We integrate data-driven ocean modeling with the stochastic Dynamically Orthogonal (DO) level-set optimization methodology to compute and study energy-optimal paths, speeds, and headings for ocean vehicles in the Middle-Atlantic Bight (MAB) region. We hindcast the energy-optimal paths from among exact time-optimal paths for the period 28 August 2006 to 9 September 2006. To do so, we first obtain a data-assimilative multiscale re-analysis, combining ocean observations with implicit two-way nested multiresolution primitive-equation simulations of the tidal-to-mesoscale dynamics in the region. Second, we solve the reduced-order stochastic DO level-set partial differential equations (PDEs) to compute the joint probability of minimum arrival-time, vehicle-speed time-series, and total energy utilized. Third, for each arrival time, we select the vehiclespeed time-series that minimize the total energy utilization from the marginal probability of vehicle-speed and total energy. The corresponding energy-optimal path and headings are obtained through a particle backtracking equation. Theoretically, the present methodology is PDE-based and provides fundamental energy-optimal predictions without heuristics. Computationally, it is three- to four-orders of magnitude faster than direct Monte Carlo methods. For the missions considered, we analyze the effects of the regional tidal currents, strong wind events, coastal jets, shelfbreak front, and other local circulations on the energy-optimal paths. Results showcase the opportunities for vehicles that intelligently utilize the ocean environment to minimize energy usage, rigorously integrating ocean forecasting with optimal control of autonomous vehicles.

In this paper a reachability-based approach is adopted to deal with the pursuit-evasion dierential game between one evader and multiple pursuers in the presence of dynamic environmental disturbances (e.g., winds, sea currents). Conditions for the game to be terminated are given in terms of reachable set inclusions. Level set equations are defined and solved to generate the forward reachable sets of the pursuers and the evader. The time-optimal trajectories and the corresponding optimal strategies are sub- sequently retrieved from these level sets. The pursuers are divided into active pursuers, guards, and redundant pursuers according to their respec- tive roles in the pursuit-evasion game. The proposed scheme is implemented on problems with both simple and realistic time-dependent flow fields, with and without obstacles.

The nonlinear Gaussian Mixture Model Dynamically Orthogonal (GMM–DO) smoother for high- dimensional stochastic fields is exemplified and contrasted with other smoothers by applications to three dynamical systems, all of which admit far-from-Gaussian distributions. The capabilities of the smoother are first illustrated using a double-well stochastic diffusion experiment. Comparisons with the original and improved versions of the ensemble Kalman smoother explain the detailed mechanics of GMM–DO smoothing and show that its accuracy arises from the joint GMM distributions across successive observation times. Next, the smoother is validated using the advection of a passive stochastic tracer by a reversible shear flow. This example admits an exact smoothed solution, whose derivation is also provided. Results show that the GMM– DO smoother accurately captures the full smoothed distributions and not just the mean states. The final example showcases the smoother in more complex nonlinear fluid dynamics caused by a barotropic jet flowing through a sudden expansion and leading to variable jets and eddies. The accuracy of the GMM–DO smoother is compared to that of the Error Subspace Statistical Estimation smoother. It is shown that even when the dynamics result in only slightly multimodal joint distributions, Gaussian smoothing can lead to a severe loss of information. The three examples show that the backward inferences of the GMM–DO smoother are skillful and efficient. Accurate evaluation of Bayesian smoothers for nonlinear high-dimensional dynamical systems is challenging in itself. The present three examples—stochastic low dimension, reversible high dimension, and irreversible high dimension—provide complementary and effective benchmarks for such evaluation.

Retrospective inference through Bayesian smoothing is indispensable in geophysics, with crucial applications in ocean and numerical weather estimation, climate dynamics, and Earth system modeling. However, dealing with the high-dimensionality and nonlinearity of geophysical processes remains a major challenge in the development of Bayesian smoothers. Addressing this issue, a novel subspace smoothing methodology for high-dimensional stochastic fields governed by general nonlinear dynamics is obtained. Building on recent Bayesian filters and classic Kalman smoothers, the fundamental equations and forward–backward algorithms of new Gaussian Mixture Model (GMM) smoothers are derived, for both the full state space and dynamic subspace. For the latter, the stochastic Dynamically Orthogonal (DO) field equations and their time-evolving stochastic subspace are employed to predict the prior subspace probabilities. Bayesian inference, both forward and backward in time, is then analytically carried out in the dominant stochastic subspace, after fitting semiparametric GMMs to joint subspace realizations. The theoretical properties, varied forms, and computational costs of the new GMM smoother equations are presented and discussed.