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...

A crucial problem concerning Tikhonov regularization is the proper choice of the regularization parameter. This paper deals with a generalization of a parameter choice rule due to Regińska (1996) [31], analyzed and algorithmically realized through a fast fixed-point method in Bazán (2008) [3], which results in a fixed-point method for multi-parameter Tikhonov regularization called MFP. Like the...

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...

In this paper we propose a variant of the Rayleigh quotient method to compute an eigenvalue and corresponding eigenvectors of a matrix. It is based on the observation that eigenvectors of a matrix with eigenvalue zero are also singular vectors corresponding to zero singular values. Instead of computing eigenvector approximations by the inverse power method, we take them to be the singular vecto...

s 6 Awad H. Al-Mohy An Improved Algorithm for the Matrix Logarithm . . . . . . . . . . . . . . . . . . . . 7 David Amsallem Interpolation on Matrix Manifolds of Reduced-Order Models and Application to On-Line Aeroelastic Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Athanasios C. Antoulas Model Reduction of Parameter-Dependent Systems . . . . . . . . . . . . . . . . . . ...

The concept of the core problem in total least squares (TLS) problems was introduced in [C. C. Paige and Z. Strakoš, SIAM J. Matrix Anal. Appl., 27, 2006, pp. 861–875]. It is based on orthogonal transformations such that the resulting problem decomposes into two independent parts, with one of the parts having trivial (zero) right-hand side and maximal dimensions, and the other part with nonzero...

Abstract When solving ill-posed inverse problems, a good choice of the prior is critical for computation reasonable solution. A common approach to include Gaussian prior, which defined by mean vector and symmetric positive definite covariance matrix, use iterative projection methods solve corresponding regularized problem. However, main challenge many these that matrix must be known fixed (up c...

In this paper, we are interested in finding an approximate solution $ \hat{\mathcal{X}} of the tensor least-squares minimization problem$ \min_{\mathcal{X}}\left\|\mathcal{X}\times_1A^{(1)}\times_2A^{(2)}\times_3\cdots\times_NA^{(N)}-\mathcal{G}\right\|$, where \mathcal{G}\in \mathbb{R}^{J_1\times J_2\times \cdots \times J_N}$ and A^{(i)}\in \mathbb{R}^{J_i\times I_i} ($ i=1,\ldots,N $) known \...