Incremental data compression for PDE-constrained optimization with a data assimilation application

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.