A Limited-Memory Multiple Shooting Method for Weakly Constrained Variational Data Assimilation

We present a limited memory method for maximum-likelihood-based state estimation of hidden Markov models. We reduce the memory storage requirements by expressing the optimal states as a function of checkpoints bounding a shooting interval. All states can then be recomputed as needed from a recursion stemming from the optimality conditions. The matching of states at checkpoints are imposed, in a multiple shooting fashion, as constraints on the optimization problem which is solved with an augmented Lagrangian method. We prove that for nonlinear systems under certain assumptions the condition number of the Hessian matrix of the augmented Lagrangian function is bounded above with respect to number of shooting intervals. Hence the method is stable for increasing time horizon. The assumptions include satisfying the observability conditions of the linearized system on a shooting interval. We also propose a recursion-based gradient evaluation algorithm for computing the gradient, which in turn allows the algorithm to proceed by storing at any time only the checkpoints and the states on a shooting interval. We demonstrate our findings with simulations in different regimes for Burgers’ equation.