The tensor Golub–Kahan–Tikhonov method applied to the solution of ill‐posed problems with a t‐product structure
Lothar Reichel, Ugochukwu O. Ugwu
Abstract
Abstract 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.