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Analyzing chaotic observations generated from some unknown nonlinear dynamics presents significant challenges for modeling the process and predicting future evolutions. We consider time series data...

In this work, we investigate a temporal decomposition approach to long-horizon dynamic optimization problems. The problems are discrete-time, linear,time-dependent and with \rev{box constraints} on the controlvariables. We prove that an overlapping domains temporaldecomposition, while inexact, approaches thesolution of the long-horizon dynamic optimization problemexponentially fast in the size of the overlap. The resultingsubproblems share no solution information and thus can becomputed independently in parallel. Our findings aredemonstrated with a small, synthetic production cost model withreal demand data.

Gaussian processes are commonly used for modeling the output of deterministic computer models. We consider the behavior of maximum likelihood estimators (MLEs) of parameters of the commonly used squared exponential covariance function when the computer model has some simple deterministic form. We prove that for regularly spaced observations on the line, the MLE of the scale parameter converges to zero if the computer model is a constant function and diverges to infinity for linear functions. When observing successive derivatives of a $p$th order monomial at zero, we find the asymptotic orders of the MLE of the scale parameter for all $p\ge 0$. For some commonly used test functions, we compare the MLE with cross validation in a prediction problem and explore the joint estimation of range and scale parameters. The correlation matrix is nearly numerically singular even when the sample size is moderate. To overcome numerical difficulties, we perform exact computation by making use of exact results for the cor...

Maximum-likelihood-based state estimation for dynamical systems with model error raises computational challenges in memory usage due to the much larger number of free variables when compared to the perfect model case. To address this challenge, we present a limited-memory method for maximum-likelihood-based estimation of state space 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 is 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 the number of shooting intervals. Hence the method is stable for increasing time horizo...