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Path Planning and Adaptive Sampling in the Coastal Ocean

Lolla, T., 2016. Path Planning and Adaptive Sampling in the Coastal Ocean. Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, February 2016.

When humans or robots operate in complex dynamic environments, the planning of paths and the collection of observations are basic, indispensable problems. In the oceanic and atmospheric environments, the concurrent use of multiple mobile sensing platforms in unmanned missions is growing very rapidly. Opportunities for a paradigm shift in the science of autonomy involve the development of fundamental theories to optimally collect information, learn, collaborate and make decisions under uncertainty while persistently adapting to and utilizing the dynamic environment. To address such pressing needs, this thesis derives governing equations and develops rigorous methodologies for optimal path planning and optimal sampling using collaborative swarms of autonomous mobile platforms. The application focus is the coastal ocean where currents can be much larger than platform speeds, but the fundamental results also apply to other dynamic environments.

We first undertake a theoretical synthesis of minimum-time control of vehicles operating in general dynamic flows. Using various ideas rooted in non-smooth calculus, we prove that an unsteady Hamilton-Jacobi equation governs the forward reachable sets in any type of Lipschitz-continuous flow. Next, we show that with a suitable modification to the Hamiltonian, the results can be rigorously generalized to perform time-optimal path planning with anisotropic motion constraints and with moving obstacles and unsafe ‘forbidden’ regions. We then derive a level-set methodology for distance-based coordination of swarms of vehicles operating in minimum time within strong and dynamic ocean currents. The results are illustrated for varied fluid and ocean flow simulations. Finally, the new path planning system is applied to swarms of vehicles operating in the complex geometry of the Philippine Archipelago, utilizing realistic multi-scale current predictions from a data-assimilative ocean modeling system.

In the second part of the thesis, we derive a theory for adaptive sampling that exploits the governing nonlinear dynamics of the system and captures the non-Gaussian structure of the random state fields. Optimal observation locations are determined by maximizing the mutual information between the candidate observations and the variables of interest. We develop a novel Bayesian smoother for high-dimensional continuous stochastic fields governed by general nonlinear dynamics. This smoother combines the adaptive reduced-order Dynamically-Orthogonal equations with Gaussian Mixture Models, extending linearized Gaussian backward pass updates to a nonlinear, non-Gaussian setting. The Bayesian information transfer, both forward and backward in time, is efficiently carried out in the evolving dominant stochastic subspace. Building on the foundations of the smoother, we then derive an efficient technique to quantify the spatially and temporally varying mutual information field in general nonlinear dynamical systems. The globally optimal sequence of future sampling locations is rigorously determined by a novel dynamic programming approach that combines this computation of mutual information fields with the predictions of the forward reachable set. All the results are exemplified and their performance is quantitatively assessed using a variety of simulated fluid and ocean flows.

The above novel theories and schemes are integrated so as to provide real-time computational intelligence for collaborative swarms of autonomous sensing vehicles. The integrated system guides groups of vehicles along predicted optimal trajectories and continuously improves field estimates as the observations predicted to be most informative are collected and assimilated. The optimal sampling locations and optimal trajectories are continuously forecast, all in an autonomous and coordinated fashion.


High Order Hybrid Discontinuous Galerkin Regional Ocean Modeling

Ueckermann, M.P., 2014. High Order Hybrid Discontinuous Galerkin Regional Ocean Modeling. Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, February 2014.

Accurate modeling of physical and biogeochemical dynamics in coastal ocean regions is required for multiple scientific and societal applications, covering a wide range of time and space scales. However, in light of the strong nonlinearities observed in coastal regions and in biological processes, such modeling is challenging. An important subject that has been largely overlooked is the numerical requirements for regional ocean simulation studies. Major objectives of this thesis are to address such computational questions for non-hydrostatic multiscale flows and for biogeochemical interactions, and to derive and develop numerical schemes that meet these requirements, utilizing the latest advances in computational fluid dynamics.

We are interested in studying nonlinear, transient, and multiscale ocean dynamics over complex geometries with steep bathymetry and intricate coastlines, from sub-mesoscales to basin-scales. These dynamical interests, when combined with our requirements for accurate, efficient and flexible ocean modeling, led us to develop new variable resolution, higher-order and non-hydrostatic ocean modeling schemes. Specifically, we derived, developed and applied new numerical schemes based on the novel hybrid discontinuous Galerkin (HDG) method in combination with projection methods.

