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This paper describes solution methods for linear discrete ill-posed problemsdefined 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 Tikhonovregularization problem for third order tensors to a problem of small size. Thedata may be represented by a laterally oriented matrix or a third order tensor,and the regularization operator is a third order tensor. The discrepancyprinciple is used to determine the regularization parameter and the number ofsteps of the t-Arnoldi process. Numerical examples compare results for severalsolution methods, and illustrate the potential superiority of solution methodsthat tensorize over solution methods that matricize linear discrete ill-posedproblems for third order tensors.

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