Figure 1.
Data layout of buffer used for transfer of the Wigner coupling coefficients and quantum numbers of irreducible tensors computed on CPU to GPU
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Computational optimization in solving the geodetic boundary value problems
Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs
| 1. | Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Trojanova 13, Praha 2,120 00, Czech Republic |
| 2. | Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA, Nuclear Physics Institute, Czech Academy of Sciences, Řež 25068, Czech Republic |
| 3. | Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA |
| 4. | Faculty of Information Technology, Czech Technical University, Prague 16000, Czech Republic, Aerospace Research and Test Establishment, Prague 19905, Czech Republic |
| 5. | Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA 70803, USA |
Starting from the matrix elements of a nucleon-nucleon potential operator provided in a basis of spherical harmonic oscillator functions, we present an algorithm for expressing a given potential operator in terms of irreducible tensors of the SU(3) and SU(2) groups. Further, we introduce a GPU-based implementation of the latter and investigate its performance compared with a CPU-based version of the same. We find that the CUDA implementation delivers speedups of 2.27x – 5.93x.
Mathematics Subject Classification:
Primary: 81V35, 81R15; Secondary: 68W10.
Citation:
Tomáš Oberhuber, Tomáš Dytrych, Kristina D. Launey, Daniel Langr, Jerry P. Draayer.
Transformation of a Nucleon-Nucleon potential operator into its SU(3) tensor form using GPUs. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020383
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References:
| [1] |
Y. Akiyama and J. P. Draayer, A user's guide to Fortran programs for Wigner and Racah coefficients of SU$_3$, Comp. Phys. Comm, 5 (1973), 405-406. Google Scholar |
| [2] |
T. Dytrych, K. D. Launey, J. P. Draayer, P. Maris, J. P. Vary, E. Saule, U. Catalyurek, M. Sosonkina, D. Langr and M. A. Caprio, Collective modes in light nuclei from first principles, Phys. Rev. Lett., 111 (2013), 252501.
doi: 10.1103/PhysRevLett.111.252501. |
| [3] |
T. Dytrych, P. Maris, K. D. Launey, J. P. Draayer, J. Vary, D. Langr, E. Saule, M. A. Caprio, U. Catalyurek and M. Sosonkina,
Efficacy of the SU(3) scheme for ab initio large-scale calculations beyond the lightest nuclei, Comp. Phys. Comm., 207 (2016), 202-210.
doi: 10.2172/1326837. |
| [4] |
H. T. Johansson and C. Forssén, Fast and accurate evaluation of Wigner $3j$, $6j$, and $9j$ symbols using prime factorization and multiword integer arithmetic, SIAM J. Sci. Comput., 38 (2016), A376–A384.
doi: 10.1137/15M1021908. |
| [5] |
K. D. Launey, T. Dytrych and J. P. Draayer,
Symmetry-guided large-scale shell-model theory, Prog. Part. Nucl. Phys., 89 (2016), 101-136.
doi: 10.1016/j.ppnp.2016.02.001. |
| [6] |
M. F. O'Reilly,
A closed formula for the product of irreducible representations of SU(3), J. Math. Phys., 23 (1982), 2022-2028.
doi: 10.1063/1.525258. |
show all references
References:
| [1] |
Y. Akiyama and J. P. Draayer, A user's guide to Fortran programs for Wigner and Racah coefficients of SU$_3$, Comp. Phys. Comm, 5 (1973), 405-406. Google Scholar |
| [2] |
T. Dytrych, K. D. Launey, J. P. Draayer, P. Maris, J. P. Vary, E. Saule, U. Catalyurek, M. Sosonkina, D. Langr and M. A. Caprio, Collective modes in light nuclei from first principles, Phys. Rev. Lett., 111 (2013), 252501.
doi: 10.1103/PhysRevLett.111.252501. |
| [3] |
T. Dytrych, P. Maris, K. D. Launey, J. P. Draayer, J. Vary, D. Langr, E. Saule, M. A. Caprio, U. Catalyurek and M. Sosonkina,
Efficacy of the SU(3) scheme for ab initio large-scale calculations beyond the lightest nuclei, Comp. Phys. Comm., 207 (2016), 202-210.
doi: 10.2172/1326837. |
| [4] |
H. T. Johansson and C. Forssén, Fast and accurate evaluation of Wigner $3j$, $6j$, and $9j$ symbols using prime factorization and multiword integer arithmetic, SIAM J. Sci. Comput., 38 (2016), A376–A384.
doi: 10.1137/15M1021908. |
| [5] |
K. D. Launey, T. Dytrych and J. P. Draayer,
Symmetry-guided large-scale shell-model theory, Prog. Part. Nucl. Phys., 89 (2016), 101-136.
doi: 10.1016/j.ppnp.2016.02.001. |
| [6] |
M. F. O'Reilly,
A closed formula for the product of irreducible representations of SU(3), J. Math. Phys., 23 (1982), 2022-2028.
doi: 10.1063/1.525258. |
Figure 2.
