Abstract
Many problems of industrial and scientific interest require the solving of tridiagonal linear systems. This paper presents several implementations for the parallel solving of large tridiagonal systems on multi-core architectures, using the OmpSs programming model. The strategy used for the parallelization is based on the combination of two different existing algorithms, PCR and Thomas. The Thomas algorithm, which cannot be parallelized, requires the fewest number of floating point operations. The PCR algorithm is the most popular parallel method, but it is more computationally expensive than Thomas. The method proposed in this paper starts applying the PCR algorithm to break down one large tridiagonal system into a set of smaller and independent ones. In a second step, these independent systems are concurrently solved using Thomas. The paper also contains an analytical study of which is the best point to switch from PCR to Thomas. Also, the paper addresses the main performance issues of combining PCR and Thomas proposing a set of alternative implementations, some of them even imply algorithmic changes. The performance evaluation shows that the best implementation achieves a peak speedup of 4 with respect to the Intel MKL counterpart routine and 2.5 with respect to a single-threaded Thomas.
| Original language | English |
|---|---|
| Article number | 8643931 |
| Pages (from-to) | 23365-23378 |
| Number of pages | 14 |
| Journal | IEEE Access |
| Volume | 7 |
| DOIs | |
| State | Published - 2019 |
Funding
This work was supported in part by the European Union’s Horizon 2020 Framework Programme for Research and Innovation under the Specific Grant Agreements Human Brain Project SGA1 and Human Brain Project SGA2 under Grant 720270 and Grant 785907, in part by the Spanish Ministry of Economy and Competitiveness under the Project Computación de Altas Prestaciones VII under Grant TIN2015-65316-P, in part by the Departament d’Innovació, Universitats i Empresa de la Generalitat de Catalunya, under project MPEXPAR: Models de Programació i Entorns d’Execució Paralůlels under Grant 2014-SGR-1051, in part by the Juan de la Cierva under Grant IJCI-2017-33511, in part by the Fujitsu under the Barcelona Supercomputing Center-Fujitsu Joint Project: Math Libraries Migration and Optimization, in part by the Ministerio de Economía, Industria y Competitividad of Spain, in part by the Fondo Europeo de Desarrollo Regional Funds of the European Union under Grant TIN2016-75845-P, and in part by the Xunta de Galicia co-founded by the European Regional Development Fund (ERDF) under the Consolidation Programme of Competitive Reference Groups under Grant ED431C 2017/04, and in part by the Centro Singular de Investigación de Galicia accreditatión 2016-2019 under Grant ED431G/01.
Keywords
- OmpSs
- Tridiagonal solve
- auto-tuning
- multi-core