  We consider determining the $\R$-minimizing solution of ill-posed problem $A
x = y$ for a bounded linear operator $A: X \to Y$ from a Banach space $X$ to a
Hilbert space $Y$, where $\R: X \to (-\infty, \infty]$ is a strongly convex
function. A dual gradient flow is proposed to approximate the sought solution
by using noisy data. Due to the ill-posedness of the underlying problem, the
flow demonstrates the semi-convergence phenomenon and a stopping time should be
chosen carefully to find reasonable approximate solutions. We consider the
choice of a proper stopping time by various rules such as the {\it a priori}
rules, the discrepancy principle, and the heuristic discrepancy principle and
establish the respective convergence results. Furthermore, convergence rates
are derived under the variational source conditions on the sought solution.
Numerical results are reported to test the performance of the dual gradient
flow.
