Browsing by Author "Knepley, Matthew G"

Now showing 1 - 3 of 3
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    A Multigrid Solver for Graph Laplacian Linear Systems on Power-Law Graphs
    (2016-05-17) Buras, Eric; Knepley, Matthew G
    The Laplacian matrix, L, of a graph, G, contains degree and edge information of a given network. Solving a Laplacian linear system Lx = b provides information about flow through the network, and in specific cases, how that information orders the nodes in the network. I propose a novel way to solve this linear system by first partitioning G into its maximum locally-connected subgraph and a small subgraph of the remaining so-called "teleportation" edges. I then apply optimal multigrid solves to the locally-connected subgraph, and linear algebra and a solve on the teleportation subgraph to solve the original linear system. I show results for this method on real-world graphs from the biological systems of the C. Elegans worm, Facebook friend networks, and the power grid of the Western United States.
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    Accurate Evaluation of Ellipsoidal Harmonics Using Tanh-Sinh Quadrature
    (2017-07-20) Klotz, Thomas S; Knepley, Matthew G
    Ellipsoidal coordinates are an orthogonal coordinate system under which the Laplace equation can be solved by separation of variables. While this has many benefits over spherical coordinates for a variety of potential problems, computation in ellipsoidal coordinates is difficult. Most notably, high-order harmonics can lack closed-form solutions and the associated normalization constants require approximating a singular integral. We provide a method for computing normalization constants to machine precision using tanh-sinh quadrature which exhibits exponential convergence for a large class of functions with singular endpoints. Combined with previous efforts to make ellipsoidal harmonics more accessible, the result is a library which makes computation in ellipsoidal coordinates as accessible as computation in spherical coordinates. Finally, we apply our implementation to the mixed-dielectric solvation problem and provide work-precision analysis for the results.
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    Computation for the Kolmogorov Superposition Theorem
    (2018-05-08) Actor, Jonas; Knepley, Matthew G
    This thesis presents the first known method to compute Lipschitz continuous inner functions for the Kolmogorov Superposition Theorem. While the inner functions of these superpositions can be Lipschitz continuous, previous algorithms that compute inner functions yield only Hölder continuous functions. These Hölder continuous functions suffer from high storage-to-evaluation complexity, thereby rendering the Kolmogorov Superposition Theorem impractical for computation. Once this concern is addressed, the Kolmogorov Superposition Theorem becomes an effective tool in dimension reduction to represent multivariate functions as univariate expressions, with applications in encryption, video compression, and image analysis. In this thesis, I posit sufficient criteria to iteratively construct a Lipschitz continuous inner function. I demonstrate that implementations of the predominant approach for such a Lipschitz construction do not satisfy these conditions. Instead, I manipulate Hölder continuous inner functions to create Lipschitz continuous reparameterizations. I show these reparameterizations meet my sufficient conditions, thereby guaranteeing that these reparameterized functions are inner functions for the Kolmogorov Superposition Theorem.