Tikhonov regularization for projected solutions of large-scale ill-posed problems is considered. The Golub-Kahan iterative bidiagonalization is used to project the problem onto a subspace and regularization then applied to find a subspace approximation to the full problem. Determination of the regularization parameter for the projected problem by unbiased predictive risk estimation, generalized...

Recent work by Kilmer and Martin, [10] and Braman [2] provides a setting in which the familiar tools of linear algebra can be extended to better understand third-order tensors. Continuing along this vein, this paper investigates further implications including: 1) a bilinear operator on the matrices which is nearly an inner product and which leads to definitions for length of matrices, angle bet...

Tikhonov regularization for projected solutions of large-scale ill-posed problems is considered. The Golub-Kahan iterative bidiagonalization is used to project the problem onto a subspace and regularization then applied to find a subspace approximation to the full problem. Determination of the regularization parameter for the projected problem by unbiased predictive risk estimation, generalized...

In this paper we study a parallel algorithm for computing the singular value decomposition SVD of a product of two matrices on message passing multiprocessors This algorithm is related to the classical Golub Kahan method for computing the SVD of a single matrix and the recent work carried out by Golub et al for computing the SVD of a general matrix product quotient The experimental results of o...

Many numerical methods for the solution of linear ill-posed problems apply Tikhonov regularization. This paper presents a modification of a numerical method proposed by Golub and von Matt for quadratically constrained least-squares problems and applies it to Tikhonov regularization of large-scale linear discrete ill-posed problems. The method is based on partial Lanczos bidiagonalization and Ga...

Generalized Cross Validation (GCV) is a popular approach to determining the regularization parameter in Tikhonov regularization. The regularization parameter is chosen by minimizing an expression, which is easy to evaluate for small-scale problems, but prohibitively expensive to compute for large-scale ones. This paper describes a novel method, based on Gauss-type quadrature, for determining up...

The Golub–Kahan–Lanczos (GKL) bidiagonal reduction generates, by recurrence, the matrix factorization of X ∈ Rm×n,m ≥ n, given by X = U BV T where U ∈ Rm×n is left orthogonal, V ∈ Rn×n is orthogonal, and B ∈ Rn×n is bidiagonal. When the GKL recurrence is implemented in finite precision arithmetic, the columns of U and V tend to lose orthogonality, making a reorthogonalization strategy necessary...