Tensor Arnoldi-Tikhonov and GMRES-type methods for ill-posed problems with a t-product structure

TL;DR

This paper introduces tensor-based Arnoldi-Tikhonov and GMRES-type methods for solving large-scale ill-posed problems using third order tensors and the t-product formalism, demonstrating their advantages over traditional matricization approaches.

Contribution

It develops a t-Arnoldi process for third order tensors, enabling efficient regularization of ill-posed problems within the tensor framework, which is a novel extension of existing matrix methods.

Findings

Tensor methods outperform matricization-based methods in numerical examples.

The t-Arnoldi process effectively reduces problem size for regularization.

Discrepancy principle guides optimal regularization parameter selection.

Abstract

This paper describes solution methods for linear discrete ill-posed problems defined by third order tensors and the t-product formalism introduced in [M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp. 641--658]. A t-product Arnoldi (t-Arnoldi) process is defined and applied to reduce a large-scale Tikhonov regularization problem for third order tensors to a problem of small size. The data may be represented by a laterally oriented matrix or a third order tensor, and the regularization operator is a third order tensor. The discrepancy principle is used to determine the regularization parameter and the number of steps of the t-Arnoldi process. Numerical examples compare results for several solution methods, and illustrate the potential superiority of solution methods that tensorize over solution methods that matricize linear discrete ill-posed problems for third order tensors.