Monotonicity of error of regularized solution and its use for parameter choice

Abstract

We consider an ill-posed equation in a Hilbert space with a noisy operator and a noisy right-hand side. The noise level information is given in a general form, as a norm of a certain operator applied to the noise. We derive the monotone error rule (ME-rule) for the choice of the regularization parameter in many methods, giving parameter such that the error is monotonically increasing for larger parameters in the Tikhonov method and for smaller stopping indices in iteration methods. Regularization methods considered include -scale regularization in (iterated) Tikhonov method and in iteration methods (Landweber method, CG type methods, semi-iterative methods). We also consider modifications of the ME-rule and show in numerical experiments (test problems from Hansen’s Regularization Toolbox, including the sideways heat equation) their advantages over the discrepancy principle.