The present paper is concerned with developing tensor iterative Krylov subspace methods to solve large multi-linear equations. We use the T-product for two tensors define tubal global Arnoldi and Golub-Kahan bidiagonalization algorithms. Furthermore, we illustrate how tensor-based approaches can be exploited ill-posed problems arising from recovering blurry multichannel (color) images videos, u...

This paper is concerned with estimating the solutions of numerically ill-posed least squares problems through Tikhonov regularization. Given a priori estimates on the covariance structure of errors in the measurement data b, and a suitable statistically-chosen σ, the Tikhonov regularized least squares functional J(σ) = ‖Ax − b‖2Wb + 1/σ 2‖D(x − x0)‖2, evaluated at its minimizer x(σ), approximat...

• First, the lower bidiagonal matrix A11 with nonzero bidiagonal elements has full column rank and its singular values are simple. Consequently, any zero singular values or repeats that A has must appear in A22. • Second, A11 has minimal dimensions, and A22 has maximal dimensions, over all orthogonal transformations giving the block structure in (2), without any additional assumptions on the st...

The reduction of a large-scale symmetric linear discrete ill-posed problem with multiple right-hand sides to smaller block tridiagonal matrix can easily be carried out by the application small number steps Lanczos method. We show that subdiagonal blocks reduced converge zero fairly rapidly increasing number. This quick convergence indicates there is little advantage in expressing solutions prob...

Given a large square matrix A and a sufficiently regular function f so that f(A) is well defined, we are interested in the approximation of the leading singular values and corresponding singular vectors of f(A), and in particular of ‖f(A)‖, where ‖ · ‖ is the matrix norm induced by the Euclidean vector norm. Since neither f(A) nor f(A)v can be computed exactly, we introduce and analyze an inexa...

The L-curve criterion is often applied to determine a suitable value of the regularization parameter when solving ill-conditioned linear systems of equations with a right-hand side contaminated by errors of unknown norm. However, the computation of the L-curve is quite costly for large problems; the determination of a point on the L-curve requires that both the norm of the regularized approxima...

The objective of this paper is to extend, in the context of multicore architectures, the concepts of algorithms-by-tiles [Buttari et al., 2007] for Cholesky, LU, QR factorizations to the family of twosided factorizations. In particular, the bidiagonal reduction of a general, dense matrix is very often used as a pre-processing step for calculating the singular value decomposition. Furthermore, i...

This paper discusses weighted tensor Golub–Kahan-type bidiagonalization processes using the t-product. product was introduced in M.E. Kilmer and C.D. Martin (2011). A few steps of a process with least squares norm are carried out to reduce large-scale linear discrete ill-posed problem small size. The weights determined by symmetric positive definite (SPD) tensors. Tikhonov regularization is app...

This paper discusses several transform-based methods for solving linear discrete ill-posed problems third order tensor equations based on a tensor-tensor product defined by an invertible transform. Linear products were first introduced in Kernfeld et al. (2015) [16]. These are applied to derive Tikhonov regularization Golub-Kahan-type bidiagonalization and Arnoldi-type processes. GMRES-type sol...