HPC‐GAP: engineering a 21st‐century high‐performance computer algebra system

Abstract

SUMMARYSymbolic computation has underpinned a number of key advances in Mathematics and Computer Science. Applications are typically large and potentially highly parallel, making them good candidates for parallel execution at a variety of scales from multi-core to high-performance computing systems. However, much existing work on parallel computing is based around numeric rather than symbolic computations. In particular, symbolic computing presents particular problems in terms of varying granularity and irregular task sizes that do not match conventional approaches to parallelisation. It also presents problems in terms of the structure of the algorithms and data. This paper describes a new implementation of the free open-source GAP computational algebra system that places parallelism at the heart of the design, dealing with the key scalability and cross-platform portability problems. We provide three system layers that deal with the three most important classes of hardware: individual shared memory multi-core nodes, mid-scale distributed clusters of (multi-core) nodes and full-blown high-performance computing systems, comprising large-scale tightly connected networks of multi-core nodes. This requires us to develop new cross-layer programming abstractions in the form of new domain-specific skeletons that allow us to seamlessly target different hardware levels. Our results show that, using our approach, we can achieve good scalability and speedups for two realistic exemplars, on high-performance systems comprising up to 32 000 cores, as well as on ubiquitous multi-core systems and distributed clusters. The work reported here paves the way towards full-scale exploitation of symbolic computation by high-performance computing systems, and we demonstrate the potential with two major case studies.

BIOS-ParallelBlast: Paralelización optimizada de alineamiento de secuencias sobre Xeon Phi

“…En la actualidad la supercomputación es un campo ampliamente estudiado, de hecho se considera uno de los tres pilares de la investigación, gracias a que mu-chos de los avances en la ciencia y la ingeniería están basados en ella (González et al, 2015) en la literatura se encuentran implementaciones realizadas por investigadores en donde buscan solucionar problemas u optimizar recursos, incluso usando máquinas virtuales (Huang et al, 2006) y para dar solución a problemáticas en campos tan variados como las matemáticas (Behrends et al, 2016), diseño de turbinas de gas (Alhatim, 2016), secuenciación (Al-Ali et al, 2016), entre otras.…”

Commutativity Degree of Crossed Modules

“…GAP (Groups, Algorithms, Programming [12]) is the leading symbolic computation system for solving computational discrete algebra problems. Symbolic computation has underpinned several key advances in Mathematics and Computer Science, for example, in number theory and coding theory (see [5] ). The system consists of a library of implementations of mathematical structures: groups, vector spaces, modules, algebras, graphs, codes, designs, etc.…”

“…gap> all48 := AllXMods( [4,8]);; gap> Length(all48); 336 gap> iso_all48 := AllXModsUpToIsomorphism(all48);; gap> Length(iso_all48); 59 gap> f_1 := IsoclinicXModFamily(iso_all48 [1],iso_all48); [ 1,3,4,6,9,10,12,27,29,31,33,35,38,40,43,55,57,59 ] gap> f_2 := IsoclinicXModFamily(iso_all48 [2],iso_all48); [ 2,5,7,8,11,13,14,28,30,32,34,36,37,39,41,42,56,58 ] gap> f_3 := IsoclinicXModFamily(iso_all48 [15],iso_all48); [ 15,18,22,25,44,47,51,53 ] gap> f_4 := IsoclinicXModFamily(iso_all48 [16],iso_all48); [ 16,17,19,…”

Current Challenges in Developing Open Source Computer Algebra Systems

“…Using ideas from the successful parallelization of GAP within the HPC-GAP project (see [4,5,6]), a multi-threaded prototype of Singular has been implemented. Considerable further efforts are needed, however, to make this accessible to users without a deep background in parallel programming.…”

“…To remedy this situation, we consider the singular locus of A, Sing(A) = {P ∈ Spec(A) | A P is not regular}, which contains the non-normal locus: N (A) ⊆ Sing(A). Since we work over a perfect field K, the Jacobian criterion tells us that Sing(A) = V (Jac(I)), where Jac(I) is the Jacobian ideal 5 of A (see [27]). Hence, if we choose J = Jac(I), the above process terminates with A m = A by the following lemma.…”