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

Author(s)

Viloche Bazan, Fermin S. ^[1] ; Borges, Leonardo S. ^[2]

Affiliation

^[1] Univ Fed Santa Catarina, Dept Math, BR-88040900 Florianopolis, SC - Brazil

^[2] Univ Estadual Campinas, IMECC UNICAMP, Dept Appl Math, BR-13081970 Campinas, SP - Brazil

• Document type: Journal article
• Source: BIT NUMERICAL MATHEMATICS; v. 50, n. 3, p. 481-507, SEP 2010.
• Web of Science Citations: 21

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 parallel to r(lambda)parallel to(2)/parallel to f(lambda)parallel to(2), where f(lambda) is the regularized solution and r. is the corresponding residual. GKB-FP determines the sought fixed-point by computing a finite sequence of fixed- points of functions parallel to r(lambda)((k))parallel to(2)/parallel to f(lambda)((k))parallel to(2), where f(lambda)((k)) approximates f. 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. (AU)