Date of Award

Summer 8-31-2019

Document Type


Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)


Mathematical Sciences

First Advisor

Richard O. Moore

Second Advisor

Wooyoung Choi

Third Advisor

David Shirokoff

Fourth Advisor

M. Ani Hsieh

Fifth Advisor

Stephen Guimond


Despite an extensive history of oceanic observation, researchers have only begun to build a complete picture of oceanic currents. Sparsity of instrumentation has created the need to maximize the information extracted from every source of data in building this picture. Within the last few decades, autonomous vehicles, or AVs, have been employed as tools to aid in this research initiative. Unmanned and self-propelled, AVs are capable of spending weeks, if not months, exploring and monitoring the oceans. However, the quality of data acquired by these vehicles is highly dependent on the paths along which they collect their observational data. The focus of this research is to find optimal sampling paths for autonomous vehicles, with the goal of building the most accurate estimate of a velocity field in the shortest time possible.

The two main numerical tools employed in this work are the level set method for time-optimal path planning, and the Kalman filter for state estimation and uncertainty quantification. Specifically, the uncertainty associated with the velocity field is defined as the trace of the covariance matrix corresponding to the Kalman filter equations. The novelty in this work is the covariance tracking algorithm, which evolves this covariance matrix along the time-optimal trajectories defined by the level set method, and determines the path expected to minimize the uncertainty corresponding to the flow field by the end of deployment. While finding optimal sampling paths using this method is straightforward for the single-vehicle problem, it becomes increasingly difficult as the number of AVs grows. As such, an iterative procedure is presented here for multi-vehicle problems, which in simple cases can be proven to find controls that collectively minimizes the expected uncertainty, assuming that such a minimum exists.

This work demonstrates the utility of combining methods from optimal control theory and estimation theory for learning uncertain fields using fleets of autonomous vehicles. Additionally, it shows that by optimizing over long-duration, continuous trajectories, superior results can be obtained when compared to ad hoc approaches such as a gradient-following control. This is demonstrated for both single-vehicle and multi-vehicle problems, and for static and evolving flow models.