Finite-dimensional regularization with nonidentity smoothing matrices

Abstract We consider the least-squares problem min x∈R n ‖Kx − y‖ 2 , where K is ill-conditioned and y contaminated with error and perturbed to y + e , with E ( e ) = 0. Regularized approximations to x of the form x α = (K T K>+αL T L) −1 K T (y+e) are considered. The properties of C ( α ) = E (‖ x − x α ‖ 2 ) are discussed. It is shown that C ( α ) has a minimum for L = I , but for more general L the existence of a minimum cannot be proved. Numerical results support the special status of L = I .