Thesis: S.M., Massachusetts Institute of Technology, Center for Computational Science & Engineering, February, 2021 / Cataloged from the official PDF version of thesis. / Includes bibliographical references (pages 55-61). / We develop an exact partial differential equation-based methodology that predicts time-energy optimal paths for autonomous vehicles navigating in dynamic environments. The differential equations solve the multi-objective optimization problem of navigating a vehicle autonomously in a dynamic flow field to any destination with the goal of minimizing travel time and energy use. Based on Hamilton-Jacobi theory for reachability and the level set method, the methodology computes the exact Pareto optimal solutions to the multi-objective path planning problem, numerically solving the equations governing time-energy reachability fronts and optimal paths. Our approach is applicable to path planning in various scenarios, however we primarily present examples of navigating in dynamic marine environments. First, we validate the methodology through a benchmark case of crossing a steady front (a highway flow) for which we compare our results to semi-analytical optimal path solutions. We then consider more complex unsteady environments and solve for time-energy optimal missions in a quasi-geostrophic double-gyre ocean flow field. / by Manan Doshi. / S.M. / S.M. Massachusetts Institute of Technology, Center for Computational Science & Engineering

Thesis: S.M., Massachusetts Institute of Technology, Center for Computational Science & Engineering, February, 2021 / Cataloged from the official PDF version of thesis. / Includes bibliographical references (pages 145-151). / Though computing power continues to grow quickly, our appetite to solve larger and larger problems grows just as fast. As a consequence, reduced-order modeling has become an essential technique in the computational scientist's toolbox. By reducing the dimensionality of a system, we are able to obtain approximate solutions to otherwise intractable problems. And because the methodology we develop is sufficiently general, we may agnostically apply it to a plethora of problems, whether the high dimensionality arises due to the sheer size of the computational domain, the fine resolution we require, or stochasticity of the dynamics. In this thesis, we develop time integration schemes, called retractions, to efficiently evolve the dynamics of a system's low-rank approximation. Through the study of differential geometry, we are able to analyze the error incurred at each time step. A novel, explicit, computationally inexpensive set of algorithms, which we call perturbative retractions, are proposed that converge to an ideal retraction that projects exactly to the manifold of fixed-rank matrices. Furthermore, each perturbative retraction itself exhibits high-order convergence to the best low-rank approximation of the full-rank solution. We show that these high-order retractions significantly reduce the numerical error incurred over time when compared to a naive Euler forward retraction. Through test cases, we demonstrate their efficacy in the cases of matrix addition, real-time data compression, and deterministic and stochastic differential equations. / by Aaron Charous. / S.M. / S.M. Massachusetts Institute of Technology, Center for Computational Science & Engineering