  We propose and analyze an inexact gradient method based on incremental proper
orthogonal decomposition (iPOD) to address the data storage difficulty in
time-dependent PDE-constrained optimization, particularly for a data
assimilation problem as a detailed demonstration for the key ideas. The
proposed method is proved robust by rigorous analysis. We first derive a sharp
data compression error estimate of the iPOD with the help of Hilbert-Schmidt
operators. Then we demonstrate a numerical PDE analysis to show how to properly
choose the Hilbert space for the iPOD data compression so that the gradient
error is under control. We further prove that for a convex problem with
appropriately bounded gradient error, the inexact gradient method achieves the
accuracy level of the optimal solution while not hurting the convergence rate
compared with the usual gradient method. Finally, numerical experiments are
provided to verify the theoretical results and validate the proposed method.
