  We consider determining $\R$-minimizing solutions of linear ill-posed
problems $A x = y$, where $A: {\mathscr X} \to {\mathscr Y}$ is a bounded
linear operator from a Banach space ${\mathscr X}$ to a Hilbert space
${\mathscr Y}$ and ${\mathcal R}: {\mathscr X} \to [0, \infty]$ 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 ${\mathcal R}$-minimizing solution using the
average of these data. By terminating the method by either an {\it 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.
