• About
    • Login
    View Item 
    •   Institutional Repository Home
    • Electronic Theses and Dissertations
    • Electronic Theses and Dissertations
    • View Item
    •   Institutional Repository Home
    • Electronic Theses and Dissertations
    • Electronic Theses and Dissertations
    • View Item
    JavaScript is disabled for your browser. Some features of this site may not work without it.

    Browse

    All of Institutional RepositoryCommunities & CollectionsBy Issue DateAuthorsTitlesSubjectsDepartmentThis CollectionBy Issue DateAuthorsTitlesSubjectsDepartment

    My Account

    LoginRegister

    Riesz energy functionals and their applications

    Vlasiuk, Oleksandr
    : https://etd.library.vanderbilt.edu/etd-08162018-100322
    http://hdl.handle.net/1803/13917
    : 2018-08-27

    Abstract

    First-order asymptotics of a class of functionals with the Riesz kernel $ |s x - s y|^{-s} $, equipped with an external field and a weight, is obtained. Characterization of the weak$ ^* $ limit of the discrete minimizers of such functionals on rectifiable sets is given, and expressed in terms of the $ Gamma $-convergence of functionals on the space of probability measures. Optimality of local covering and separation of the discrete minimizers is proved. Several properties of the minimizers of the unweighted energy functional are established in the case of optimization on a self-similar fractal, rather than a rectifiable set; in particular, the weak$ ^* $ limit of a sequence with the lowest asymptotics is characterized, and the structure of sequences achieving different asymptotics on a self-similar fractal with equal scaling coefficients is described. A number of numerical examples involving distribution of discrete configurations on surfaces are given, as well as inside domains of full dimension, with both smooth and non-smooth boundaries. Applications of the Riesz energy functionals to constructing RBF-FD stencils are described and characterized in several cases, including an instance of a configuration for geo-scale atmospheric modeling.
    Show full item record

    Files in this item

    Icon
    Name:
    dissertationVlasiuk.pdf
    Size:
    7.580Mb
    Format:
    PDF
    View/Open

    This item appears in the following collection(s):

    • Electronic Theses and Dissertations

    Connect with Vanderbilt Libraries

    Your Vanderbilt

    • Alumni
    • Current Students
    • Faculty & Staff
    • International Students
    • Media
    • Parents & Family
    • Prospective Students
    • Researchers
    • Sports Fans
    • Visitors & Neighbors

    Support the Jean and Alexander Heard Libraries

    Support the Library...Give Now

    Gifts to the Libraries support the learning and research needs of the entire Vanderbilt community. Learn more about giving to the Libraries.

    Become a Friend of the Libraries

    Quick Links

    • Hours
    • About
    • Employment
    • Staff Directory
    • Accessibility Services
    • Contact
    • Vanderbilt Home
    • Privacy Policy