TY - JOUR
T1 - Towards Massively Parallel Computations in Algebraic Geometry
AU - Böhm, Janko
AU - Decker, Wolfram
AU - Frühbis-Krüger, Anne
AU - Pfreundt, Franz-Josef
AU - Rahn, Mirko
AU - Ristau, Lukas
N1 - Publisher Copyright:
© 2020, SFoCM.
PY - 2021/6
Y1 - 2021/6
N2 - 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.
AB - 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.
KW - Computational algebraic geometry
KW - Computer algebra
KW - Distributed computing
KW - GPI-Space
KW - Hironaka desingularization
KW - Petri nets
KW - Singular
KW - Smoothness test
KW - Surfaces of general type
UR - http://www.scopus.com/inward/record.url?scp=85087557425&partnerID=8YFLogxK
U2 - 10.1007/s10208-020-09464-x
DO - 10.1007/s10208-020-09464-x
M3 - Article
VL - 21
SP - 767
EP - 806
JO - Foundations of Computational Mathematics
JF - Foundations of Computational Mathematics
SN - 1615-3375
IS - 3
ER -