Multi-core Computation of Transfer Matrices for Strip Lattices in the Potts Model

Navarro, Cristóbal A.; Hitschfeld, Nancy; Canfora, Fabrizio

doi:10.1109/hpcc.and.euc.2013.27

Cited by 1 publication

(6 citation statements)

References 26 publications

(35 reference statements)

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“…Note: P F T refers to the actual Parallel Family Trees strategy and P CM to the Parallel Catalan Method from [19].…”

Section: Performance Resultsmentioning

confidence: 99%

“…We have realized performance tests for the parallel transfer matrix method implemented with MPI for both shared and distributed memory scenarios. The experimental design consists of measuring the main performance metrics (i.e., running time, speedup, efficiency, knee) of the implementation by computing the compressed transfer matrix several times, each time varying the number of processors p. We also compute the improvement factor with respect to previous work [19]. The experiments are divided into two categories; (1) multi-core and (2) cluster.…”

Section: Performance Resultsmentioning

confidence: 99%

“…Computation of the family trees can be performed completely in parallel because family trees are independent from each other, and the configuration space is generated a priori, removing any potential time-dependence. We have compared the experimental results of PFT and indeed it runs exponentially faster than the Catalan Parallel Method (CPM) [19], both in sequential and parallel execution. The resulting matrix of PFT is a compressed structure based on systems of linear equations.…”

Section: Discussionmentioning

confidence: 99%

“…Research on transfer matrices for strip lattices in the Potts model have not reported experimental results on the parallel performance, except for a prior work of the authors [19] that consists of a parallel method for computing general (q, v) transfer matrices using the Catalan approach, which will be named the Catalan Parallel Method (CPM) for the ease of referencing it later on. The CPM method was successfully used to study new widths of the kagome strip [31] with generic (q, v).…”

Section: Related Workmentioning

confidence: 99%

“…Section 6 provides details about the implementation while in section 7 we present detailed results for running time, speedup, efficiency and knee, using different amount of processors. We also compare performance against the Catalan Parallel Method (CPM) [19]. Section 8 is devoted to the validation of the algorithm by computing some physical results; from limiting curves to energy and specific heat, and comparing them to the results obtained by other authors.…”

Section: Introductionmentioning

confidence: 99%

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Parallel family trees for transfer matrices in the Potts model

Navarro¹,

Canfora²,

Hitschfeld³

et al. 2015

Computer Physics Communications

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The computational cost of transfer matrix methods for the Potts model is related to the question into how many ways can two layers of a lattice be connected?. Answering the question leads to the generation of a combinatorial set of lattice configurations. This set defines the configuration space of the problem, and the smaller it is, the faster the transfer matrix can be computed. The configuration space of generic (q, v) transfer matrix methods for strips is in the order of the Catalan numbers, which grows asymptotically as O(4 m ) where m is the width of the strip. Other transfer matrix methods with a smaller configuration space indeed exist but they make assumptions on the temperature, number of spin states, or restrict the structure of the lattice. In this paper we propose a parallel algorithm that uses a sub-Catalan configuration space of O(3 m ) to build the generic (q, v) transfer matrix in a compressed form. The improvement is achieved by grouping the original set of Catalan configurations into a forest of family trees, in such a way that the solution to the problem is now computed by solving the root node of each family. As a result, the algorithm becomes exponentially faster than the Catalan approach while still highly parallel. The resulting matrix is stored in a compressed form using O(3 m × 4 m ) of space, making numerical evaluation and decompression to be faster than evaluating the matrix in its O(4 m × 4 m ) uncompressed form. Experimental results for different sizes of strip lattices show that the parallel family trees (PFT) strategy indeed runs exponentially faster than the Catalan Parallel Method (CPM), especially when dealing with dense transfer matrices. In terms of parallel performance, we report strongscaling speedups of up to 5.7X when running on an 8-core shared memory machine and 28X for a 32-core cluster. The best balance of speedup and efficiency for the multi-core machine was achieved when using p = 4 processors, while for the cluster scenario it was in the range p ∈ [8, 10]. Because of the parallel capabilities of the algorithm, a large-scale execution of the parallel family trees strategy in a supercomputer could contribute to the study of wider strip lattices.

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“…Note: P F T refers to the actual Parallel Family Trees strategy and P CM to the Parallel Catalan Method from [19].…”

Section: Performance Resultsmentioning

confidence: 99%

Section: Performance Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%