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Decomposing Solution Sets of Polynomial Systems: A New Parallel Monodromy Breakup Algorithm

Anton Leykin, Jan Verschelde

TL;DR

The paper addresses numerical irreducible decomposition of positive-dimensional polynomial-solution sets by leveraging monodromy around singularities and linear-trace certification. It introduces a novel monodromy breakup algorithm with two generic slices and an adaptive, workload-aware loop strategy, coupled with a parallel master/servant implementation that interleaves trace certification with path tracking. Key contributions include a serial and a parallel version of the algorithm, a bipartite-graph viewpoint to minimize path tracking, and experimental evidence showing improved performance and scalability on large-degree systems (e.g., a degree-$256$, 18-variable example with 34 irreducible components). The results demonstrate more predictable, efficient, and scalable numerical irreducible decomposition using PHCpack/MPI for distributed computing, enabling factorization of larger and more complex solution sets in practice.

Abstract

We consider the numerical irreducible decomposition of a positive dimensional solution set of a polynomial system into irreducible factors. Path tracking techniques computing loops around singularities connect points on the same irreducible components. The computation of a linear trace for each factor certifies the decomposition. This factorization method exhibits a good practical performance on solution sets of relative high degrees. Using the same concepts of monodromy and linear trace, we present a new monodromy breakup algorithm. It shows a better performance than the old method which requires construction of permutations of witness points in order to break up the solution set. In contrast, the new algorithm assumes a finer approach allowing us to avoid tracking unnecessary homotopy paths. As we designed the serial algorithm keeping in mind distributed computing, an additional advantage is that its parallel version can be easily built. Synchronization issues resulted in a performance loss of the straightforward parallel version of the old algorithm. Our parallel implementation of the new approach bypasses these issues, therefore, exhibiting a better performance, especially on solution sets of larger degree.

Decomposing Solution Sets of Polynomial Systems: A New Parallel Monodromy Breakup Algorithm

TL;DR

The paper addresses numerical irreducible decomposition of positive-dimensional polynomial-solution sets by leveraging monodromy around singularities and linear-trace certification. It introduces a novel monodromy breakup algorithm with two generic slices and an adaptive, workload-aware loop strategy, coupled with a parallel master/servant implementation that interleaves trace certification with path tracking. Key contributions include a serial and a parallel version of the algorithm, a bipartite-graph viewpoint to minimize path tracking, and experimental evidence showing improved performance and scalability on large-degree systems (e.g., a degree-, 18-variable example with 34 irreducible components). The results demonstrate more predictable, efficient, and scalable numerical irreducible decomposition using PHCpack/MPI for distributed computing, enabling factorization of larger and more complex solution sets in practice.

Abstract

We consider the numerical irreducible decomposition of a positive dimensional solution set of a polynomial system into irreducible factors. Path tracking techniques computing loops around singularities connect points on the same irreducible components. The computation of a linear trace for each factor certifies the decomposition. This factorization method exhibits a good practical performance on solution sets of relative high degrees. Using the same concepts of monodromy and linear trace, we present a new monodromy breakup algorithm. It shows a better performance than the old method which requires construction of permutations of witness points in order to break up the solution set. In contrast, the new algorithm assumes a finer approach allowing us to avoid tracking unnecessary homotopy paths. As we designed the serial algorithm keeping in mind distributed computing, an additional advantage is that its parallel version can be easily built. Synchronization issues resulted in a performance loss of the straightforward parallel version of the old algorithm. Our parallel implementation of the new approach bypasses these issues, therefore, exhibiting a better performance, especially on solution sets of larger degree.

Paper Structure

This paper contains 12 sections, 5 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Problem Statement
  3. Numerical Homotopies define Loops around Singularities
  4. Parallel Algorithms
  5. USING MONODROMY MORE EFFICIENTLY
  6. A NEW ALGORITHM
  7. Serial Version
  8. Parallel Version
  9. EXPERIMENTAL RESULTS
  10. Plain Parallel Path Tracking
  11. Performance of the new Algorithm
  12. CONCLUSIONS

Figures (4)

  • Figure 1: The bipartite graph $W_1\leftrightarrow W_2$ for cyclic4
  • Figure 2: Permutation (12)(34) for cyclic4
  • Figure 3: A 6-edge graph for cyclic4
  • Figure 4: The flow chart of our new Monodromy Breakup Algorithm