The new numerical schemes are first derived for the Navier-Stokes equations. To ensure mass conservation, we define numerical fluxes that are consistent with the discrete divergence equation. To improve stability and accuracy, we derive a consistent HDG stability parameter for the pressure-correction equation. We also apply a new boundary condition for the pressure-corrector, and show the form and origin of the projection method’s time-splitting error for a case with implicit diffusion and explicit advection. Our scheme is implemented for arbitrary, mixed-element unstructured grids using a novel quadrature-free integration method for a nodal basis, which is consistent with the HDG method. To prevent numerical oscillations, we design a selective high-order nodal limiter. We demonstrate the correctness of our new schemes using a tracer advection benchmark, a manufactured solution for the steady diffusion and stokes equations, and the 2D lock-exchange problem.

These numerical schemes are then extended for non-hydrostatic, free-surface, variable-density regional ocean dynamics. The time-splitting procedure using projection methods is derived for non-hydrostatic or hydrostatic, and nonlinear free-surface or rigid-lid, versions of the model. We also derive consistent HDG stability parameters for the free-surface and non-hydrostatic pressure-corrector equations to ensure stability and accuracy. New boundary conditions for the free-surface-corrector and pressure-corrector are also introduced. We prove that these conditions lead to consistent boundary conditions for the free-surface and pressure proper. To ensure discrete mass conservation with a moving free-surface, we use an arbitrary LagrangianEulerian (ALE) moving mesh algorithm. These schemes are again verified, this time using a tidal flow problem with analytical solutions and a 3D lock-exchange benchmark.

We apply our new numerical schemes to evaluate the numerical requirements of the coupled biological-physical dynamics. We find that higher-order schemes are more accurate at the same efficiency compared to lower-order (e.g. second-order) accurate schemes when modeling a biological patch. Due to decreased numerical dissipation, the higher-order schemes are capable of modeling biological patchiness over a sustained duration, while the lower-order schemes can lose significant biomass after a few non-dimensional times and can thus solve erroneous nonlinear dynamics.

Finally, inspired by Stellwagen Bank in Massachusetts Bay, we study the effect of non-hydrostatic physics on biological productivity and phytoplankton fields for tidally-driven flows over an idealized bank. We find that the non-hydrostatic pressure and flows are important for biological dynamics, especially when flows are supercritical. That is, when the slope of the topography is larger than the slope of internal wave rays at the tidal frequency. The non-hydrostatic effects increase with increasing nonlinearity, both when the internal Froude number and criticality parameter increase. Even in cases where the instantaneous biological productivity is not largely modified, we find that the total biomass, spatial variability and patchiness of phytoplankton can be significantly altered by non-hydrostatic processes.

Our ultimate dynamics motivation is to allow quantitative simulation studies of fundamental nonlinear biological-physical dynamics in coastal regions with complex bathymetric features such as straits, sills, ridges and shelfbreaks. This thesis develops the necessary numerical schemes that meet the stringent accuracy requirements for these types of flows and dynamics.


Dynamically Orthogonal Field Equations for Stochastic Fluid Flows and Particle Dynamics

Sapsis, Themis, 2011. Dynamically Orthogonal Field Equations for Stochastic Fluid Flows and Particle Dynamics. Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, February 2011.

In the past decades an increasing number of problems in continuum theory have been treated using stochastic dynamical theories. This is because dynamical systems governing real processes always contain some elements characterized by uncertainty or stochasticity. Uncertainties may arise in the system parameters, the boundary and initial conditions, and also in the external forcing processes. Also, many problems are treated through the stochastic framework due to the incomplete or partial understanding of the governing physical laws. In all of the above cases the existence of random perturbations, combined with the com- plex dynamical mechanisms of the system often leads to their rapid growth which causes distribution of energy to a broadband spectrum of scales both in space and time, making the system state particularly complex. Such problems are mainly described by Stochastic Partial Differential Equations and they arise in a number of areas including fluid mechanics, elasticity, and wave theory, describing phenomena such as turbulence, random vibrations, flow through porous media, and wave propagation through random media. This is but a partial listing of applications and it is clear that almost any phenomenon described by a field equation has an important subclass of problems that may profitably be treated from a stochastic point of view.

