Conjugate gradient regularization under general smoothness and noise assumptions

Abstract

Abstract We study noisy linear operator equations in Hilbert space under a self-adjoint operator. Approximate solutions are sought by conjugate gradient type iteration, given as Krylov-subspace minimizers under a general weight function. Solution smoothness is given in terms of general source conditions. The noise may be controlled in stronger norm. We establish conditions under which stopping according to a modified discrepancy principle yields optimal regularization of the iteration. The present analysis extends much of the known theory and reveals some intrinsic features which are hidden when studying standard conjugate gradient type regularization under standard smoothness assumptions. In particular, under a non-self adjoint operator, regularization of the associated normal equation is a direct consequence from the main result and does not require a separate treatment.