This paper discusses an application of partial tensor Golub-Kahan bidiagonalization to the solution of large-scale linear discrete ill-posed problems based on the t-product formalism for third-order tensors proposed by Kilmer and Martin (M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp. 641-658). The solution methods presented first reduce a given (large-scale) problem to a problem of small size by application of a few steps of tensor Golub-Kahan bidiagonalization and then regularize the reduced problem by Tikhonov's method. The regularization operator is a third-order tensor, and the data may be represented by a matrix, that is, a tensor slice, or by a general third-order tensor. A regularization parameter is determined by the discrepancy principle. This results in fully automatic solution methods that neither require a user to choose the number of bidiagonalization steps nor the regularization parameter. The methods presented extend available methods for the solution for linear discrete ill-posed problems defined by a matrix operator to linear discrete ill-posed problems defined by a third-order tensor operator. An interlacing property of singular tubes for third-order tensors is shown and applied. Several algorithms are presented. Computed examples illustrate the advantage of the tensor t-product approach, in comparison with solution methods that are based on matricization of the tensor equation. K E Y W O R D Sdiscrepancy principle, discrete ill-posed problem, t-product, tensor Golub-Kahan bidiagonalization, Tikhonov regularization 1 where  = [a ijk ] 𝓁,m,n i,j,k=1 ∈ R 𝓁×m×n is a third-order tensor, ⃗  ∈ R m×1×n and ⃗  ∈ R 𝓁×1×n are lateral slices of third-order tensors and may be thought of as laterally oriented matrices. The operation * denotes the tensor t-product introduced in the seminal work by Kilmer and Martin 1 and applied to image deblurring problems by Kilmer et al. 1,2 The t-product between

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This paper describes and compares some structure preserving techniques for the solution of linear discrete ill-posed problems with the t-product. A new randomized tensor singular value decomposition (R-tSVD) with a t-product is presented for low tubal rank tensor approximations. Regularization of linear inverse problems by truncated tensor eigenvalue decomposition (T-tEVD), truncated tSVD (T-tSVD), randomized T-tSVD (RT-tSVD), t-product Golub-Kahan bidiagonalization (tGKB) process, and t-product Lanczos (t-Lanczos) process are considered. A solution method that is based on reusing tensor Krylov subspaces generated by the tGKB process is described. The regularization parameter is the number of iterations required by each method. The discrepancy principle is used to determine this parameter. Solution methods that are based on truncated iterations are compared with solution methods that combine Tikhonov regularization with the tGKB and t-Lanczos processes. Computed examples illustrate the performance of these methods when applied to image and gray-scale video restorations. Our new RT-tSVD method is seen to require less CPU time and yields restorations of higher quality than the T-tSVD method.

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.