A discrepancy principle for the Landweber iteration based on risk minimization

In this paper we propose a criterion based on risk minimization to stop the Landweber algorithm for estimating the solution of a linear system with noisy data. Under the hypothesis of white Gaussian noise, we provide an unbiased estimator of the risk and we use it for defining a variant of the classical discrepancy principle. Moreover, we prove that the proposed variant satisfies the regularization property in expectation. Finally, we perform some numerical simulations when the signal formation model is given by a convolution or a Radon transform, to show that the proposed method is numerically reliable and furnishes slightly better solutions than classical estimators based on the predictive risk, namely the Unbiased Predictive Risk Estimator and the Generalized Cross Validation.