Paramotopy: Parameter Homotopies in Parallel

Abstract

Numerical algebraic geometry provides a number of efficient tools for approximating the solutions of polynomial systems. One such tool is the parameter homotopy, which can be an extremely efficient method to solve numerous polynomial systems that differ only in coefficients, not monomials. This technique is frequently used for solving a parameterized family of polynomial systems at multiple parameter values. Parameter homotopies have recently been useful in several areas of application and have been implemented in at least two software packages. This article describes Paramotopy, a new, parallel, optimized implementation of this technique, making use of the Bertini software package. The novel features of this implementation, not available elsewhere, include allowing for the simultaneous solutions of arbitrary polynomial systems in a parameterized family on an automatically generated (or manually provided) mesh in the parameter space of coefficients, front ends and back ends that are easily specialized to particular classes of problems, and adaptive techniques for solving polynomial systems near singular points in the parameter space. This last feature automates and simplifies a task that is important but often misunderstood by non-experts.1 In fact, this technique applies when the coefficients are holomorphic functions of the parameters [28], but we restrict to the case of polynomials as Bertini is restricted to polynomials.

Computing the Feasible Spaces of Optimal Power Flow Problems

“…Future work includes computational improvements, such as integration with the software Paramotopy [38] to reduce overhead and exploitation of network structure [40], [41] to reduce the initial solution time for the parameterized NPHC algorithm. Future work also includes applying the proposed algorithm to other test cases to further characterize the physical features that are associated with challenging OPF problems.…”

“…The guarantees inherent to the NPHC algorithm ensure the capturing of the entire OPF feasible space. The proposed algorithm is similar to that used in the software Paramotopy [38] for visualizing the effects of parameter variation in general polynomial systems.…”

Towards a complete mass spectrum of type-IIB flux vacua at large complex structure

“…In order to perform a numerical exploration of the flux landscape of the octic, we have used Paramotopy [110]. This software uses a numerical technique known as the Polynomial Homotopy Continuation (PHC) method [111,112], which efficiently finds all roots of non-linear polynomial systems, such as the no-scale equations (2.20) (see appendix B).…”

“…Many different implementations of the PHC method can be found in the literature, such as phcpy [133], StringVacua [134], and Bertini [135]. In this work, we have used Paramotopy 22 [110], a highly efficient PHC-based algorithm specially suited for polynomial systems like (B.1) which depend on parameter tuples.…”

Decomposing the Parameter Space of Biological Networks via a Numerical Discriminant Approach

“…Recently, it has become possible to compute all solutions over the complex numbers C to a system of polynomial equations using homotopy continuation and, more generally, numerical algebraic geometry, (see (10)(11)(12)). Such methods have been implemented in software packages including Bertini (13), HOM4PS-3 (14), and PHCpack (15) with Paramtopy (16) extending Bertini to study the solutions at many points in parameter space. Typically, these methods work over C while the solutions of interest in biological models are in a subset of the real numbers R, e.g., one is interested in steady-states in the positive orthant where the variables are biologically meaningful.…”