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  1. Free, publicly-accessible full text available June 1, 2023
  2. Free, publicly-accessible full text available April 1, 2023
  3. We study discretizations of Hamiltonian systems on the probability density manifold equipped with the L 2 L^2 -Wasserstein metric. Based on discrete optimal transport theory, several Hamiltonian systems on a graph (lattice) with different weights are derived, which can be viewed as spatial discretizations of the original Hamiltonian systems. We prove consistency of these discretizations. Furthermore, by regularizing the system using the Fisher information, we deduce an explicit lower bound for the density function, which guarantees that symplectic schemes can be used to discretize in time. Moreover, we show desirable long time behavior of these symplectic schemes, and demonstrate theirmore »performance on several numerical examples. Finally, we compare the present approach with the standard viscosity methodology.« less
    Free, publicly-accessible full text available March 14, 2023
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  5. Free, publicly-accessible full text available December 1, 2022
  6. A bounded cost path planning method is developed for underwater vehicles assisted by a data-driven flow modeling method. The modeled flow field is partitioned as a set of cells of piece-wise constant flow speed. A flow partition algorithm and a parameter estimation algorithm are proposed to learn the flow field structure and parameters with justified convergence. A bounded cost path planning algorithm is developed taking advantage of the partitioned flow model. An extended potential search method is proposed to determine the sequence of partitions that the optimal path crosses. The optimal path within each partition is then determined by solvingmore »a constrained optimization problem. Theoretical justification is provided for the proposed extended potential search method generating the optimal solution. The path planned has the highest probability to satisfy the bounded cost constraint. The performance of the algorithms is demonstrated with experimental and simulation results, which show that the proposed method is more computationally efficient than some of the existing methods.« less