Discrete cosine transform LSQR and GMRES methods for multidimensional ill-posed problems

TL;DR

This paper introduces tensor Krylov subspace methods based on discrete cosine transforms for efficiently solving multidimensional ill-posed linear tensor problems, such as in image restoration.

Contribution

It develops tensor versions of GMRES, Golub-Kahan bidiagonalization, and LSQR methods utilizing tensor discrete cosine transforms for faster computations.

Findings

Methods are computationally fast.

Achieve good accuracy in tensor ill-posed problems.

Effective for color and video image restoration.

Abstract

In the present work, we propose new tensor Krylov subspace method for ill posed linear tensor problems such as in color or video image restoration. Those methods are based on the tensor-tensor discrete cosine transform that gives fast tensor-tensor product computations. In particular, we will focus on the tensor discrete cosine versions of GMRES, Golub-Kahan bidiagonalisation and LSQR methods. The presented numerical tests show that the methods are very fast and give good accuracies when solving some linear tensor ill-posed problems.