GCV for Tikhonov regularization via global Golub-Kahan decomposition

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 upper and lower bounds for the desired expression. These bounds are used to determine the regularization parameter for large scale problems. Computed examples illustrate the performance of the proposed method and demonstrate its competitiveness