  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.
