Global Golub-Kahan bidiagonalization applied to large discrete ill-posed problems

We consider the solution of large linear systems of equations that arise from the discretization of ill-posed problems. The matrix has a Kronecker product structure and the right-hand side is contaminated by measurement error. Problems of this kind arise, for instance, from the discretization of Fredholm integral equations of the first kind in two space-dimensions with a separable kernel and in image restoration problems. Regularization methods, such as Tikhonov regularization, have to be employed to reduce the propagation of the error in the right-hand side into the computed solution. We investigate the use of the global Golub–Kahan bidiagonalization method to reduce the given large problem to a small one. The small problem is solved by employing Tikhonov regularization. A regularization parameter determines the amount of regularization. The connection between global Golub–Kahan bidiagonalization and Gauss-type quadrature rules is exploited to inexpensively compute bounds that are useful for determining the regularization Email addresses: [email protected] (A. H. Bentbib), [email protected] (M. El Guide), [email protected] (K. Jbilou), [email protected] (L. Reichel) Preprint submitted to Journal of Computational and Applied Mathematics (JCAM)March 17, 2017 parameter by the discrepancy principle.