We present a low-memory approach for the best-state estimate (data assimilation) of hidden Markov models where model error is considered. In particular, our findings apply for the 4D- Var framework. The novelty of our approach resides in the fact that the storage needed by our estimation framework, while including model error, is dramatically reduced from O(number of time steps) to O(1). The main insight is that we can restate the objective function of the state estimation (the likelihood function) from a function of all states to a function of the initial state only. We do so by restricting the other states by recursively enforcing the optimality conditions. This results in a regular nonlinear equation or an optimization problem for which a descent direction can be computed using only a forward sweep. In turn, the best estimate can be obtained by limited-memory quasi-Newton algorithms that need only O(1) storage with respect to the time steps. Our findings are demonstrated by numerical experiments on Burgers' equations.
Preprint ANL/MCS-P1919-0711, Argonne National Laboratory, Mathematics and Computer Science Division