Robust and efficient solution of the drum problem via Nystrom approximation of the Fredholm determinant

The drum problem-finding the eigenvalues and eigenfunctions of the Laplacian with Dirichlet boundary condition-has many applications, yet remains challenging for general domains when high accuracy or high frequency is needed. Boundary integral equations are appealing for large-scale problems, yet certain difficulties have limited their use. We introduce two ideas to remedy this: 1) We solve the resulting nonlinear eigenvalue problem using Boyd's method for analytic root-finding applied to the Fredholm determinant. We show that this is many times faster than the usual iterative minimization of a singular value. 2) We fix the problem of spurious exterior resonances via a combined field representation. This also provides the first robust boundary integral eigenvalue method for non-simply-connected domains. We implement the new method in two dimensions using spectrally accurate Nystrom product quadrature. We prove exponential convergence of the determinant at roots for domains with analytic boundary. We demonstrate 13-digit accuracy, and improved efficiency, in a variety of domain shapes including ones with strong exterior resonances.Comment: 21 pages, 7 figures, submitted to SIAM Journal of Numerical Analysis. Updated a duplicated picture. All results unchange

Algorithms for Optimal Paths of One, Many, and an Infinite Number of Agents

In this dissertation, we provide efficient algorithms for modeling the behavior of a single agent, multiple agents, and a continuum of agents. For a single agent, we combine the modeling framework of optimal control with advances in optimization splitting in order to efficiently find optimal paths for problems in very high-dimensions, thus providing alleviation from the curse of dimensionality. For a multiple, but finite, number of agents, we take the framework of multi-agent reinforcement learning and utilize imitation learning in order to decentralize a centralized expert, thus obtaining optimal multi-agents that act in a decentralized fashion. For a continuum of agents, we take the framework of mean-field games and use two neural networks, which we train in an alternating scheme, in order to efficiently find optimal paths for high-dimensional and stochastic problems. These tools cover a wide variety of use-cases that can be immediately deployed for practical applications

Fractal scattering dynamics of the three-dimensional HOCl molecule

We compare the 2D and 3D classical fractal scattering dynamics of Cl and HO for energies just above dissociation of the HOCl molecule, using a realistic potential energy surface for the HOCl molecule and techniques developed to analyze 3D chaotic scattering processes. For parameter regimes where the HO dimer initially has small vibrational energy, only small intervals of initial conditions show fractal scattering behavior and the scattering process is well described by a 2D model. For parameter regimes where the HO dimer initially has large vibrational energy, the scattering process is fully 3D and is dominated by fractal behavior.Robert A. Welch Foundation F-1051CONACyT 79988DGAPA IN110110Physic