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• 1

Dataset

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
• 2

Dataset

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
• 3

Dataset

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
• 4

Dataset

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
• 5

Dataset

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
• Creator(s):
Margaret Regan, Jonathan Hauenstein
Description:

Three key aspects of applying homotopy continuation to parameterized systems of polynomial equations are investigated. First, for parameterized systems which are homogenized with solutions in projective space, we investigate options for selecting the affine patch where computations are performed. Second, for parameterized systems which are overdetermined, we investigate options for randomizing the system for improving the numerically stability of the computations. Finally, since one is typica…

Date Created:
2017-07-06
• Creator(s):
Jonathan Hauenstein
Description:

The Kuramoto model describes synchronization behavior among coupled oscillators and enjoys successful application in a wide variety of fields. Many of these applications seek phase-coherent solutions, i.e., equilibria of the model. Historically, research has focused on situations where the number of oscillators, n, is extremely large and can be treated as being infinite. More recently, however, applications have arisen in areas such as electrical engineering with more modest values of n. For…

Date Created:
2017-04-07
• 8

Video

Description(s):
A movie showing the change in the number of stable steady-state solutions as a function of the cell-to-cell communication. Software code using Matlab and Bertini is also provided which was used to generate the frames of this movie.
Creator(s):
Jonathan Hauenstein
Date Published:
2016
• Creator(s):
Jonathan Hauenstein
Description:

Stewart-Gough platforms are mechanisms which consist of two rigid objects, a base and a platform, connected by six legs via spherical joints. For fixed leg lengths, a generic Stewart-Gough platform is rigid with 40 assembly configurations (over the complex numbers) while exceptional Stewart-Gough platforms have infinitely many assembly configurations and thus have self-motion. We define a family of exceptional Stewart-Gough platforms called Segre-dependent Stewart-Gough platforms which aris…

Date Created:
2017-01-17
• 10

Dataset

Creator(s):
Jonathan Hauenstein
Description:

We define tensors, most of which correspond with cubic polynomials, which have the same exponent w as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor defined on an nxn matrix A by trace(A^3). The use of polynomials enables the introduction of additional techniques from algebraic geometry in the study of the matrix multiplication exponent w.

Date Created:
2017-01-07
• 11

Article

Author(s):
Daniel Bates, Andrew Sommese, Jonathan Hauenstein
Abstract:

Numerical algebraic geometry is the area devoted to the solution and manipulation of polynomial systems by numerical methods, which are mainly based on continuation. Due to the extreme intrinsic parallelism of continuation, polynomial systems may be successfully dealt with that are much larger than is possible with other methods. Singular solutions require special numerical methods called endgames, and the endgames currently used do not take advantage of parallelism. This article gives an ove…

• 12

Article

Author(s):
Charles Wampler II, Daniel Bates, Andrew Sommese, Jonathan Hauenstein
Abstract:

Dedicated to our collaborator, mentor, and friend, Andrew Sommese, by Bates, Hauenstein, and Wampler on the occasion of his sixtieth birthday.

When numerically tracking implicitly-defined paths, such as is required for homotopy continuation methods, efficiency and reliability are enhanced by using adaptive stepsize and adaptive multiprecision methods. Both efficiency and reliability can be further improved by adapting precision and stepsize simultaneously. This paper presents a strategy fo…