Numerical algebraic geometry is focused on the development of numerical algorithms for solving systems of nonlinear polynomial equations, implementations of these algorithms, and their use in solving systems arising in applications. This talk will introduce common themes in numerical algebraic geometry, particularly those related to software, and serve as a gentle introduction to the research talks in this session.
It has long been known in the numerical algebraic geometry community that parameter homotopies provide a powerful tool for solving parametrized polynomial systems at a large number of parameter values. Paramotopy is a new software package dedicated to the efficient, parallel processing of a large number of parameter homotopies. Paramotopy makes use of Bertini's implementation of user-defined homotopies, paired with Boost and other libraries for the efficient handling of large numbers of parameter values and solutions. It was initially developed to solve problems from redundant kinematics - see joint work with D. Brake (University of Notre Dame), A. Maciejewski (Colorado State University), and V. Putkaradze (University of Alberta).
This talk will include a brief introduction to parameter homotopies, a short tutorial on Paramotopy, and details about an application from biochemistry. In particular, the number of equilibria of a biochemical reaction network depends on the choice of a number of parameter values. With D. Brake, J. Gunawardena (Harvard Medical School), B. Gyori (National University of Singapore), and K-M Nam (Swarthmore College), we have devised a number of methods for better understanding the geography of the parameter space for such problems, with Paramotopy as a key ingredient.
Paramotopy is joint work with D. Brake and M. Niemerg (NIMS/Simons Institute for the Theory of Computing).
The need to numerically solve systems of polynomial equations occurs frequently in various fields of mathematics, science, and engineering. Homotopy continuation methods has been proved to be an efficient and reliable class of numerical methods for solving these systems. Hom4PS-3 is a software package for solving systems of polynomial equations that implements many different numerical homotopy methods including the Polyhedral Homotopy continuation method. Based on the successful software package Hom4PS-2.0, Hom4PS-3 has a new fully modular design which allows it to be easily extended. Furthermore, it is capable of carrying out computation in parallel on a wide range of hardware architectures including multi-core systems, computer clusters, distributed environments, and GPUs with great efficiency and scalability. Designed to be user-friendly, it includes interfaces to a variety of existing mathematical software and programming languages such as Python, Ruby, Octave, and Matlab. This talk will include a short tutorial on Hom4PS-3 and a brief introduction to the new features of Hom4PS-3. Hom4PS-3 is a joint work with Tsung-Lin Lee (National Sun Yat-sen University) and Tien-Yien Li (Michigan State University)
Many fields of study wish to compute real surfaces, often as components of algebraic varieties. This has typically been challenging, as numerical methods for polynomial systems work over the complex numbers rather than the real, and such systems may have such difficulties as positive dimensional singular sets.
In this talk, we will present Bertini Real, an implementation of the Cell Decomposition method for Algebraic Surfaces of Besana, et al. The starting point for this method is the witness set, obtained from numerical irreducible decomposition. Bertini Real works by performing repeated Curve Decompositions on specially constructed systems, such as `critical' curves, sphere intersection curves, singular curves, and slices, using isosingular deflation to reduce multiplicity when necessary. By connecting the midpoints of edges from these curves, we obtain a set of singularity-free faces, over which an implicit parametrization holds. This software package is enabled by Bertini's homotopy tracker, and produces a complete skeletal structure for real surfaces, which one can use to obtain an arbitrary sampling, perhaps for 3D printing or numerical integration, among other purposes. Bertini Real is joint work with Dan Bates (Colorado State University), Jon Hauenstein (University of Notre Dame), Charles Wampler (General Motors), Andrew Sommese (Notre Dame), and Wenrui Hao (Mathematical Biosciences Institute).
Numerical algebraic geometry is the field of computational mathematics concerning the numerical solution of polynomial systems of equations. Bertini, a popular software package for computational applications of this field, includes implementations of a variety of algorithms based on polynomial homotopy continuation.
The Macaulay2 package Bertini.m2 provides an interface to Bertini, making it possible to access the core run modes of Bertini in Macaulay2. With these run modes, users can find approximate solutions to zero-dimensional systems and positive-dimensional systems, test numerically whether a point lies on a variety, sample numerically from a variety, and perform parameter homotopy runs.
In this talk, a short tutorial will be given for the Macaulay2 package Bertini.m2 and details to an application in algbebraic statistics involving maximum likelihood estimation. This is joint work with D. Bates (Colorado State University), E. Gross (North Carolina State University), and A. Leykin (Georgia Institute of Technology).
When solving polynomial systems with homotopy continuation, the fundamental numerical linear algebra computations become inaccurate when two paths are in close proximity. The current best defense against this ill-conditioning is the use of adaptive precision. While sufficiently high precision indeed overcomes any such loss of accuracy, high precision can be very expensive. We describe a simple heuristic rooted in monodromy that can be used to try to avoid the use of high precision. This is joint work with D. Bates (Colorado State University).