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pfd-parallel, a Singular/GPI-Space package for massively parallel multivariate partial fractioning

Dominik Bendle, Janko Boehm, Murray Heymann, Rourou Ma, Mirko Rahn, Lukas Ristau, Marcel Wittmann, Zihao Wu, Yang Zhang

TL;DR

This work addresses the challenge of simplifying enormous sets of multivariate rational functions arising in scattering amplitudes by implementing a massively parallel partial fractioning framework. It fuses an improved Leinartas algorithm with MultivariateApart inside the Singular/GPI-Space ecosystem, enabling parallelism across coefficient batches and within each coefficient’s decomposition via Petri-net–based workflows. The authors demonstrate substantial compression and performance gains on demanding problems, including IBP reduction coefficients for a two-loop five-point diagram and analytic two-loop Wγ+j amplitudes, achieved on HPC resources with open-source tools. This approach offers scalable, modular, and reproducible reductions critical for advancing multiloop computations in high-energy physics. The framework’s Petri-net–driven parallelization and modular design hold promise for broader adoption in modular algebraic computations beyond scattering amplitudes.

Abstract

Multivariate partial fractioning is a powerful tool for simplifying rational function coefficients in scattering amplitude computations. Since current research problems lead to large sets of complicated rational functions, performance of the partial fractioning as well as size of the obtained expressions are a prime concern. We develop a large scale parallel framework for multivariate partial fractioning, which implements and combines an improved version of Leinartas' algorithm and the {\sc MultivariateApart} algorithm. Our approach relies only on open source software. It combines parallelism over the different rational function coefficients with parallelism for individual expressions. The implementation is based on the \textsc{Singular}/\textsc{GPI-Space framework} for massively parallel computer algebra, which formulates parallel algorithms in terms of Petri nets. The modular nature of this approach allows for easy incorporation of future algorithmic developments into our package. We demonstrate the performance of our framework by simplifying expressions arising from current multiloop scattering amplitude problems.

pfd-parallel, a Singular/GPI-Space package for massively parallel multivariate partial fractioning

TL;DR

This work addresses the challenge of simplifying enormous sets of multivariate rational functions arising in scattering amplitudes by implementing a massively parallel partial fractioning framework. It fuses an improved Leinartas algorithm with MultivariateApart inside the Singular/GPI-Space ecosystem, enabling parallelism across coefficient batches and within each coefficient’s decomposition via Petri-net–based workflows. The authors demonstrate substantial compression and performance gains on demanding problems, including IBP reduction coefficients for a two-loop five-point diagram and analytic two-loop Wγ+j amplitudes, achieved on HPC resources with open-source tools. This approach offers scalable, modular, and reproducible reductions critical for advancing multiloop computations in high-energy physics. The framework’s Petri-net–driven parallelization and modular design hold promise for broader adoption in modular algebraic computations beyond scattering amplitudes.

Abstract

Multivariate partial fractioning is a powerful tool for simplifying rational function coefficients in scattering amplitude computations. Since current research problems lead to large sets of complicated rational functions, performance of the partial fractioning as well as size of the obtained expressions are a prime concern. We develop a large scale parallel framework for multivariate partial fractioning, which implements and combines an improved version of Leinartas' algorithm and the {\sc MultivariateApart} algorithm. Our approach relies only on open source software. It combines parallelism over the different rational function coefficients with parallelism for individual expressions. The implementation is based on the \textsc{Singular}/\textsc{GPI-Space framework} for massively parallel computer algebra, which formulates parallel algorithms in terms of Petri nets. The modular nature of this approach allows for easy incorporation of future algorithmic developments into our package. We demonstrate the performance of our framework by simplifying expressions arising from current multiloop scattering amplitude problems.

Paper Structure

This paper contains 16 sections, 21 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Partial fraction decomposition
  3. Nullstellensatz decomposition
  4. Algebraic dependence decomposition
  5. Numerator decomposition
  6. The resulting algorithm
  7. A simple example
  8. Parallel structures in the algorithm
  9. Parallel computing using the Singular/GPI-Space framework
  10. GPI-Space and Singular
  11. Algorithmic realization of the partial fraction decomposition based on Petri nets
  12. Instructions on how to use the massively parallel PFD application
  13. Examples for partial fraction via large scale parallelizations
  14. Simplifying IBP reduction coefficients
  15. Simplifying analytic multi-loop scattering amplitudes
  16. ...and 1 more sections

Figures (11)

  • Figure 1: The WaitAll Petri net to parallelize over an input tuple
  • Figure 2: The subnet calculating the partial fraction decomposition.
  • Figure 3: The Fork/Join subnet to parallelize over the decomposition of terms.
  • Figure 4: The WaitAll Petri net with pre-sorting by input file size
  • Figure 5: Two-loop five-point nonplanar "double pentagon" diagram
  • ...and 6 more figures

Theorems & Definitions (5)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Remark 3.4
  • Remark 4.1