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.

Incremental tensor regularized least squares with multiple right-hand sides

“…where A ∈ R m×n×p , B ∈ R m×c×p , X ∈ R n×c×p with c > 1 and λ > 0 is a regularization parameter. Several forms of the regularization operator L are presented in [12,14] and here we focus on L = I, an identity tensor throughout this paper. The operator * denotes the tensor-tensor t-product introduced in the seminal work [6,7], which has been proved to be a useful tool with a large number of applications, such as image processing [6,11,16,18], signal processing [1,9,10], tensor recovery and robust tensor PCA [8,9], data completion and denoising [4,10,19].…”

“…in which the severe ill-conditioning of A and the error in B may cause a large propagated error in computing the solution, solving a nearby problem (1.1) can give a much more meaningful approximation [12,14]. Several methods, such as tensor Golub-Kahan method [2,[13][14][15], tensor Arnoldi method [12,15], tensor GMRES algorithm [2,15] and tensor generalized singular value decomposition algorithm [22] are researched for the t-RLS problem.…”