color image restoration.

## Contribution

The paper proposes a novel tensor bidiagonalization algorithm and explores its theoretical properties and practical applications in high-dimensional tensor problems.

## Key findings

- The TGKB algorithm effectively stabilizes solutions to ill-posed tensor equations.
- Numerical experiments confirm the algorithm's applicability to color image restoration.
- Theoretical analysis reveals the conditioning of tensor equations and the utility of TGKB.

## Abstract

This paper is concerned with solving ill-posed tensor linear equations. These kinds of equations may appear from finite difference discretization of high-dimensional convection-diffusion problems or when partial differential equations in many dimensions are discretized by collocation spectral methods. Here, we propose the Tensor Golub--Kahan bidiagonalization (TGKB) algorithm in conjunction with the well known Tikhonov regularization method to solve the mentioned problems. Theoretical results are presented to discuss on conditioning of the Stein tensor equation and to reveal that how the TGKB process can be exploited for general tensor equations. In the last section, some classical test problems are examined to numerically illustrate the feasibility of proposed algorithms and also applications for color image restoration are considered.