Efficient Algorithms for High-Dimensional Hamilton-Jacobi Partial Differential Equations and Optimal Control ProblemsdissertationsChen, PaulacreatorDarbon, JeromeAdvisorShu, Chi-WangReaderHewer, GaryReaderBrown University. Department of Applied Mathematicssponsor2023xii, 238 p.born digitalThesis (Ph. D.)--Brown University, 2023thesesHamilton-Jacobi partial differential equations (HJ PDEs) have deep connections to a wide range of scientific disciplines including optimal control, differential games, imaging sciences, and machine learning, among many others. In this dissertation, we focus on connections between HJ PDEs, optimal control, and machine learning. In the first part of this dissertation, we utilize the well-studied connection between optimal control problems and HJ PDEs to derive representation formulas for two classes of optimal control problems with non-quadratic, state-dependent running costs or non-smooth constraints on the control and their corresponding HJ PDEs. We leverage our derived representation formulas to develop efficient numerical algorithms based on optimization techniques to solve these problems in high dimensions. We then present implementations of our algorithms on both central processing units (CPUs) and field-programmable gate arrays (FPGAs) to highlight the promising computational benefits of using FPGAs for high-performance scientific computing. In the second part of this dissertation, we establish a novel theoretical connection between the multi-time Hopf formula, which provides a representation of the solution to certain multi-time HJ PDEs, and specific optimization problems arising in machine learning. Through this connection, we increase the interpretability of the training process of certain machine learning applications by showing that when we solve these learning problems, we also solve a multi-time HJ PDE and, by extension, its corresponding optimal control problem. We then leverage our theoretical connection to adapt existing efficient numerical algorithms from optimal control to design new training approaches for machine learning.Machine LearningField programmable gate arraysRegression analysisoptimal controlHamilton-Jacobi equationsRiccati equationEnglishRPB20230602