In this work, we develop a new methodology for the representation and evolution of the complete probabilistic response of infinite-dimensional, random, dynamical systems. More specifically, we derive an exact, closed set of evolution equations for general nonlinear continuous stochastic fields described by a Stochastic Partial Differential Equation. The derivation is based on a novel condition, the Dynamical Orthogonality (DO), on the representation of the solution. This condition is the “key” to overcome the redundancy issues of the full representation used while it does not restrict its generic features. Based on the DO condition we derive a system of field equations consisting of a Partial Differential Equation (PDE) for the mean field, a family of PDEs for the orthonormal basis that describe the stochastic subspace where uncertainty “lives” as well as a system of Stochastic Differential Equations that defines how the uncertainty evolves in the time varying stochastic subspace. If additional restrictions are assumed on the form of the representation, we recover both the Proper-Orthogonal-Decomposition (POD) equations and the generalized Polynomial- Chaos (PC) equations; thus the new methodology generalizes these two approaches. For the efficient treatment of the strongly transient character on the systems described above we derive adaptive criteria for the variation of the stochastic dimensionality that characterizes the system response. Those criteria follow directly from the dynamical equations describing the system.

We illustrate and validate this novel technique by solving the 2D stochastic Navier-Stokes equations in various geometries and compare with direct Monte Carlo simulations. We also apply the derived framework for the study of the statistical responses of an idealized “double gyre” model, which has elements of ocean, atmospheric and climate instability behaviors.

Finally, we use our new stochastic description for flow fields to study the motion of inertial particles in flows with uncertainties. Inertial or finite-size particles in fluid flows are commonly encountered in nature (e.g., contaminant dispersion in the ocean and atmosphere) as well as in technological applications (e.g., chemical systems involving particulate reactant mixing). As it has been observed both numerically and experimentally, their dynamics can differ markedly from infinitesimal particle dynamics. Here we use recent results from stochastic singular perturbation theory in combination with the DO representation of the random flow, in order to derive a reduced order inertial equation that will describe efficiently the stochastic dynamics of inertial particles in arbitrary random flows.


Adaptive Rapid Environmental Assessment

Ding Wang, 2007. Adaptive Rapid Environmental Assessment. Ph.D. Thesis, Massachusetts Institute of Technology, Department of Mechanical Engineering, September 2007 (Co-supervised with Prof. Henrik Schmidt).

In shallow water, a large part of underwater acoustic prediction uncertainties are in- duced by sub-meso-to-small scale oceanographic variabilities. Conventional oceano- graphic measurements for capturing such ocean-acoustic environmental variabilities face the classical conflict between resolution and coverage. The Adaptive Rapid En- vironmental Assessment (AREA) project was proposed to resolve this conflict by optimizing the location of in-situ measurements in an adaptive manner. In this thesis, ideas, concepts and performance limits in AREA are clarified. Both an engineering and a mathematical model for AREA are developed. A modularized AREA simulator was developed and implemented in C++. Philosophies in AREA are discussed. Presumptions about the ocean are made to bridge the gap between the viewpoint in the oceanography community, where the ocean environment is consid- ered to be a deterministic but very complicated system, and that of the underwater acoustic community, where the ocean environment is treated as a random system. At present, how to optimally locate the in-situ measurements made by a single AUV carrying a CTD (conductivity, temperature and depth) sensor is considered in AREA. In this thesis, the AUV path planning is modeled as a Shortest Path problem. However, due to the sound velocity correlation effect, the size of this problem can be very large. A method is developed to simplify the graph for a fast solution. As a significant step, a linear approximation for acoustic Transmission Loss (TL) is investigated numerically and analytically. In addition to following a predetermined path, an AUV can also adaptively gener- ate its path on-board. This adaptive on-board AUV routing problem is modeled using Dynamic Programming (DP) in this thesis. A method based on an optimized prede- termined path is developed to reduce the size of the DP problem and approximately yet efficiently solve it using Pattern Recognition. As a special case, a thermocline- oriented AUV yoyo control and control parameter optimization methods for AREA are also developed. 2 Finally, some AUV control algorithms for capturing fronts are developed. A frame- work for real-time TL forecasts is developed. This is the first time that TL forecasts have been linked with ocean forecasts in real-time. All of the above ideas and methods developed were tested in two experiments, FAF05 in the northern Tyrrhenian Sea in 2005 and MB06 in Monterey Bay, CA in 2006. The latter MB06 sea exercise was a major field experiment sponsored by the Office of Naval Research and the thesis compiles significant findings from this effort.