Data layout of buffer holding records with triplets of references to records in buffer $ B_W $ . Each such triplet thus represents one irreducible tensor
Table 1.
Performance results obtained on the Blue Waters system. Notation $ +1 $ in the column with MPI processes denotes one master process which just assigns HO shells to other MPI processes. The master process is not taken into account for efficiency evaluation
| MPI procs. | CPU only | CPU+GPU | |||
| Time [s] | Efficiency | Time [s] | Speed-up | ||
| 7+1 | 295.3 | – | 75.3 | 3.92 | |
| 15+1 | 137.6 | 1 | 36.8 | 3.73 | |
| 31+1 | 66.5 | 1 | 17.5 | 3.79 | |
| 63+1 | 39.4 | 0.83 | 9.23 | 4.26 | |
| 127+1 | 32.8 | 0.56 | 6.35 | 5.17 | |
| 255+1 | 31.0 | 0.52 | 5.22 | 5.94 | |
| 7+1 | 2219 | – | 648 | 3.42 | |
| 15+1 | 1034 | 1 | 318 | 3.24 | |
| 31+1 | 499 | 1 | 151 | 3.28 | |
| 63+1 | 248 | 0.99 | 74 | 3.32 | |
| 127+1 | 165 | 0.74 | 43 | 3.75 | |
| 255+1 | 138 | 0.59 | 32 | 4.25 | |
| 7+1 | 13083 | – | 4493 | 2.91 | |
| 15+1 | 6097 | 1 | 2116 | 2.88 | |
| 31+1 | 2943 | 1 | 1054 | 2.79 | |
| 63+1 | 1447 | 1 | 515 | 2.80 | |
| 127+1 | 776 | 0.92 | 269 | 2.88 | |
| 255+1 | 565 | 0.68 | 169 | 3.33 | |
| 5+1 | 64865 | – | 26104 | 2.48 | |
| 15+1 | 30227 | 1 | 12204 | 2.47 | |
| 31+1 | 14596 | 1 | 5944 | 2.45 | |
| 63+1 | 7179 | 1 | 2932 | 2.44 | |
| 127+1 | 3581 | 0.99 | 1461 | 2.45 | |
| 255+1 | 2142 | 0.83 | 838 | 2.55 | |
| MPI procs. | CPU only | CPU+GPU | |||
| Time [s] | Efficiency | Time [s] | Speed-up | ||
| 7+1 | 295.3 | – | 75.3 | 3.92 | |
| 15+1 | 137.6 | 1 | 36.8 | 3.73 | |
| 31+1 | 66.5 | 1 | 17.5 | 3.79 | |
| 63+1 | 39.4 | 0.83 | 9.23 | 4.26 | |
| 127+1 | 32.8 | 0.56 | 6.35 | 5.17 | |
| 255+1 | 31.0 | 0.52 | 5.22 | 5.94 | |
| 7+1 | 2219 | – | 648 | 3.42 | |
| 15+1 | 1034 | 1 | 318 | 3.24 | |
| 31+1 | 499 | 1 | 151 | 3.28 | |
| 63+1 | 248 | 0.99 | 74 | 3.32 | |
| 127+1 | 165 | 0.74 | 43 | 3.75 | |
| 255+1 | 138 | 0.59 | 32 | 4.25 | |
| 7+1 | 13083 | – | 4493 | 2.91 | |
| 15+1 | 6097 | 1 | 2116 | 2.88 | |
| 31+1 | 2943 | 1 | 1054 | 2.79 | |
| 63+1 | 1447 | 1 | 515 | 2.80 | |
| 127+1 | 776 | 0.92 | 269 | 2.88 | |
| 255+1 | 565 | 0.68 | 169 | 3.33 | |
| 5+1 | 64865 | – | 26104 | 2.48 | |
| 15+1 | 30227 | 1 | 12204 | 2.47 | |
| 31+1 | 14596 | 1 | 5944 | 2.45 | |
| 63+1 | 7179 | 1 | 2932 | 2.44 | |
| 127+1 | 3581 | 0.99 | 1461 | 2.45 | |
| 255+1 | 2142 | 0.83 | 838 | 2.55 | |
Table 2.
Performance results obtained for 32 MPI processes running on two 16-core IBM Power9 CPUs and a single NVIDIA V100 GPU
| CPU only | CPU+GPU | ||
| Time [s] | Time [s] | Speed-up | |
| 8 | 41.65 | 15.78 | 2.63 |
| 10 | 274.14 | 97.99 | 2.79 |
| 12 | 1649.7 | 611.1 | 2.69 |
| 14 | 7761.9 | 3407.6 | 2.27 |
| CPU only | CPU+GPU | ||
| Time [s] | Time [s] | Speed-up | |
| 8 | 41.65 | 15.78 | 2.63 |
| 10 | 274.14 | 97.99 | 2.79 |
| 12 | 1649.7 | 611.1 | 2.69 |
| 14 | 7761.9 | 3407.6 | 2.27 |
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