The quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions. Such situations are common in multi-scale, intermittent and non-homogeneous fluid and ocean flows, and other non-linear dynamical systems. The Dynamically Orthogonal (DO) field equations provide an efficient time- dependent adaptive methodology to predict the probability density functions of such dynamics. The present work derives efficient computational schemes for the DO methodology applied to unsteady stochastic Navier- Stokes and Boussinesq equations, and illustrates and studies the numerical aspects of these schemes. Semi- implicit projection methods are developed for the mean and for the orthonormal modes that define a basis for the evolving DO subspace, and time-marching schemes of first to fourth order are used for the stochastic coefficients. Conservative second-order nite-volumes are employed in physical space with new advection schemes based on Total Variation Diminishing methods. Other results specific to the DO equations include: (i) the definition of pseudo-stochastic pressures to obtain a number of pressure equations that is linear in the subspace size instead of quadratic; (ii) symmetric advection schemes for the stochastic velocities; (iii) the use of generalized inversion to deal with singular subspace covariances or deterministic modes; and (iv) schemes to maintain orthonormal modes at the numerical level. While (i) and (ii) are specific to fluid flows, (iii) and (iv) are important for any system of equations discretized using the DO methodology. To verify the correctness of our implementation and study the properties of our schemes and their variations, a set of stochastic flow benchmarks are defined including asymmetric Dirac and symmetric lock-exchange flows, lid-driven cavity flows, and flows past objects in a confined channel. Different Reynolds number and Grashof number regimes are employed to illustrate robustness. Optimal convergence under both time and space refinements is shown as well as the convergence of the probability density functions with the number of stochastic realizations.

Data assimilation, as presented in this thesis, is the statistical merging of sparse observational data with computational models so as to optimally improve the probabilistic description of the field of interest, thereby reducing uncertainties. The centerpiece of this thesis is the introduction of a novel such scheme that overcomes prior shortcomings observed within the community. Adopting techniques prevalent in Machine Learning and Pattern Recognition, and building on the foundations of classical assimilation schemes, we introduce the GMM-DO filter: Data Assimilation with Gaussian mixture models using the Dynamically Orthogonal field equations.

We combine the use of Gaussian mixture models, the EM algorithm and the Bayesian Information Criterion to accurately approximate distributions based on Monte Carlo data in a framework that allows for efficient Bayesian inference. We give detailed descriptions of each of these techniques, supporting their application by recent literature. One novelty of the GMM-DO filter lies in coupling these concepts with an efficient representation of the evolving probabilistic description of the uncertain dynamical field: the Dynamically Orthogonal field equations. By limiting our attention to a dominant evolving stochastic subspace of the total state space, we bridge an important gap previously identified in the literature caused by the dimensionality of the state space.

We successfully apply the GMM-DO filter to two test cases: (1) the Double Well Diffusion Experiment and (2) the Sudden Expansion fluid flow. With the former, we prove the validity of utilizing Gaussian mixture models, the EM algorithm and the Bayesian Information Criterion in a dynamical systems setting. With the application of the GMM-DO filter to the two-dimensional Sudden Expansion fluid flow, we further show its applicability to realistic test cases of non-trivial dimensionality. The GMM-DO filter is shown to consistently capture and retain the far-from-Gaussian statistics that arise, both prior and posterior to the assimilation of data, resulting in its superior performance over contemporary filters. We present the GMM-DO filter as an efficient, data-driven assimilation scheme, focused on a dominant evolving stochastic subspace of the total state space, that respects nonlinear dynamics and captures non-Gaussian statistics, obviating the use of heuristic arguments.

From naval operations to ocean science missions, the importance of autonomous vehicles is increasing with the advances in underwater robotics technology. Due to the dynamic and intermittent underwater environment and the physical limitations of autonomous underwater vehicles, feasible and optimal path planning is crucial for autonomous underwater operations. The objective of this thesis is to develop and demonstrate an efficient underwater path planning algorithm based on the level set method. Specifically, the goal is to compute the paths of autonomous vehicles which minimize travel time in the presence of ocean currents. The approach is to either utilize or avoid any type of ocean flows, while allowing for currents that are much larger than the nominal vehicle speed and for three-dimensional currents which vary with time. Existing path planning methods for the fields of ocean science and robotics are first reviewed, and the advantages and disadvantages of each are discussed. The underpinnings of the level set and fast marching methods are then reviewed, including their new extension and application to underwater path planning. Finally, a new feasible and optimal time-dependent underwater path planning algorithm is derived and presented. In order to demonstrate the capabilities of the algorithm, a set of idealized test-cases of increasing complexity are first presented and discussed. A real three-dimensional path planning example, involving strong current conditions, is also illustrated. This example utilizes four-dimensional ocean flows from a realistic ocean prediction system which simulate the ocean response to the passage of a tropical storm in the Middle Atlantic Bight region.