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  • Author:
    Evercita Cuevas Eugenio
    Advisory Committee:
    Ick-Hoon Jin, Fang Liu, Lizhen Lin
    Degree Area:
    Applied and Computational Mathematics and Statistics
    Degree:
    Doctor of Philosophy
    Defense Date:
    2019-06-04
  • Author:
    Kelsey DiPietro
    Advisory Committee:
    Alan E. Lindsay
    Degree Area:
    Applied and Computational Mathematics and Statistics
    Degree:
    Doctor of Philosophy
    Defense Date:
    2019-05-02
  • Author:
    Ryan Pyle
    Advisory Committee:
    Robert J. Rosenbaum, Alan Lindsay, Lizhen Lin
    Degree Area:
    Applied and Computational Mathematics and Statistics
    Degree:
    Doctor of Philosophy
    Defense Date:
    2019-04-16
  • Creator(s):
    Dan Bates, David Eklund, Jonathan Hauenstein, Chris Peterson
    Description:

    A fundamental problem in algebraic geometry is to decompose the solution set of a given polynomial system. A numerical description of this solution set is called a numerical irreducible decomposition and currently all standard algorithms use a sequence of homotopies forming a dimension-by-dimension approach. In this article, we pair a classical result to compute a smooth point on every irreducible component in every dimension using a single homotopy together with the theory of isosingular s…

    Date Created:
    2019-04-26
  • Author:
    Kirsten Marie Kozlovsky
    Advisory Committee:
    Steven Schmid
    Degree Area:
    Aerospace and Mechanical Engineering
    Degree:
    Doctor of Philosophy
    Defense Date:
    2019-03-26
  • Author:
    Christopher L. Ebsch
    Advisory Committee:
    Robert Rosenbaum
    Degree Area:
    Applied and Computational Mathematics and Statistics
    Degree:
    Doctor of Philosophy
    Defense Date:
    2019-03-29
  • Author:
    Martin P. Barron
    Advisory Committee:
    Fang Liu, Steven Buechler, Jun Li
    Degree Area:
    Applied and Computational Mathematics and Statistics
    Degree:
    PhD
    Defense Date:
    2018-06-14
  • Creator(s):
    Jonathan Hauenstein, Alan Liddell, Yi Zhang
    Description:

    Standard interior point methods in semidefinite programming track a solution path for a homotopy defined by a system of polynomial equations. By viewing this in the context of numerical algebraic geometry, we are able to employ techniques to handle various cases which can arise. Adaptive precision path tracking techniques can help navigate around ill-conditioned areas. When the optimizer is singular with respect to the first-order optimality conditions, endgames can be used to efficiently and…

    Date Created:
    2018-04-10
  • Author:
    Claire McKay Bowen
    Advisory Committee:
    Fang Liu
    Degree Area:
    Applied and Computational Mathematics and Statistics
    Degree:
    PhD
    Defense Date:
    2018-03-27
  • Creator(s):
    Jonathan Hauenstein
    Description:

    A common problem when analyzing models, such as a mathematical modeling of a biological process, is to determine if the unknown parameters of the model can be determined from given input-output data. Identifiable models are models such that the unknown parameters can be determined to have a finite number of values given input-output data. The total number of such values over the complex numbers is called the identifiability degree of the model. Unidentifiable models are models such that th…

    Date Created:
    2018-03-09
  • Creator(s):
    Samantha Sherman, Jonathan Hauenstein
    Description:

    Computational tools in numerical algebraic geometry can be used to numerically approximate solutions to a system of polynomial equations. If the system is well-constrained (i.e., square), Newton’s method is locally quadratically convergent near each nonsingular solution. In such cases, Smale’s alpha theory can be used to certify that a given point is in the quadratic convergence basin of some solution. This was extended to certifiably determine the reality of the corresponding sol…

    Date Created:
    2018-03-08
  • Creator(s):
    Margaret Regan, Jonathan Hauenstein
    Description:

    A common computational problem is to compute topological information about a real surface defined by a system of polynomial equations. Our software, called polyTop, leverages numerical algebraic geometry computations from Bertini and Bertini_real with topological computations in javaPlex to compute the Euler characteristic, genus, Betti numbers, and generators of the fundamental group of a real surface. Several examples are used to demonstrate this new software.

    Date Created:
    2018-03-02