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

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.