GKB-FP: an algorithm for large-scale discrete ill-posed problems

Abstract

We describe an algorithm for large-scale discrete ill-posed problems, called GKB-FP, which combines the Golub-Kahan bidiagonalization algorithm with Tikhonov regularization in the generated Krylov subspace, with the regularization parameter for the projected problem being chosen by the fixed-point method by Bazan (Inverse Probl. 24(3), 2008). The fixed-point method selects as regularization parameter a fixed-point of the function ‖rλ‖2/‖fλ‖2, where fλ is the regularized solution and rλ is the corresponding residual. GKB-FP determines the sought fixed-point by computing a finite sequence of fixed-points of functions \(\|r_{\lambda}^{(k)}\|_{2}/\|f_{\lambda}^{(k)}\|_{2}\), where \(f_{\lambda}^{(k)}\) approximates fλ in a k-dimensional Krylov subspace and \(r_{\lambda}^{(k)}\) is the corresponding residual. Based on this and provided the sought fixed-point is reached, we prove that the regularized solutions \(f_{\lambda}^{(k)}\) remain unchanged and therefore completely insensitive to the number of iterations. This and the performance of the method when applied to well-known test problems are illustrated numerically.