High-Performance Computer Algebra: A Hecke Algebra Case Study

Maier, Patrick; Livesey, Daria; Loidl, Hans-Wolfgang; Trinder, Phil

doi:10.1007/978-3-319-09873-9_35

Cited by 5 publications

(12 citation statements)

References 20 publications

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“…Our results with a challenging symbolic computation application on up to 1024 cores of HECToR show good scalability for a program with complex data dependencies (Maier et al, 2014a). However, we anticipate that larger, hierarchical systems will highlight the need for improved control of locality, hierarchical workload management, and other mechanisms of work and data distribution that we have discussed above.…”

Section: Limitations and Further Workmentioning

confidence: 67%

PAEAN: Portable and scalable runtime support for parallel Haskell dialects

Berthold

Loidl²,

Hammond

2016

J. Funct. Prog.

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Over time, several competing approaches to parallel Haskell programming have emerged. Different approaches support parallelism at various different scales, ranging from small multicores to massively parallel high-performance computing systems. They also provide varying degrees of control, ranging from completely implicit approaches to ones providing full programmer control. Most current designs assume a shared memory model at the programmer, implementation and hardware levels. This is, however, becoming increasingly divorced from the reality at the hardware level. It also imposes significant unwanted runtime overheads in the form of garbage collection synchronisation etc. What is needed is an easy way to abstract over the implementation and hardware levels, while presenting a simple parallelism model to the programmer.The PAEAN (PArallEl shAred Nothing) runtime system design aims to provide a portable and high-level shared-nothing implementation platform for parallel Haskell dialects. It abstracts over major issues such as work distribution and data serialisation, consolidating existing, successful designs into a single framework. It also provides an optional virtual shared-memory programming abstraction for (possibly) shared-nothing parallel machines, such as modern multicore/manycore architectures or cluster/cloud computing systems. It builds on, unifies, and extends, existing well-developed support for shared-memory parallelism that is provided by the widely-used GHC Haskell compiler. This paper summarises the state-of-the-art in shared-nothing parallel Haskell implementations, introduces the PAEAN abstractions, shows how they can be used to implement three distinct parallel Haskell dialects, and demonstrates that good scalability can be obtained on recent parallel machines. * Corresponding author. Reported work performed while at the University of Copenhagen (DIKU).

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Section: Limitations and Further Workmentioning

confidence: 67%

PAEAN: Portable and scalable runtime support for parallel Haskell dialects

Berthold

Loidl²,

Hammond

2016

J. Funct. Prog.

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“…Actually, apart from our own experiences, this kind of approach is pursued in [20], and the figures on timings and memory consumption given there seem to support the above comments. But it should be stressed that the emphasis of [20] is on parallelizing this kind of computations, which we here do not consider at all.…”

Section: Inverse Given a Matrixmentioning

confidence: 79%

“…In view of the complexity of the task, and the experiences made elsewhere with explicit computations in type E 8 (see the references cited above), it was clear that developing efficient methods would not be a standard, let alone press-button application of existing tools from computer algebra. Maier et al [20] proposed an approach based on parallel techniques, but type E 8 still seems to be a major challenge there. Hence one of the purposes of this paper is to give a systematic description of the (serial) methods we have used for the computation of Gram matrices of invariant bilinear forms for Iwahori-Hecke algebras.…”

Section: Introductionmentioning

confidence: 99%

Invariant Bilinear Forms on W-Graph Representations and Linear Algebra Over Integral Domains

Geck

Müller

2017

Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory

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Lie-theoretic structures of type E 8 (e.g., Lie groups and algebras, Iwahori-Hecke algebras and Kazhdan-Lusztig cells, . . .) are considered to serve as a "gold standard" when it comes to judging the effectiveness of a general algorithm for solving a computational problem in this area. Here, we address a problem that occurred in our previous work on decomposition numbers of Iwahori-Hecke algebras, namely, the computation of invariant bilinear forms on so-called W -graph representations. We present a new algorithmic solution which makes it possible to produce and effectively use the main results in further applications.It should be clear from the above description that to pursue our novel approach we had to solve quite a few tasks for which there was no pre-existing implementation, let alone in one and the same computer algebra system. To develop the necessary new code, as our computational platform we have chosen the computer algebra system GAP [4]. This system provides efficient arithmetics for the various basic objects we need: (i) rational integers and rational numbers, which in turn are handled by the GMP library [13]; (ii) row vectors and matrices over the integers, the rationals or (small) finite fields, where in this context the entries of row vectors are actually treated as immediate objects; (iii) floating point numbers, where the limited built-in facilities are sufficient for our purposes. Moreover, the necessary input data on Iwahori-Hecke algebras and their representations is provided by the computer algebra system CHEVIE [21], which conveniently is a branch of GAP.

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“…At first glance, it may seem that the sequential Orbit algorithm can be parallelised by carrying out step 5 for all generators and all points in U independently. However, synchronisation is required to avoid duplicates (steps [6][7][8], and this requires communication among the tasks that apply generators to different elements of U . Our Parallel Orbit skeleton uses two different kinds of threads:…”

Section: Definition 1 (Orbit Of An Element)mentioning

confidence: 99%

“…In the following, we will only sketch this case study and present the main per- formance results; the reader is referred to [6] for a detailed report and to [62] for background on the problem. An n-dimensional representation of H is an R-algebra homomorphism from H to M n .R/, the R-algebra of n n matrices over R D ZOEx; x 1 , the ring of Laurent polynomials in indeterminate x.…”

Section: Hecke Algebras In Sgp2mentioning

confidence: 99%

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

Behrends

Hammond

Janjić

et al. 2016

Concurrency and Computation

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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.

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High-Performance Computer Algebra: A Hecke Algebra Case Study

Cited by 5 publications

References 20 publications

PAEAN: Portable and scalable runtime support for parallel Haskell dialects

PAEAN: Portable and scalable runtime support for parallel Haskell dialects

Invariant Bilinear Forms on W-Graph Representations and Linear Algebra Over Integral Domains

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

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