Towards Massively Parallel Computations in Algebraic Geometry

Abstract

Introducing parallelism and exploring its use is still a fundamental challenge for the computer algebra community. In high-performance numerical simulation, on the other hand, transparent environments for distributed computing which follow the principle of separating coordination and computation have been a success story for many years. In this paper, we explore the potential of using this principle in the context of computer algebra. More precisely, we combine two well-established systems: The mathematics we are interested in is implemented in the computer algebra system Singular, whose focus is on polynomial computations, while the coordination is left to the workflow management system GPI-Space, which relies on Petri nets as its mathematical modeling language and has been successfully used for coordinating the parallel execution (autoparallelization) of academic codes as well as for commercial software in application areas such as seismic data processing. The result of our efforts is a major step towards a framework for massively parallel computations in the application areas of Singular, specifically in commutative algebra and algebraic geometry. As a first test case for this framework, we have modeled and implemented a hybrid smoothness test for algebraic varieties which combines ideas from Hironaka’s celebrated desingularization proof with the classical Jacobian criterion. Applying our implementation to two examples originating from current research in algebraic geometry, one of which cannot be handled by other means, we illustrate the behavior of the smoothness test within our framework and investigate how the computations scale up to 256 cores.

IBP reduction coefficients made simple

“…A version relying on parallelism implemented via the Singular/GPI-Space framework[75] is under development. This version will allow the use of our algorithm on HPC clusters.…”

Module Intersection for the Integration-by-Parts Reduction of Multi-Loop Feynman Integrals

“…In this section we discuss our parallel implementations in the workflow management system GPI-S [72] in combination with the computer algebra system S [71] using the framework developed in [70]. In the computation of Feynman integrals, GPI-S is used to parallelize the Gaussian reduction of the linear system which is derived from the IBP identities.…”

“…The most efficient method for performing this task is to utilize rational reconstruction and interpolation [54,[67][68][69], where by sampling different numerical points one can obtain the analytical form of the IBP coefficients. In our implementation of this idea, we rely on the S -GPI-S framework [70] for massively parallel computation in computer algebra, which combines the computer algebra system S [71] with the workflow management system GPI-S [72], and thus allows us to massively parallelize the reduction and interpolation part of the algorithm. This framework relies on Petri nets to coordinate the computation in GPI-S , while S is used as the computational back end.…”

Massively Parallel Computations in Algebraic Geometry

“…Using the flexibility of GPI-Space, we are also working on using other computer algebra systems as workers and front end. This Singular/GPI-Space approach has led to various recent success stories, for example, addressing a smoothness certificate for algebraic schemes [6], resolution of singularities [14], computation of GIT-fans in geometric invariant theory [8,18], computation of tropical varieties [4], integration-by-parts reduction of Feynman integrals [1,2], and partial fraction decomposition of integration-byparts coefficients [3]. We will give some examples in the subsequent section.…”

“…From an approach of this kind, fields like numerical simulation have already benefited significantly in recent years. Using the computer algebra system Singular as the computational back-end within the framework of the workflow management system GPI-space, which employs Petri nets to model the respective algorithm in the coordination layer, the necessary infrastructure has been developed in the PhD thesis of Lukas Ristau, see also [8]. So far, we have addressed three sample applications which illustrate the benefits of our approach in computer algebra and will be discussed in this note.…”