Dual gradient method for ill-posed problems using multiple repeated measurement data

Abstract

We consider determining -minimizing solutions of linear ill-posed problems , where is a bounded linear operator from a Banach space to a Hilbert space and is a proper strongly convex penalty function. Assuming that multiple repeated independent identically distributed unbiased data of y are available, we consider a dual gradient method to reconstruct the -minimizing solution using the average of these data. By terminating the method by either an a priori stopping rule or a statistical variant of the discrepancy principle, we provide the convergence analysis and derive convergence rates when the sought solution satisfies certain variational source conditions. Various numerical results are reported to test the performance of the method.